Review ArticleThe Fractal Geometry of the Cosmic Web and Its Formation

Jose Gaite

Applied Physics Dept ETSIAE Univ Politecnica de Madrid E-28040 Madrid Spain

Correspondence should be addressed to Jose Gaite josegaiteupmes

Received 29 November 2018 Revised 15 March 2019 Accepted 14 April 2019 Published 2 May 2019

Academic Editor Geza Kovacs

Copyright copy 2019 Jose Gaite This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The cosmic web structure is studied with the concepts and methods of fractal geometry employing the adhesion model ofcosmological dynamics as a basic referenceThe structures of matter clusters and cosmic voids in cosmological N-body simulationsor the Sloan Digital Sky Survey are elucidated by means of multifractal geometry A nonlacunar multifractal geometry canencompass three fundamental descriptions of the cosmic structure namely the web structure hierarchical clustering and halodistributions Furthermore it explains our present knowledge of cosmic voids In this way a unified theory of the large-scalestructure of the universe seems to emergeThemultifractal spectrum thatwe obtain significantly differs from the one of the adhesionmodel and conforms better to the laws of gravityThe formation of the cosmic web is best modeled as a type of turbulent dynamicsgeneralizing the known methods of Burgers turbulence

1 Introduction

The evolution of the universe at large is ruled by gravityAlthough the Einstein equations of the theory of generalrelativity are nonlinear and difficult to solve the early evo-lution of the universe can be described by an exact solutionnamely the FLRW solution plus its linear perturbationsThese perturbations grow and they grow faster on smallerscales becoming nonlinear Then large overdensities arisethat decouple from the global expansionThis signals the endof a dust-like description of thematter dynamics and the needfor a finer description of it in which dissipative processes inparticular play a prominent roleThe dissipation is essentiallya transfer of kinetic energy from one scale to another smallerscale due to nonlinear mode coupling as is characteristic offluid turbulence

Indeed the successful adhesion model of large-scalestructure formation is actually a model of turbulence inirrotational pressureless flow [1 2] This model generatesthe well-known cosmic web structure The geometry andformation of this web structure are the subject of this studyThe cosmic web is a foam-like structure formed by a web ofsheets surrounding voids of multiple sizes In fact the rangeof sizes is so large that we can speak of a self-similar structureThis motivates its study by means of fractal geometry

In fact fractal models of the universe predate the dis-covery of the cosmic web structure and arose from the ideaof a hierarchy of galaxy clusters that continues indefinitelytowards the largest scales an idea championed byMandelbrot[3] In spite of the work of many cosmologists along severaldecades eg Peebles [4 5] the debate about the scaleof transition to homogeneity is not fully settled [4ndash10]However the magnitude of the final value of the scale ofhomogeneity shall not diminish the importance of the fractalstructure on lower scales namely in the highly nonlinearclustering regime [11] At any rate the notion of a hierarchyof disconnected matter clusters has turned out to be naiveMandelbrot [3] already considered the early signs of cosmicvoids and of filamentary structure in the universe and withthis motivation he proposed the study of fractal textureHowever only the study of the cosmological dynamics ofstructure formation can unveil the precise nature of theintricate cosmic web geometry

The geometry of the cosmic web has been observed ingalaxy surveys [12ndash14] and cosmological119873-body simulations[15ndash17] In addition the fractal geometry of the large-scalestructure has been also studied in the distribution of galaxies[18ndash20] and in cosmological 119873-body simulations [21ndash23]Recent 119873-body simulations have very good resolution andthey clearly show both the morphological and self-similar

HindawiAdvances in AstronomyVolume 2019 Article ID 6587138 25 pageshttpsdoiorg10115520196587138

2 Advances in Astronomy

aspects of the cosmic web However while there seems tobe a consensus about the reality and importance of thistype of structure a comprehensive geometrical model isstill missing In particular although our present knowledgesupports a multifractal model its spectrum of dimensions isonly partially known as will be discussed in this paper

We must also consider halo models [24] as models oflarge-scale structure that are constructed from statisticalrather than geometrical principles In the basic halo modelthe matter distribution is separated into a distribution withinldquohalosrdquo with given density profiles and a distribution ofhalo centers in space Halo models have gained popularityas models suitable for analyzing the results of cosmological119873-body simulations and can even be employed for designingsimulations [25] Halomodels are also employed for the studyof the small-scale problems of the Lambda cold dark matter(LCDM) cosmology [26] Remarkably the basic halo modelwith some modifications can be combined with fractalmodels [27ndash31] At any rate halo models have questionableaspects and a careful analysis of the mass distribution withinhalos shows that it is too influenced by discreteness effectsintrinsic to119873-body simulations [31 32]

We review in this paper a relevant portion of the effortsto understand the cosmic structure with a bias towardsmethods of fractal geometry as regards the description ofthe structure and towards nonlinear methods of the theoryof turbulence as regards the formation of the structureSo this review does not have as broad a scope as forexample Sahni and Colesrsquo review of nonlinear gravitationalclustering [33] inwhich one canfind information about othertopics such as the Press-Schechter approach the Voronoifoam model and the BBGKY statistical hierarchy Generalaspects of the cosmic web geometry have been reviewed byvan de Weygaert [34] Here we shall initially focus on theadhesion approximation as a convenientmodel of the cosmicweb geometry and of the formation of structures accordingto methods of the theory of fluid turbulence namely thetheory of Burgers turbulence in pressureless fluids [35 36]However the adhesion model is as we shall see somewhatsimplistic for the geometry as well as for the dynamics Forthe former we need more sophisticated fractal models andfor the latter we have to consider first a stochastic versionof the adhesion model and eventually the full nonlineargravitational dynamics

2 The Cosmic Web Geometry

In this section we study the geometry of the universe onmiddle to large scales in the present epoch and from adescriptive point of view As already mentioned there arethree main paradigms in this respect the cosmic web thatarises from the Zeldovich approximation and the adhesionmodel [1 2] the fractal model [3] and the halo model [24]These paradigms have different origins and motivations andhave often been in conflict Notably a fractal model with notransition to homogeneity as Mandelbrot proposed [3] isin conflict with the standard FLRW cosmology and hencewith the other two paradigms Indeed Peeblesrsquo cosmologytextbook [5] places the description of ldquoFractal Universerdquo in

the chapter of Alternative Cosmologies However a fractalnonlinear regime in the FLRW cosmology does not generateany conflict with the standard cosmology theory [11] andactually constitutes an important aspect of the cosmic webgeometry Furthermore we propose that the three paradigmscan be unified and that they represent from amature point ofview just different aspects of the geometry of the universe onmiddle to large scales Here we present the three paradigmsseparately but attempting to highlight the many connectionsbetween them and suggesting how to achieve a unifiedpicture

21 Self-Similarity of the Cosmic Web Although the cosmicweb structure has an obviously self-similar appearance thisaspect of it was not initially realized and it was insteadassumed that there is a ldquocellular structurerdquo with a limitedrange of cell sizes [37] However the web structure is gener-ated by Burgers turbulence and in the study of turbulenceand in particular of Burgers turbulence it is natural toassume the self-similarity of velocity correlation functions[35 36] Therefore it is natural to look for scaling in thecosmic web structure The study of gravitational clusteringwith methods of the theory of turbulence is left for Section 3and we now introduce some general notions useful tounderstand the geometrical features of the cosmic web andto connect with the fractal and halo models

The FLRW solution is unstable because certain perturba-tions growThis growth means for dust (pressureless) matterfollowing geodesics that close geodesics either diverge orconverge In the latter case the geodesics eventually joinand matter concentrates on caustic surfaces In fact from ageneral relativity standpoint caustics are just a geometricalphenomenon that always appears in the attempt to constructa synchronous reference frame from a three-dimensionalCauchy surface and a family of geodesics orthogonal to thissurface [38] But what are in general nonreal singularitiesowing to the choice of coordinates become real densitysingularities in the irrotational flow of dust matter

One can consider the study and classification of thepossible types of caustics as a purely geometrical problemconnected with a broad class of problems in the theory ofsingularities or catastrophes [39] To be specific the causticsurfaces are singularities of potential flows and are calledLagrangian singularities [39 ch 9] A caustic is a critical pointof the Lagrangian map which maps initial to final positionsAfter a caustic forms the inverse Lagrangian map becomesmultivalued that is to say the flow has several streams(initially just three) Although this is a general process adetailed study of the formation of singularities in cosmologyhas only been carried out in the Zeldovich approximationwhich is a Newtonian and quasi-linear approximation of thedynamics [1 33] Recently Hidding Shandarin and van deWeygaert [40] have described the geometry of all genericsingularities formed in theZeldovich approximation display-ing some useful graphics that show the patterns of foldingshell crossing and multistreaming Since the classification ofLagrangian singularities is universal and so is the resultinggeometry those patterns are not restricted to the Zeldovichapproximation

Advances in Astronomy 3

Unfortunately the real situation in cosmology is morecomplicated the initial condition is not compatible witha smooth flow so the theory of singularities of smoothmaps can only be employed as an approximation suitablefor smoothened initial conditions This approach yieldssome results [33] However a nonsmooth initial conditiongives rise to a random distribution of caustics of all sizeswith an extremely complex distribution of multistreamingflows whose geometry is mostly unexplored Besides theZeldovich approximation fails after shell crossing The realmultistreaming flow generated by Newtonian gravity hasbeen studied in simplified situations for example in onedimension (which can describe the dimension transverseto caustics) Early 119873-body simulations showed that thethickness of an isolated multistreaming zone grows slowlyas if gravity makes particles stick together [1] The analyticaltreatment of Gurevich and Zybin [41] for smooth initialconditions proves that multistreaming gives rise to power-law mass concentrations One-dimensional simulations withcosmological initial conditions (uniformdensity and randomvelocities) [42ndash44] show that particles tend to concentratein narrow multistreaming zones pervading the full spatialdomain Furthermore both the space and phase-space dis-tributions tend to have self-similar properties

At any rate before any deep study of multistreamingwas undertaken the notion of ldquogravitational stickingrdquo ofmatter suggested a simple modification of the Zeldovichapproximation that suppresses multistreaming It is naturalto replace the collisionless particles by small volume elementsand hence to assume that where they concentrate andproduce an infinite density it is necessary to take into accountlower-scale processes The simplest way to achieve this isembodied in the adhesion model which supplements theZeldovich approximation with a small viscosity giving riseto the (three-dimensional) Burgers equation [1] One mustnot identify this viscosity with the ordinary gas viscosity butwith an effect of coarse graining of the collisionlessNewtoniandynamics [45] Even a vanishing viscosity is sufficient to pre-vent multistreaming Actually the vanishing viscosity limitis in dimensionless variables the high Reynolds-numberlimit which gives rise to Burgers turbulence Although wepostpone the study of cosmic turbulence to Section 3 wesummarize here some pertinent results of the adhesionmodel

A deep study of the geometry of the mass distributiongenerated by the adhesion model has been carried out byVergassola et al [46] Their motivation was to compare theprediction of the adhesion model with the result of thePress-Schechter approach to the mass function of collapsedobjects (this approach is explained in [33]) But Vergassolaet al actually show how a self-similar mass distributionarises (including the analytical proof for a particular case)The adhesion model (with vanishing viscosity) has an exactgeometrical solution in terms of the convex hull of theLagrangian potential [46] In one dimension andwith cosmo-logical initial conditions the adhesion process transforms theLagrangian map to a random devilrsquos staircase with an infinitenumber of steps whose lengths correspond to the amountsof stuck mass (Figure 1 left-hand side) To be precise if the

initial condition consists of a uniform density and a Gaussianvelocity field with a power-law energy spectrum then atany time 119905 gt 0 singularities arise in the velocity field andmass condensates on them giving rise to the steps of thedevilrsquos staircase Such condensations initially contain verysmall mass but grow as mass sticking proceeds preserving aself-similar distribution in space and time

Let us explain in more detail the properties of the self-similar mass distribution that arises from a scale-invariantinitial velocity field such that

119906 (120582119903) = 120582ℎ119906 (119903) ℎ isin (0 1) (1)

namely a fractional Brownian field with Hurst exponent ℎ[46] These fields are Gaussian random fields that generalizethe Brownian field with ℎ = 12 and independent incre-ments in such a way that for ℎ gt 12 same-sign incrementsare likely (persistence) whereas for ℎ lt 12 opposite-signincrements are likely (antipersistence) [3 ch IX] If ℎ 997888rarr 1then the field tends to be smooth whereas if ℎ 997888rarr 0 thenthe field tends to be discontinuous A fractional Brownianvelocity field corresponds to Gaussian density fluctuationswith power-law Fourier spectrum of exponent 119899 = 2(1 minusℎ) minus 119889 in 119889 dimensions There is a scale 119871 prop 1199051(1minusℎ) calledldquocoalescence lengthrdquo by Vergassola et al [46] such that itseparates small scales where mass condensation has givenrise to an inhomogeneous distribution from large scaleswhere the initial conditions hold and the distribution isstill practically homogeneous (119905 is a redefined time variablenamely the growth factor of perturbations [1 46]) In onedimension we have a devilrsquos staircase with steps that followa power-law so that the cumulative mass function is

119873(119872 gt 119898) prop 119898minusℎ (2)

However the number of steps longer than 119871 that is tosay of large mass condensations has an exponential decay(like in the Press-Schechter approach) These results can begeneralized to higher dimensions in which the web structureis manifest (Figure 1 right-hand side) However in higherdimensions the full geometrical construction is very complexand hard to visualize [47]

Even though the results of the adhesion model are veryappealing wemust notice its shortcomings On the one handit is based on the Zeldovich approximation which is so simplethat the mass function is directly given by the initial energyspectrum The Zeldovich approximation is not arbitrary andis actually the first order of the Lagrangian perturbationtheory of collisionless Newtonian dynamics as thoroughlystudied by Buchert (eg [48]) but some nice properties of theZeldovich approximation are not preserved in higher ordersAt any rate the Zeldovich approximation is exact in onedimension yet the one-dimensional adhesion model is notbecause the instantaneous adhesion of matter is not realisticand differs from what is obtained in calculations or 119873-bodysimulations namely power-law mass concentrations insteadof zero-size mass condensations [41ndash44 49]The calculationsand simulations also show that the final mass distribution isnot simply related to the initial conditions

4 Advances in Astronomy

00 01 02 03 04 0500

01

02

03

04

05

Figure 1 Realizations in one or two dimensions of the exact solution of the adhesionmodel with scale-invariant initial velocity field (ℎ = 12)In both the length scale is arbitrary due to the self-similarity so that the relevant scale is the scale of the largest structures Left-hand side inone dimension Lagrangianmap from initial to final positions withmultistreaming and devilrsquos staircase formed bymass sticking Right-handside web structure of final positions in two dimensions

Nevertheless it can be asserted that the locations ofpower-law mass concentrations in cosmological 119873-bodysimulations in one or more dimensions form a self-similardistribution that looks like the cosmic web of the adhesionmodel Indeed all these mass distributions are actuallyparticular types of multifractal distributions [27ndash30 43] Wenow study fractal models of large-scale structure from ageneral standpoint

22 Fractal Geometry of Cosmic Structure The concept ofcosmic web structure has its origin in the adhesion modelwhich is based on the study of cosmological dynamics Incontrast fractal models of large-scale structure have mostlyobservational origins combined with ancient theoreticalmotivations (the solution of the Olbers paradox the ideaof nested universes etc) Actually the ideas about thestructure of the universe have swung between the principle ofhomogeneity and the principle of hierarchical structure [5]The cosmological principle is the modern formulation of theformer but it admits a conditional formulation that supportsthe latter [3 sect22]There is no real antithesis and the synthesisis provided by the understanding of how a fractal nonlinearregime develops below some homogeneity scale consequentto the instability of the FLRWmodelThis synthesis is alreadypresent in the adhesion model in which the homogeneityscale is the above-mentioned coalescence length119871

The notion of hierarchical clustering namely the ideaof clusters of galaxies that are also clustered in superclustersand so onwards inspired the early fractal models of theuniverse This idea can be simply realized in some three-dimensional and random generalization of the Cantor setwith the adequate fractal dimension [3] In mathematicalterms the model places the focus on the geometry of

fractal sets characterized by the Hausdorff dimension Arandom fractal set model for the distribution of galaxieshas been supported by measures of the reduced two-pointcorrelation function of galaxies which is long known to bewell approximated as the power-law

120585 (119903) = ⟨984858 (119903) 984858 (0)⟩⟨984858⟩2 minus 1 = (1199030119903 )

120574 (3)

with exponent 120574 ≃ 18 and correlation length1199030 ≃ 6 Mpc asstandard but perhaps questionable values [3ndash10] In fact thename correlation length for 1199030 is essentially a misnomer [9]and 1199030 is to be identified with the present-time value of 119871 inthe adhesion model that is to say with the current scale oftransition to homogeneity Although the value of 1199030 has beenvery debated (3) implies for any value of 1199030 that there arevery large fluctuations of the density when 119903 ≪ 1199030 and indeedthat the distribution is fractal with Hausdorff dimension 3minus120574[3ndash11]

It would be convenient that fractals were described onlyby its Hausdorff dimension but it was soon evident that onesole number cannot fully characterize the richness of complexfractal structures and in particular of the observed large-scale structure of the universe Indeed fractal sets with thesameHausdorff dimension can look very different and shouldbe distinguishable from a purely mathematical standpointThis is obvious in two or higher dimensions where arbitrarysubsets possess topological invariants such as connectivitythat are certainly relevant Furthermore it is also true inone dimension in spite of the fact that any (compact) fractalsubset of R is totally disconnected and has trivial topology

In an effort to complement the concept of fractal dimen-sion Mandelbrot was inspired by the appearance of galaxy

Advances in Astronomy 5

clusters and superclusters and introduced the notion oflacunarity without providing a mathematical definition ofit [3 sect34] Lacunarity is loosely defined as the property ofhaving large gaps or voids (lacuna) This property gainedimportance in cosmology with the discovery of large voidsin galaxy redshift surveys which came to be considereda characteristic feature of the large-scale structure of theuniverse [13 14]The precise nature of cosmic voids is indeedimportant to characterize the geometry of the cosmicweb andwill be studied below

In general for sets of a given Hausdorff dimension thereshould be a number of parameters that specify their appear-ance what Mandelbrot calls texture [3 ch X] Lacunarity isthe first one and can be applied to sets in any dimensionincluding one dimension Others are only applicable inhigher dimension for example parameters to measure theexistence and extension of subsets of a fractal set that aretopologically equivalent to line segments an issue related topercolation [3 sect34 and 35]

In fact the methods of percolation theory have beenemployed in the study of the cosmic web [1] Those methodsare related to the mathematical notion of path connectionof sets which is just one topological property among manyothers Topological properties are certainly useful but topol-ogy does not discriminate enough in fractal geometry asshown by the fact that all (compact) fractal subsets of Rare topologically equivalent According to Falconer [50]fractal geometry can be defined in analogy with topologyreplacing bicontinuous transformations by a subset of themnamely bilipschitz transformationsTheHausdorff dimensionis invariant under these transformations although it isnot invariant under arbitrary bicontinuous transformationsTherefore the set of parameters characterizing a fractalset includes on the one hand topological invariants andon the other hand properly fractal parameters beginningwith the Hausdorff dimension Unfortunately there has beenlittle progress in the definition of these parameters beyondMandelbrotrsquos heuristic definition of texture parameters

Besides the need for a further development of the geom-etry of fractal sets we must realize that the concept of fractalset is not sufficient to deal with the complexity of cosmicgeometry This is easily perceived by reconsidering the massdistribution generated by the adhesion model in particularin one dimension In the geometry of a devilrsquos staircase wemust distinguish the set of lengths of the steps from thelocations of these steps which are their ordinates in the graphon the left-hand side of Figure 1 These locations constitutea dense set that is to say they leave no interval empty [46]The set is nevertheless a countable set and therefore has zeroHausdorff dimension But more important than the locationsof mass condensations is the magnitude of these masses (thelengths of the steps) To wit we are not just dealing with theset of locations butwith the fullmass distributionMandelbrot[3] does not emphasize the distinction between these twoconcepts and actually gives no definition ofmass distributionbut other authors do for example Falconer includes anintroduction to measure theory [50] which is useful toprecisely define the Hausdorff dimension as well as to definethe concept of mass distribution A mass distribution is a

finite measure on a bounded set of R119899 a definition thatrequires some mathematical background [50] although theintuitive notion is nonetheless useful

The geometry of a generic mass distribution on R iseasy to picture because we can just consider the geometryof the devilrsquos staircase and generalize it It is not difficult tosee that the cumulative mass that corresponds to a devilrsquosstaircase interval is given by switching its axes for examplethe totalmass in the interval [0 119909] is given by the length of theinitial interval that transforms to [0 119909] under the Lagrangianmap The switching of axes can be performed mentallyin Figure 1 (left) The resulting cumulative mass functionis monotonic but not continuous A general cumulativemass function needs only to be monotonic and it may ormay not be continuous its discontinuities represent masscondensations (of zero size) Given the cumulative massfunction we could derive the mass density by differentiationin principle This operation should present no problemsbecause a monotonic function is almost everywhere differ-entiable However the inverse devilrsquos staircase has a denseset of discontinuities where it is certainly nondifferentiableThis does not constitute a contradiction because the set ofpositions of discontinuities is countable and therefore haszero (Lebesgue) measure but it shows the complexity ofthe situation Moreover in the complementary set (of fullmeasure) where the inverse devilrsquos staircase is differentiablethe derivative vanishes [46]

In summary we have found that our first model ofcosmological mass distribution is such that the mass densityis either zero or infinity We might attribute this singularbehavior to the presence of a dense set of zero-size mass con-densations due to the adhesion approximation Thereforewe could expect that a generic mass distribution should benonsingular Of course themeaning of ldquogenericrdquo is indefiniteFrom amathematical standpoint a generic mass distributionis one selected at random according to a natural probabilitydistribution (a probability distribution on the space of massdistributions)This apparently abstract problem is of interestin probability theory and has been studied with the resultthat the standard methods of randomly generating massdistributions indeed produce strictly singular distributions[51] This type of distributions has no positive finite densityanywhere like our discontinuous example but the resultalso includes continuous distributions In fact the massdistributions obtained in cosmological calculations or sim-ulations [41 43] seem to be continuous but strictly singularnonetheless namely they seem to contain dense sets ofpower-law mass concentrations

Let us consider the example of random mass functiondisplayed in Figure 2 which is generated with one of themethods referred to in [51] Actually the real mass distribu-tion namely the mass density is obtained as the derivativeof the mass function and is not a proper function because ittakes only the values zero or infinity The graphs in Figure 2are the result of coarse graining with a small length butit is evident that the values of the density are either verylarge or very small Furthermore the large or small valuesare clustered respectively They form mass clusters or voidsthe former occupying vanishing total length and the latter

6 Advances in Astronomy

00 02 04 06 08 1000

02

04

06

08

10

00 02 04 06 08 10

Figure 2 Left one example of random mass distribution function generated as a random homeomorphism of the interval [0 1] Right thederivative of this function giving the mass distribution

occupying the full total length in the limit of vanishingcoarse-graining length Unlike the inverse devilrsquos staircasethe present mass distribution function is continuous soits graph contains no vertical segments in spite of beingvery steep and actually nondifferentiable at many pointsTherefore there are no zero-size mass condensations but justmass concentrationsThesemass concentrations are clustered(in our example as a consequence of the generating process)In view of its characteristics this random mass functionillustrates the type of one-dimensional mass distributions incosmology which can be considered generic Some furtherproperties of generic strictly singular mass distributions areworth noticing

To any mass distribution is associated a particular setnamely its supporting set which is the smallest closed setthat contains all the mass [50] This set is the full interval inthe examples of Figure 1 or Figure 2 but it can be smallerin strictly singular mass distributions and it can actually bea fractal set namely a set with Hausdorff dimension strictlysmaller than one In fact if Mandelbrot [3] seldom distin-guishes between fractal sets and fractal mass distributionsit is probably because in most of his examples the supportis a fractal set and the mass is uniformly distributed on it Inthis simple situation the distinction is superfluous Howeverwe have to consider singular mass distributions in general forthe cosmic geometry and take into account the fact that theobserved clustering may not be a property of the supportingset but a property of themass distribution on itThis aspect ofclustering is directly related to the nature of voids in a fractalset the set of voids is simply the complementary set whichis totally empty whereas the voids in a mass distributionwith full support have tiny mass concentrations inside (seeFigures 1 and 2)The latter option describes better our presentknowledge of cosmic voids [28 52 53]

In conclusion the appropriate geometrical setting forthe study of the large-scale structure of the universe is

the geometry of mass distributions as a generalization ofthe geometry of sets Naturally a generic mass distributionis more complex in three-dimensional space than in one-dimensional space In particular one has to consider prop-erties such as connectivity which are important for the webstructure (Figure 1) Since the supporting set of a geometricalmodel of cosmic structure is probably trivial namely the fullvolume one cannot formulate the morphological propertiesin terms of the topology of that set So it is necessary toproperly define the topology of a mass distribution This canbe done in terms of the topology of sets of particular massconcentrations but is beyond the scope of this work Theclassification of mass concentrations is certainly useful as aprevious step and by itself The standard way of classifyingmass concentrations constitutes the multifractal analysis ofmeasures [50 ch 17]

23 Multifractal Analysis of Mass Distributions The originalidea of hierarchical clustering of galaxies considers galaxiesas equivalent entities and constructs a sort of fractal setas the prolongation of the hierarchy towards higher scales(up to the scale of homogeneity) and towards lower scales(the prolongation to lower and indeed infinitesimal scales isnecessary to actually define a mathematical fractal set) Thechief observable is the fractal dimension which has beengenerally obtained through (3) namely from the value of 120574estimated through the correlation function of galaxy positions[3ndash6] By using the galaxy positions the density 984858 in (3) isthe galaxy number density instead of the real mass densityBoth densities should give similar results if the mass weredistributed uniformly over the galaxy positions HoweverPietronero [18] noticed that galaxies are not equivalent to oneanother and that in fact their masses span a broad rangeTherefore he argued that the fractal dimension based ongalaxy positions would not be enough and one should extend

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

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[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 3

Unfortunately the real situation in cosmology is morecomplicated the initial condition is not compatible witha smooth flow so the theory of singularities of smoothmaps can only be employed as an approximation suitablefor smoothened initial conditions This approach yieldssome results [33] However a nonsmooth initial conditiongives rise to a random distribution of caustics of all sizeswith an extremely complex distribution of multistreamingflows whose geometry is mostly unexplored Besides theZeldovich approximation fails after shell crossing The realmultistreaming flow generated by Newtonian gravity hasbeen studied in simplified situations for example in onedimension (which can describe the dimension transverseto caustics) Early 119873-body simulations showed that thethickness of an isolated multistreaming zone grows slowlyas if gravity makes particles stick together [1] The analyticaltreatment of Gurevich and Zybin [41] for smooth initialconditions proves that multistreaming gives rise to power-law mass concentrations One-dimensional simulations withcosmological initial conditions (uniformdensity and randomvelocities) [42ndash44] show that particles tend to concentratein narrow multistreaming zones pervading the full spatialdomain Furthermore both the space and phase-space dis-tributions tend to have self-similar properties

At any rate before any deep study of multistreamingwas undertaken the notion of ldquogravitational stickingrdquo ofmatter suggested a simple modification of the Zeldovichapproximation that suppresses multistreaming It is naturalto replace the collisionless particles by small volume elementsand hence to assume that where they concentrate andproduce an infinite density it is necessary to take into accountlower-scale processes The simplest way to achieve this isembodied in the adhesion model which supplements theZeldovich approximation with a small viscosity giving riseto the (three-dimensional) Burgers equation [1] One mustnot identify this viscosity with the ordinary gas viscosity butwith an effect of coarse graining of the collisionlessNewtoniandynamics [45] Even a vanishing viscosity is sufficient to pre-vent multistreaming Actually the vanishing viscosity limitis in dimensionless variables the high Reynolds-numberlimit which gives rise to Burgers turbulence Although wepostpone the study of cosmic turbulence to Section 3 wesummarize here some pertinent results of the adhesionmodel

A deep study of the geometry of the mass distributiongenerated by the adhesion model has been carried out byVergassola et al [46] Their motivation was to compare theprediction of the adhesion model with the result of thePress-Schechter approach to the mass function of collapsedobjects (this approach is explained in [33]) But Vergassolaet al actually show how a self-similar mass distributionarises (including the analytical proof for a particular case)The adhesion model (with vanishing viscosity) has an exactgeometrical solution in terms of the convex hull of theLagrangian potential [46] In one dimension andwith cosmo-logical initial conditions the adhesion process transforms theLagrangian map to a random devilrsquos staircase with an infinitenumber of steps whose lengths correspond to the amountsof stuck mass (Figure 1 left-hand side) To be precise if the

initial condition consists of a uniform density and a Gaussianvelocity field with a power-law energy spectrum then atany time 119905 gt 0 singularities arise in the velocity field andmass condensates on them giving rise to the steps of thedevilrsquos staircase Such condensations initially contain verysmall mass but grow as mass sticking proceeds preserving aself-similar distribution in space and time

Let us explain in more detail the properties of the self-similar mass distribution that arises from a scale-invariantinitial velocity field such that

119906 (120582119903) = 120582ℎ119906 (119903) ℎ isin (0 1) (1)

namely a fractional Brownian field with Hurst exponent ℎ[46] These fields are Gaussian random fields that generalizethe Brownian field with ℎ = 12 and independent incre-ments in such a way that for ℎ gt 12 same-sign incrementsare likely (persistence) whereas for ℎ lt 12 opposite-signincrements are likely (antipersistence) [3 ch IX] If ℎ 997888rarr 1then the field tends to be smooth whereas if ℎ 997888rarr 0 thenthe field tends to be discontinuous A fractional Brownianvelocity field corresponds to Gaussian density fluctuationswith power-law Fourier spectrum of exponent 119899 = 2(1 minusℎ) minus 119889 in 119889 dimensions There is a scale 119871 prop 1199051(1minusℎ) calledldquocoalescence lengthrdquo by Vergassola et al [46] such that itseparates small scales where mass condensation has givenrise to an inhomogeneous distribution from large scaleswhere the initial conditions hold and the distribution isstill practically homogeneous (119905 is a redefined time variablenamely the growth factor of perturbations [1 46]) In onedimension we have a devilrsquos staircase with steps that followa power-law so that the cumulative mass function is

119873(119872 gt 119898) prop 119898minusℎ (2)

However the number of steps longer than 119871 that is tosay of large mass condensations has an exponential decay(like in the Press-Schechter approach) These results can begeneralized to higher dimensions in which the web structureis manifest (Figure 1 right-hand side) However in higherdimensions the full geometrical construction is very complexand hard to visualize [47]

Even though the results of the adhesion model are veryappealing wemust notice its shortcomings On the one handit is based on the Zeldovich approximation which is so simplethat the mass function is directly given by the initial energyspectrum The Zeldovich approximation is not arbitrary andis actually the first order of the Lagrangian perturbationtheory of collisionless Newtonian dynamics as thoroughlystudied by Buchert (eg [48]) but some nice properties of theZeldovich approximation are not preserved in higher ordersAt any rate the Zeldovich approximation is exact in onedimension yet the one-dimensional adhesion model is notbecause the instantaneous adhesion of matter is not realisticand differs from what is obtained in calculations or 119873-bodysimulations namely power-law mass concentrations insteadof zero-size mass condensations [41ndash44 49]The calculationsand simulations also show that the final mass distribution isnot simply related to the initial conditions

4 Advances in Astronomy

00 01 02 03 04 0500

01

02

03

04

05

Figure 1 Realizations in one or two dimensions of the exact solution of the adhesionmodel with scale-invariant initial velocity field (ℎ = 12)In both the length scale is arbitrary due to the self-similarity so that the relevant scale is the scale of the largest structures Left-hand side inone dimension Lagrangianmap from initial to final positions withmultistreaming and devilrsquos staircase formed bymass sticking Right-handside web structure of final positions in two dimensions

Nevertheless it can be asserted that the locations ofpower-law mass concentrations in cosmological 119873-bodysimulations in one or more dimensions form a self-similardistribution that looks like the cosmic web of the adhesionmodel Indeed all these mass distributions are actuallyparticular types of multifractal distributions [27ndash30 43] Wenow study fractal models of large-scale structure from ageneral standpoint

22 Fractal Geometry of Cosmic Structure The concept ofcosmic web structure has its origin in the adhesion modelwhich is based on the study of cosmological dynamics Incontrast fractal models of large-scale structure have mostlyobservational origins combined with ancient theoreticalmotivations (the solution of the Olbers paradox the ideaof nested universes etc) Actually the ideas about thestructure of the universe have swung between the principle ofhomogeneity and the principle of hierarchical structure [5]The cosmological principle is the modern formulation of theformer but it admits a conditional formulation that supportsthe latter [3 sect22]There is no real antithesis and the synthesisis provided by the understanding of how a fractal nonlinearregime develops below some homogeneity scale consequentto the instability of the FLRWmodelThis synthesis is alreadypresent in the adhesion model in which the homogeneityscale is the above-mentioned coalescence length119871

The notion of hierarchical clustering namely the ideaof clusters of galaxies that are also clustered in superclustersand so onwards inspired the early fractal models of theuniverse This idea can be simply realized in some three-dimensional and random generalization of the Cantor setwith the adequate fractal dimension [3] In mathematicalterms the model places the focus on the geometry of

fractal sets characterized by the Hausdorff dimension Arandom fractal set model for the distribution of galaxieshas been supported by measures of the reduced two-pointcorrelation function of galaxies which is long known to bewell approximated as the power-law

120585 (119903) = ⟨984858 (119903) 984858 (0)⟩⟨984858⟩2 minus 1 = (1199030119903 )

120574 (3)

with exponent 120574 ≃ 18 and correlation length1199030 ≃ 6 Mpc asstandard but perhaps questionable values [3ndash10] In fact thename correlation length for 1199030 is essentially a misnomer [9]and 1199030 is to be identified with the present-time value of 119871 inthe adhesion model that is to say with the current scale oftransition to homogeneity Although the value of 1199030 has beenvery debated (3) implies for any value of 1199030 that there arevery large fluctuations of the density when 119903 ≪ 1199030 and indeedthat the distribution is fractal with Hausdorff dimension 3minus120574[3ndash11]

It would be convenient that fractals were described onlyby its Hausdorff dimension but it was soon evident that onesole number cannot fully characterize the richness of complexfractal structures and in particular of the observed large-scale structure of the universe Indeed fractal sets with thesameHausdorff dimension can look very different and shouldbe distinguishable from a purely mathematical standpointThis is obvious in two or higher dimensions where arbitrarysubsets possess topological invariants such as connectivitythat are certainly relevant Furthermore it is also true inone dimension in spite of the fact that any (compact) fractalsubset of R is totally disconnected and has trivial topology

In an effort to complement the concept of fractal dimen-sion Mandelbrot was inspired by the appearance of galaxy

Advances in Astronomy 5

clusters and superclusters and introduced the notion oflacunarity without providing a mathematical definition ofit [3 sect34] Lacunarity is loosely defined as the property ofhaving large gaps or voids (lacuna) This property gainedimportance in cosmology with the discovery of large voidsin galaxy redshift surveys which came to be considereda characteristic feature of the large-scale structure of theuniverse [13 14]The precise nature of cosmic voids is indeedimportant to characterize the geometry of the cosmicweb andwill be studied below

In general for sets of a given Hausdorff dimension thereshould be a number of parameters that specify their appear-ance what Mandelbrot calls texture [3 ch X] Lacunarity isthe first one and can be applied to sets in any dimensionincluding one dimension Others are only applicable inhigher dimension for example parameters to measure theexistence and extension of subsets of a fractal set that aretopologically equivalent to line segments an issue related topercolation [3 sect34 and 35]

In fact the methods of percolation theory have beenemployed in the study of the cosmic web [1] Those methodsare related to the mathematical notion of path connectionof sets which is just one topological property among manyothers Topological properties are certainly useful but topol-ogy does not discriminate enough in fractal geometry asshown by the fact that all (compact) fractal subsets of Rare topologically equivalent According to Falconer [50]fractal geometry can be defined in analogy with topologyreplacing bicontinuous transformations by a subset of themnamely bilipschitz transformationsTheHausdorff dimensionis invariant under these transformations although it isnot invariant under arbitrary bicontinuous transformationsTherefore the set of parameters characterizing a fractalset includes on the one hand topological invariants andon the other hand properly fractal parameters beginningwith the Hausdorff dimension Unfortunately there has beenlittle progress in the definition of these parameters beyondMandelbrotrsquos heuristic definition of texture parameters

Besides the need for a further development of the geom-etry of fractal sets we must realize that the concept of fractalset is not sufficient to deal with the complexity of cosmicgeometry This is easily perceived by reconsidering the massdistribution generated by the adhesion model in particularin one dimension In the geometry of a devilrsquos staircase wemust distinguish the set of lengths of the steps from thelocations of these steps which are their ordinates in the graphon the left-hand side of Figure 1 These locations constitutea dense set that is to say they leave no interval empty [46]The set is nevertheless a countable set and therefore has zeroHausdorff dimension But more important than the locationsof mass condensations is the magnitude of these masses (thelengths of the steps) To wit we are not just dealing with theset of locations butwith the fullmass distributionMandelbrot[3] does not emphasize the distinction between these twoconcepts and actually gives no definition ofmass distributionbut other authors do for example Falconer includes anintroduction to measure theory [50] which is useful toprecisely define the Hausdorff dimension as well as to definethe concept of mass distribution A mass distribution is a

finite measure on a bounded set of R119899 a definition thatrequires some mathematical background [50] although theintuitive notion is nonetheless useful

The geometry of a generic mass distribution on R iseasy to picture because we can just consider the geometryof the devilrsquos staircase and generalize it It is not difficult tosee that the cumulative mass that corresponds to a devilrsquosstaircase interval is given by switching its axes for examplethe totalmass in the interval [0 119909] is given by the length of theinitial interval that transforms to [0 119909] under the Lagrangianmap The switching of axes can be performed mentallyin Figure 1 (left) The resulting cumulative mass functionis monotonic but not continuous A general cumulativemass function needs only to be monotonic and it may ormay not be continuous its discontinuities represent masscondensations (of zero size) Given the cumulative massfunction we could derive the mass density by differentiationin principle This operation should present no problemsbecause a monotonic function is almost everywhere differ-entiable However the inverse devilrsquos staircase has a denseset of discontinuities where it is certainly nondifferentiableThis does not constitute a contradiction because the set ofpositions of discontinuities is countable and therefore haszero (Lebesgue) measure but it shows the complexity ofthe situation Moreover in the complementary set (of fullmeasure) where the inverse devilrsquos staircase is differentiablethe derivative vanishes [46]

In summary we have found that our first model ofcosmological mass distribution is such that the mass densityis either zero or infinity We might attribute this singularbehavior to the presence of a dense set of zero-size mass con-densations due to the adhesion approximation Thereforewe could expect that a generic mass distribution should benonsingular Of course themeaning of ldquogenericrdquo is indefiniteFrom amathematical standpoint a generic mass distributionis one selected at random according to a natural probabilitydistribution (a probability distribution on the space of massdistributions)This apparently abstract problem is of interestin probability theory and has been studied with the resultthat the standard methods of randomly generating massdistributions indeed produce strictly singular distributions[51] This type of distributions has no positive finite densityanywhere like our discontinuous example but the resultalso includes continuous distributions In fact the massdistributions obtained in cosmological calculations or sim-ulations [41 43] seem to be continuous but strictly singularnonetheless namely they seem to contain dense sets ofpower-law mass concentrations

Let us consider the example of random mass functiondisplayed in Figure 2 which is generated with one of themethods referred to in [51] Actually the real mass distribu-tion namely the mass density is obtained as the derivativeof the mass function and is not a proper function because ittakes only the values zero or infinity The graphs in Figure 2are the result of coarse graining with a small length butit is evident that the values of the density are either verylarge or very small Furthermore the large or small valuesare clustered respectively They form mass clusters or voidsthe former occupying vanishing total length and the latter

6 Advances in Astronomy

00 02 04 06 08 1000

02

04

06

08

10

00 02 04 06 08 10

Figure 2 Left one example of random mass distribution function generated as a random homeomorphism of the interval [0 1] Right thederivative of this function giving the mass distribution

occupying the full total length in the limit of vanishingcoarse-graining length Unlike the inverse devilrsquos staircasethe present mass distribution function is continuous soits graph contains no vertical segments in spite of beingvery steep and actually nondifferentiable at many pointsTherefore there are no zero-size mass condensations but justmass concentrationsThesemass concentrations are clustered(in our example as a consequence of the generating process)In view of its characteristics this random mass functionillustrates the type of one-dimensional mass distributions incosmology which can be considered generic Some furtherproperties of generic strictly singular mass distributions areworth noticing

To any mass distribution is associated a particular setnamely its supporting set which is the smallest closed setthat contains all the mass [50] This set is the full interval inthe examples of Figure 1 or Figure 2 but it can be smallerin strictly singular mass distributions and it can actually bea fractal set namely a set with Hausdorff dimension strictlysmaller than one In fact if Mandelbrot [3] seldom distin-guishes between fractal sets and fractal mass distributionsit is probably because in most of his examples the supportis a fractal set and the mass is uniformly distributed on it Inthis simple situation the distinction is superfluous Howeverwe have to consider singular mass distributions in general forthe cosmic geometry and take into account the fact that theobserved clustering may not be a property of the supportingset but a property of themass distribution on itThis aspect ofclustering is directly related to the nature of voids in a fractalset the set of voids is simply the complementary set whichis totally empty whereas the voids in a mass distributionwith full support have tiny mass concentrations inside (seeFigures 1 and 2)The latter option describes better our presentknowledge of cosmic voids [28 52 53]

In conclusion the appropriate geometrical setting forthe study of the large-scale structure of the universe is

the geometry of mass distributions as a generalization ofthe geometry of sets Naturally a generic mass distributionis more complex in three-dimensional space than in one-dimensional space In particular one has to consider prop-erties such as connectivity which are important for the webstructure (Figure 1) Since the supporting set of a geometricalmodel of cosmic structure is probably trivial namely the fullvolume one cannot formulate the morphological propertiesin terms of the topology of that set So it is necessary toproperly define the topology of a mass distribution This canbe done in terms of the topology of sets of particular massconcentrations but is beyond the scope of this work Theclassification of mass concentrations is certainly useful as aprevious step and by itself The standard way of classifyingmass concentrations constitutes the multifractal analysis ofmeasures [50 ch 17]

23 Multifractal Analysis of Mass Distributions The originalidea of hierarchical clustering of galaxies considers galaxiesas equivalent entities and constructs a sort of fractal setas the prolongation of the hierarchy towards higher scales(up to the scale of homogeneity) and towards lower scales(the prolongation to lower and indeed infinitesimal scales isnecessary to actually define a mathematical fractal set) Thechief observable is the fractal dimension which has beengenerally obtained through (3) namely from the value of 120574estimated through the correlation function of galaxy positions[3ndash6] By using the galaxy positions the density 984858 in (3) isthe galaxy number density instead of the real mass densityBoth densities should give similar results if the mass weredistributed uniformly over the galaxy positions HoweverPietronero [18] noticed that galaxies are not equivalent to oneanother and that in fact their masses span a broad rangeTherefore he argued that the fractal dimension based ongalaxy positions would not be enough and one should extend

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

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[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

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[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Submit your manuscripts atwwwhindawicom

4 Advances in Astronomy

00 01 02 03 04 0500

01

02

03

04

05

Figure 1 Realizations in one or two dimensions of the exact solution of the adhesionmodel with scale-invariant initial velocity field (ℎ = 12)In both the length scale is arbitrary due to the self-similarity so that the relevant scale is the scale of the largest structures Left-hand side inone dimension Lagrangianmap from initial to final positions withmultistreaming and devilrsquos staircase formed bymass sticking Right-handside web structure of final positions in two dimensions

Nevertheless it can be asserted that the locations ofpower-law mass concentrations in cosmological 119873-bodysimulations in one or more dimensions form a self-similardistribution that looks like the cosmic web of the adhesionmodel Indeed all these mass distributions are actuallyparticular types of multifractal distributions [27ndash30 43] Wenow study fractal models of large-scale structure from ageneral standpoint

22 Fractal Geometry of Cosmic Structure The concept ofcosmic web structure has its origin in the adhesion modelwhich is based on the study of cosmological dynamics Incontrast fractal models of large-scale structure have mostlyobservational origins combined with ancient theoreticalmotivations (the solution of the Olbers paradox the ideaof nested universes etc) Actually the ideas about thestructure of the universe have swung between the principle ofhomogeneity and the principle of hierarchical structure [5]The cosmological principle is the modern formulation of theformer but it admits a conditional formulation that supportsthe latter [3 sect22]There is no real antithesis and the synthesisis provided by the understanding of how a fractal nonlinearregime develops below some homogeneity scale consequentto the instability of the FLRWmodelThis synthesis is alreadypresent in the adhesion model in which the homogeneityscale is the above-mentioned coalescence length119871

The notion of hierarchical clustering namely the ideaof clusters of galaxies that are also clustered in superclustersand so onwards inspired the early fractal models of theuniverse This idea can be simply realized in some three-dimensional and random generalization of the Cantor setwith the adequate fractal dimension [3] In mathematicalterms the model places the focus on the geometry of

fractal sets characterized by the Hausdorff dimension Arandom fractal set model for the distribution of galaxieshas been supported by measures of the reduced two-pointcorrelation function of galaxies which is long known to bewell approximated as the power-law

120585 (119903) = ⟨984858 (119903) 984858 (0)⟩⟨984858⟩2 minus 1 = (1199030119903 )

120574 (3)

with exponent 120574 ≃ 18 and correlation length1199030 ≃ 6 Mpc asstandard but perhaps questionable values [3ndash10] In fact thename correlation length for 1199030 is essentially a misnomer [9]and 1199030 is to be identified with the present-time value of 119871 inthe adhesion model that is to say with the current scale oftransition to homogeneity Although the value of 1199030 has beenvery debated (3) implies for any value of 1199030 that there arevery large fluctuations of the density when 119903 ≪ 1199030 and indeedthat the distribution is fractal with Hausdorff dimension 3minus120574[3ndash11]

It would be convenient that fractals were described onlyby its Hausdorff dimension but it was soon evident that onesole number cannot fully characterize the richness of complexfractal structures and in particular of the observed large-scale structure of the universe Indeed fractal sets with thesameHausdorff dimension can look very different and shouldbe distinguishable from a purely mathematical standpointThis is obvious in two or higher dimensions where arbitrarysubsets possess topological invariants such as connectivitythat are certainly relevant Furthermore it is also true inone dimension in spite of the fact that any (compact) fractalsubset of R is totally disconnected and has trivial topology

In an effort to complement the concept of fractal dimen-sion Mandelbrot was inspired by the appearance of galaxy

Advances in Astronomy 5

clusters and superclusters and introduced the notion oflacunarity without providing a mathematical definition ofit [3 sect34] Lacunarity is loosely defined as the property ofhaving large gaps or voids (lacuna) This property gainedimportance in cosmology with the discovery of large voidsin galaxy redshift surveys which came to be considereda characteristic feature of the large-scale structure of theuniverse [13 14]The precise nature of cosmic voids is indeedimportant to characterize the geometry of the cosmicweb andwill be studied below

In general for sets of a given Hausdorff dimension thereshould be a number of parameters that specify their appear-ance what Mandelbrot calls texture [3 ch X] Lacunarity isthe first one and can be applied to sets in any dimensionincluding one dimension Others are only applicable inhigher dimension for example parameters to measure theexistence and extension of subsets of a fractal set that aretopologically equivalent to line segments an issue related topercolation [3 sect34 and 35]

In fact the methods of percolation theory have beenemployed in the study of the cosmic web [1] Those methodsare related to the mathematical notion of path connectionof sets which is just one topological property among manyothers Topological properties are certainly useful but topol-ogy does not discriminate enough in fractal geometry asshown by the fact that all (compact) fractal subsets of Rare topologically equivalent According to Falconer [50]fractal geometry can be defined in analogy with topologyreplacing bicontinuous transformations by a subset of themnamely bilipschitz transformationsTheHausdorff dimensionis invariant under these transformations although it isnot invariant under arbitrary bicontinuous transformationsTherefore the set of parameters characterizing a fractalset includes on the one hand topological invariants andon the other hand properly fractal parameters beginningwith the Hausdorff dimension Unfortunately there has beenlittle progress in the definition of these parameters beyondMandelbrotrsquos heuristic definition of texture parameters

Besides the need for a further development of the geom-etry of fractal sets we must realize that the concept of fractalset is not sufficient to deal with the complexity of cosmicgeometry This is easily perceived by reconsidering the massdistribution generated by the adhesion model in particularin one dimension In the geometry of a devilrsquos staircase wemust distinguish the set of lengths of the steps from thelocations of these steps which are their ordinates in the graphon the left-hand side of Figure 1 These locations constitutea dense set that is to say they leave no interval empty [46]The set is nevertheless a countable set and therefore has zeroHausdorff dimension But more important than the locationsof mass condensations is the magnitude of these masses (thelengths of the steps) To wit we are not just dealing with theset of locations butwith the fullmass distributionMandelbrot[3] does not emphasize the distinction between these twoconcepts and actually gives no definition ofmass distributionbut other authors do for example Falconer includes anintroduction to measure theory [50] which is useful toprecisely define the Hausdorff dimension as well as to definethe concept of mass distribution A mass distribution is a

finite measure on a bounded set of R119899 a definition thatrequires some mathematical background [50] although theintuitive notion is nonetheless useful

The geometry of a generic mass distribution on R iseasy to picture because we can just consider the geometryof the devilrsquos staircase and generalize it It is not difficult tosee that the cumulative mass that corresponds to a devilrsquosstaircase interval is given by switching its axes for examplethe totalmass in the interval [0 119909] is given by the length of theinitial interval that transforms to [0 119909] under the Lagrangianmap The switching of axes can be performed mentallyin Figure 1 (left) The resulting cumulative mass functionis monotonic but not continuous A general cumulativemass function needs only to be monotonic and it may ormay not be continuous its discontinuities represent masscondensations (of zero size) Given the cumulative massfunction we could derive the mass density by differentiationin principle This operation should present no problemsbecause a monotonic function is almost everywhere differ-entiable However the inverse devilrsquos staircase has a denseset of discontinuities where it is certainly nondifferentiableThis does not constitute a contradiction because the set ofpositions of discontinuities is countable and therefore haszero (Lebesgue) measure but it shows the complexity ofthe situation Moreover in the complementary set (of fullmeasure) where the inverse devilrsquos staircase is differentiablethe derivative vanishes [46]

In summary we have found that our first model ofcosmological mass distribution is such that the mass densityis either zero or infinity We might attribute this singularbehavior to the presence of a dense set of zero-size mass con-densations due to the adhesion approximation Thereforewe could expect that a generic mass distribution should benonsingular Of course themeaning of ldquogenericrdquo is indefiniteFrom amathematical standpoint a generic mass distributionis one selected at random according to a natural probabilitydistribution (a probability distribution on the space of massdistributions)This apparently abstract problem is of interestin probability theory and has been studied with the resultthat the standard methods of randomly generating massdistributions indeed produce strictly singular distributions[51] This type of distributions has no positive finite densityanywhere like our discontinuous example but the resultalso includes continuous distributions In fact the massdistributions obtained in cosmological calculations or sim-ulations [41 43] seem to be continuous but strictly singularnonetheless namely they seem to contain dense sets ofpower-law mass concentrations

Let us consider the example of random mass functiondisplayed in Figure 2 which is generated with one of themethods referred to in [51] Actually the real mass distribu-tion namely the mass density is obtained as the derivativeof the mass function and is not a proper function because ittakes only the values zero or infinity The graphs in Figure 2are the result of coarse graining with a small length butit is evident that the values of the density are either verylarge or very small Furthermore the large or small valuesare clustered respectively They form mass clusters or voidsthe former occupying vanishing total length and the latter

6 Advances in Astronomy

00 02 04 06 08 1000

02

04

06

08

10

00 02 04 06 08 10

Figure 2 Left one example of random mass distribution function generated as a random homeomorphism of the interval [0 1] Right thederivative of this function giving the mass distribution

occupying the full total length in the limit of vanishingcoarse-graining length Unlike the inverse devilrsquos staircasethe present mass distribution function is continuous soits graph contains no vertical segments in spite of beingvery steep and actually nondifferentiable at many pointsTherefore there are no zero-size mass condensations but justmass concentrationsThesemass concentrations are clustered(in our example as a consequence of the generating process)In view of its characteristics this random mass functionillustrates the type of one-dimensional mass distributions incosmology which can be considered generic Some furtherproperties of generic strictly singular mass distributions areworth noticing

To any mass distribution is associated a particular setnamely its supporting set which is the smallest closed setthat contains all the mass [50] This set is the full interval inthe examples of Figure 1 or Figure 2 but it can be smallerin strictly singular mass distributions and it can actually bea fractal set namely a set with Hausdorff dimension strictlysmaller than one In fact if Mandelbrot [3] seldom distin-guishes between fractal sets and fractal mass distributionsit is probably because in most of his examples the supportis a fractal set and the mass is uniformly distributed on it Inthis simple situation the distinction is superfluous Howeverwe have to consider singular mass distributions in general forthe cosmic geometry and take into account the fact that theobserved clustering may not be a property of the supportingset but a property of themass distribution on itThis aspect ofclustering is directly related to the nature of voids in a fractalset the set of voids is simply the complementary set whichis totally empty whereas the voids in a mass distributionwith full support have tiny mass concentrations inside (seeFigures 1 and 2)The latter option describes better our presentknowledge of cosmic voids [28 52 53]

In conclusion the appropriate geometrical setting forthe study of the large-scale structure of the universe is

the geometry of mass distributions as a generalization ofthe geometry of sets Naturally a generic mass distributionis more complex in three-dimensional space than in one-dimensional space In particular one has to consider prop-erties such as connectivity which are important for the webstructure (Figure 1) Since the supporting set of a geometricalmodel of cosmic structure is probably trivial namely the fullvolume one cannot formulate the morphological propertiesin terms of the topology of that set So it is necessary toproperly define the topology of a mass distribution This canbe done in terms of the topology of sets of particular massconcentrations but is beyond the scope of this work Theclassification of mass concentrations is certainly useful as aprevious step and by itself The standard way of classifyingmass concentrations constitutes the multifractal analysis ofmeasures [50 ch 17]

23 Multifractal Analysis of Mass Distributions The originalidea of hierarchical clustering of galaxies considers galaxiesas equivalent entities and constructs a sort of fractal setas the prolongation of the hierarchy towards higher scales(up to the scale of homogeneity) and towards lower scales(the prolongation to lower and indeed infinitesimal scales isnecessary to actually define a mathematical fractal set) Thechief observable is the fractal dimension which has beengenerally obtained through (3) namely from the value of 120574estimated through the correlation function of galaxy positions[3ndash6] By using the galaxy positions the density 984858 in (3) isthe galaxy number density instead of the real mass densityBoth densities should give similar results if the mass weredistributed uniformly over the galaxy positions HoweverPietronero [18] noticed that galaxies are not equivalent to oneanother and that in fact their masses span a broad rangeTherefore he argued that the fractal dimension based ongalaxy positions would not be enough and one should extend

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

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[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

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[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

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[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

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[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

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24 Advances in Astronomy

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[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

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[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 5

clusters and superclusters and introduced the notion oflacunarity without providing a mathematical definition ofit [3 sect34] Lacunarity is loosely defined as the property ofhaving large gaps or voids (lacuna) This property gainedimportance in cosmology with the discovery of large voidsin galaxy redshift surveys which came to be considereda characteristic feature of the large-scale structure of theuniverse [13 14]The precise nature of cosmic voids is indeedimportant to characterize the geometry of the cosmicweb andwill be studied below

In general for sets of a given Hausdorff dimension thereshould be a number of parameters that specify their appear-ance what Mandelbrot calls texture [3 ch X] Lacunarity isthe first one and can be applied to sets in any dimensionincluding one dimension Others are only applicable inhigher dimension for example parameters to measure theexistence and extension of subsets of a fractal set that aretopologically equivalent to line segments an issue related topercolation [3 sect34 and 35]

In fact the methods of percolation theory have beenemployed in the study of the cosmic web [1] Those methodsare related to the mathematical notion of path connectionof sets which is just one topological property among manyothers Topological properties are certainly useful but topol-ogy does not discriminate enough in fractal geometry asshown by the fact that all (compact) fractal subsets of Rare topologically equivalent According to Falconer [50]fractal geometry can be defined in analogy with topologyreplacing bicontinuous transformations by a subset of themnamely bilipschitz transformationsTheHausdorff dimensionis invariant under these transformations although it isnot invariant under arbitrary bicontinuous transformationsTherefore the set of parameters characterizing a fractalset includes on the one hand topological invariants andon the other hand properly fractal parameters beginningwith the Hausdorff dimension Unfortunately there has beenlittle progress in the definition of these parameters beyondMandelbrotrsquos heuristic definition of texture parameters

Besides the need for a further development of the geom-etry of fractal sets we must realize that the concept of fractalset is not sufficient to deal with the complexity of cosmicgeometry This is easily perceived by reconsidering the massdistribution generated by the adhesion model in particularin one dimension In the geometry of a devilrsquos staircase wemust distinguish the set of lengths of the steps from thelocations of these steps which are their ordinates in the graphon the left-hand side of Figure 1 These locations constitutea dense set that is to say they leave no interval empty [46]The set is nevertheless a countable set and therefore has zeroHausdorff dimension But more important than the locationsof mass condensations is the magnitude of these masses (thelengths of the steps) To wit we are not just dealing with theset of locations butwith the fullmass distributionMandelbrot[3] does not emphasize the distinction between these twoconcepts and actually gives no definition ofmass distributionbut other authors do for example Falconer includes anintroduction to measure theory [50] which is useful toprecisely define the Hausdorff dimension as well as to definethe concept of mass distribution A mass distribution is a

finite measure on a bounded set of R119899 a definition thatrequires some mathematical background [50] although theintuitive notion is nonetheless useful

The geometry of a generic mass distribution on R iseasy to picture because we can just consider the geometryof the devilrsquos staircase and generalize it It is not difficult tosee that the cumulative mass that corresponds to a devilrsquosstaircase interval is given by switching its axes for examplethe totalmass in the interval [0 119909] is given by the length of theinitial interval that transforms to [0 119909] under the Lagrangianmap The switching of axes can be performed mentallyin Figure 1 (left) The resulting cumulative mass functionis monotonic but not continuous A general cumulativemass function needs only to be monotonic and it may ormay not be continuous its discontinuities represent masscondensations (of zero size) Given the cumulative massfunction we could derive the mass density by differentiationin principle This operation should present no problemsbecause a monotonic function is almost everywhere differ-entiable However the inverse devilrsquos staircase has a denseset of discontinuities where it is certainly nondifferentiableThis does not constitute a contradiction because the set ofpositions of discontinuities is countable and therefore haszero (Lebesgue) measure but it shows the complexity ofthe situation Moreover in the complementary set (of fullmeasure) where the inverse devilrsquos staircase is differentiablethe derivative vanishes [46]

In summary we have found that our first model ofcosmological mass distribution is such that the mass densityis either zero or infinity We might attribute this singularbehavior to the presence of a dense set of zero-size mass con-densations due to the adhesion approximation Thereforewe could expect that a generic mass distribution should benonsingular Of course themeaning of ldquogenericrdquo is indefiniteFrom amathematical standpoint a generic mass distributionis one selected at random according to a natural probabilitydistribution (a probability distribution on the space of massdistributions)This apparently abstract problem is of interestin probability theory and has been studied with the resultthat the standard methods of randomly generating massdistributions indeed produce strictly singular distributions[51] This type of distributions has no positive finite densityanywhere like our discontinuous example but the resultalso includes continuous distributions In fact the massdistributions obtained in cosmological calculations or sim-ulations [41 43] seem to be continuous but strictly singularnonetheless namely they seem to contain dense sets ofpower-law mass concentrations

Let us consider the example of random mass functiondisplayed in Figure 2 which is generated with one of themethods referred to in [51] Actually the real mass distribu-tion namely the mass density is obtained as the derivativeof the mass function and is not a proper function because ittakes only the values zero or infinity The graphs in Figure 2are the result of coarse graining with a small length butit is evident that the values of the density are either verylarge or very small Furthermore the large or small valuesare clustered respectively They form mass clusters or voidsthe former occupying vanishing total length and the latter

6 Advances in Astronomy

00 02 04 06 08 1000

02

04

06

08

10

00 02 04 06 08 10

Figure 2 Left one example of random mass distribution function generated as a random homeomorphism of the interval [0 1] Right thederivative of this function giving the mass distribution

occupying the full total length in the limit of vanishingcoarse-graining length Unlike the inverse devilrsquos staircasethe present mass distribution function is continuous soits graph contains no vertical segments in spite of beingvery steep and actually nondifferentiable at many pointsTherefore there are no zero-size mass condensations but justmass concentrationsThesemass concentrations are clustered(in our example as a consequence of the generating process)In view of its characteristics this random mass functionillustrates the type of one-dimensional mass distributions incosmology which can be considered generic Some furtherproperties of generic strictly singular mass distributions areworth noticing

To any mass distribution is associated a particular setnamely its supporting set which is the smallest closed setthat contains all the mass [50] This set is the full interval inthe examples of Figure 1 or Figure 2 but it can be smallerin strictly singular mass distributions and it can actually bea fractal set namely a set with Hausdorff dimension strictlysmaller than one In fact if Mandelbrot [3] seldom distin-guishes between fractal sets and fractal mass distributionsit is probably because in most of his examples the supportis a fractal set and the mass is uniformly distributed on it Inthis simple situation the distinction is superfluous Howeverwe have to consider singular mass distributions in general forthe cosmic geometry and take into account the fact that theobserved clustering may not be a property of the supportingset but a property of themass distribution on itThis aspect ofclustering is directly related to the nature of voids in a fractalset the set of voids is simply the complementary set whichis totally empty whereas the voids in a mass distributionwith full support have tiny mass concentrations inside (seeFigures 1 and 2)The latter option describes better our presentknowledge of cosmic voids [28 52 53]

In conclusion the appropriate geometrical setting forthe study of the large-scale structure of the universe is

the geometry of mass distributions as a generalization ofthe geometry of sets Naturally a generic mass distributionis more complex in three-dimensional space than in one-dimensional space In particular one has to consider prop-erties such as connectivity which are important for the webstructure (Figure 1) Since the supporting set of a geometricalmodel of cosmic structure is probably trivial namely the fullvolume one cannot formulate the morphological propertiesin terms of the topology of that set So it is necessary toproperly define the topology of a mass distribution This canbe done in terms of the topology of sets of particular massconcentrations but is beyond the scope of this work Theclassification of mass concentrations is certainly useful as aprevious step and by itself The standard way of classifyingmass concentrations constitutes the multifractal analysis ofmeasures [50 ch 17]

23 Multifractal Analysis of Mass Distributions The originalidea of hierarchical clustering of galaxies considers galaxiesas equivalent entities and constructs a sort of fractal setas the prolongation of the hierarchy towards higher scales(up to the scale of homogeneity) and towards lower scales(the prolongation to lower and indeed infinitesimal scales isnecessary to actually define a mathematical fractal set) Thechief observable is the fractal dimension which has beengenerally obtained through (3) namely from the value of 120574estimated through the correlation function of galaxy positions[3ndash6] By using the galaxy positions the density 984858 in (3) isthe galaxy number density instead of the real mass densityBoth densities should give similar results if the mass weredistributed uniformly over the galaxy positions HoweverPietronero [18] noticed that galaxies are not equivalent to oneanother and that in fact their masses span a broad rangeTherefore he argued that the fractal dimension based ongalaxy positions would not be enough and one should extend

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

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[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

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[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

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[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

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[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

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[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

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[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

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Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

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[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

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[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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The Scientific World Journal

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Acoustics and VibrationAdvances in

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6 Advances in Astronomy

00 02 04 06 08 1000

02

04

06

08

10

00 02 04 06 08 10

Figure 2 Left one example of random mass distribution function generated as a random homeomorphism of the interval [0 1] Right thederivative of this function giving the mass distribution

occupying the full total length in the limit of vanishingcoarse-graining length Unlike the inverse devilrsquos staircasethe present mass distribution function is continuous soits graph contains no vertical segments in spite of beingvery steep and actually nondifferentiable at many pointsTherefore there are no zero-size mass condensations but justmass concentrationsThesemass concentrations are clustered(in our example as a consequence of the generating process)In view of its characteristics this random mass functionillustrates the type of one-dimensional mass distributions incosmology which can be considered generic Some furtherproperties of generic strictly singular mass distributions areworth noticing

To any mass distribution is associated a particular setnamely its supporting set which is the smallest closed setthat contains all the mass [50] This set is the full interval inthe examples of Figure 1 or Figure 2 but it can be smallerin strictly singular mass distributions and it can actually bea fractal set namely a set with Hausdorff dimension strictlysmaller than one In fact if Mandelbrot [3] seldom distin-guishes between fractal sets and fractal mass distributionsit is probably because in most of his examples the supportis a fractal set and the mass is uniformly distributed on it Inthis simple situation the distinction is superfluous Howeverwe have to consider singular mass distributions in general forthe cosmic geometry and take into account the fact that theobserved clustering may not be a property of the supportingset but a property of themass distribution on itThis aspect ofclustering is directly related to the nature of voids in a fractalset the set of voids is simply the complementary set whichis totally empty whereas the voids in a mass distributionwith full support have tiny mass concentrations inside (seeFigures 1 and 2)The latter option describes better our presentknowledge of cosmic voids [28 52 53]

In conclusion the appropriate geometrical setting forthe study of the large-scale structure of the universe is

the geometry of mass distributions as a generalization ofthe geometry of sets Naturally a generic mass distributionis more complex in three-dimensional space than in one-dimensional space In particular one has to consider prop-erties such as connectivity which are important for the webstructure (Figure 1) Since the supporting set of a geometricalmodel of cosmic structure is probably trivial namely the fullvolume one cannot formulate the morphological propertiesin terms of the topology of that set So it is necessary toproperly define the topology of a mass distribution This canbe done in terms of the topology of sets of particular massconcentrations but is beyond the scope of this work Theclassification of mass concentrations is certainly useful as aprevious step and by itself The standard way of classifyingmass concentrations constitutes the multifractal analysis ofmeasures [50 ch 17]

23 Multifractal Analysis of Mass Distributions The originalidea of hierarchical clustering of galaxies considers galaxiesas equivalent entities and constructs a sort of fractal setas the prolongation of the hierarchy towards higher scales(up to the scale of homogeneity) and towards lower scales(the prolongation to lower and indeed infinitesimal scales isnecessary to actually define a mathematical fractal set) Thechief observable is the fractal dimension which has beengenerally obtained through (3) namely from the value of 120574estimated through the correlation function of galaxy positions[3ndash6] By using the galaxy positions the density 984858 in (3) isthe galaxy number density instead of the real mass densityBoth densities should give similar results if the mass weredistributed uniformly over the galaxy positions HoweverPietronero [18] noticed that galaxies are not equivalent to oneanother and that in fact their masses span a broad rangeTherefore he argued that the fractal dimension based ongalaxy positions would not be enough and one should extend

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Submit your manuscripts atwwwhindawicom

Advances in Astronomy 7

the concept of fractal to that of multifractal with a spectrumof different exponents [18]

This argument was a good motivation for consideringmass distributions other than Mandelbrotrsquos typical examplenamely a self-similar fractal set with mass uniformly dis-tributed on itThis type of distribution characterized by onlyone dimension is usually called unifractal or monofractalin contrast with the general case of multifractal distribu-tions characterized by many dimensions However one canhave a sample of equivalent points from an arbitrary massdistribution as is obvious by considering this distributionas a probability distribution Actually several researcherswere soon making multifractal analyses of the distributionof galaxy positions without considering their masses whichwere not available [19 20] This approach is not mathemati-cally wrong but it does not reveal the properties of the realmass distribution We show in Section 24 the result of aproper multifractal analysis of the real mass distribution

Let us introduce a practical method of multifractalanalysis that employs a coarse-grained mass distribution andis called ldquocoarsemultifractal analysisrdquo [50] A cube that coversthe mass distribution is divided into a lattice of cells (boxes)of edge-length 119897 Fractional statistical moments are defined as

M119902 (119897) = sum119894

(119898119894119872)119902 (4)

where the index 119894 runs over the set of nonempty cells 119898119894 isthe mass in the cell 119894119872 = sum119894119898119894 is the total mass and 119902 isin RThe restriction to nonempty cells is superfluous for 119902 gt 0 butcrucial for 119902 le 0The power-law behavior ofM119902(119897) for 119897 997888rarr 0is given by the exponent 120591(119902) such that

M119902 (119897) sim 119897120591(119902) (5)

where 119897 is the box-size The function 120591(119902) is a globalmeasureof scaling

On the other hand mass concentrates on each box witha different ldquostrengthrdquo 120572 such that the mass in the box is119898 sim 119897120572 (if 119897 lt 1 then the smaller 120572 is the larger 119898 is andthe greater the strength is) In the limit of vanishing 119897 theexponent 120572 becomes a local fractal dimension Points with 120572larger than the ambient space dimension are mass depletionsand not mass concentrations It turns out that the spectrumof local dimensions is related to the function 120591(119902) by 120572(119902) =1205911015840(119902) [50] Besides every set of points with common localdimension 120572 forms a fractal set with Hausdorff dimension119891(120572) given by the Legendre transform 119891(120572) = 119902 120572 minus 120591(119902)

Of course a coarse multifractal analysis only yields anapproximated 119891(120572) which however has the same mathemat-ical properties as the exact spectrumThe exact spectrum canbe obtained by computing 119891(120572) for a sequence of decreasing119897 and establishing its convergence to a limit function

For the application of coarse multifractal analysis tocosmology it is convenient to assume that the total cube oflength 119871 covers a homogeneity volume that is to say 119871 isthe scale of transition to homogeneity The integral momentsM119899(119897) 119899 isin N are connected with the 119899-point correlationfunctions which are generally employed in cosmology [4]

(we can understand the nonintegral moments with 119902 gt 0as an interpolation) A straightforward calculation startingfrom (4) shows thatM119902 for 119902 gt 0 can be expressed in termsof the coarse-grained mass density as

M119902 (119897) = ⟨984858119902⟩119897⟨984858⟩119902 (

119897119871)3(119902minus1)

(6)

where ⟨984858119902⟩119897 is the average corresponding to coarse-graininglength 119897 In particularM2 is connected with the density two-point correlation function introduced in Section 22 In factwe can use (3) to integrate 120585 in a cell and calculate ⟨9848582⟩119897Assuming that 120585 ≫ 1 we obtain

⟨9848582⟩119897

⟨984858⟩2 sim (119871119897 )120574 (7)

ThereforeM2 sim 1198973minus120574 and comparing with (5) 120591(2) = 3 minus 120574In a monofractal (or unifractal) this is the only dimensionneeded and actually 120591(119902) = (119902 minus 1)120591(2) which implies 120572(119902) =1205911015840(119902) = 120591(2) and 119891(120572) = 119902 120572 minus 120591(119902) = 120591(2) In this case wecan write (5) as

M119902 sim M119902minus12 (8)

This relation restricted to integral moments is known incosmology as a hierarchical relation of moments [4 7]Actually if all119898119894 are equal for some 119897 in (4) thenM119902 = M

1minus1199020

for all 119902 and thereforeM119902 = M119902minus12 Let us notice that when

⟨984858119902⟩119897 ≃ ⟨984858⟩119902 (6) implies that 120591(119902) = 3(119902 minus 1) that is to saywe have a ldquomonofractalrdquo of dimension 3 In cosmology thishappens in the homogeneous regime namely for 119897 997888rarr 119871 butnot for 119897 997888rarr 0

The multifractal spectrum 119891(120572) provides a classificationof mass concentrations and 119891(120572) is easy to compute through(4) and (5) followed by a Legendre transform Its computa-tion is equally simple for mass distributions in one or severaldimensions Actually we know what type of multifractalspectrum to expect for self-similar multifractals it spans aninterval [120572min 120572max] is concave (from below) and fulfills119891(120572) le 120572 [50] Furthermore the equality 119891(120572) = 120572 isreached at one point which represents the set of singularmass concentrations that contains the bulk of the mass andis called the ldquomass concentraterdquoThemaximum value of 119891(120572)is the dimension of the support of the mass distribution Wededuce that 119891(120572) is tangent to the diagonal segment from(0 0) to (max119891max119891) and to the horizontal half line thatstarts at 120572 = max119891 and extends to the right The valuesof 119902 = 1198911015840(120572) at these two points of tangency are 1 and 0respectively Beyond the maximum of 119891(120572) we have 119902 lt 0Notice that the restriction to 119902 gt 0 and in particular to thecalculation of only integral moments misses information onthe structure corresponding to a considerable range of massconcentrations (possibly mass depletions) An example ofthis type of multifractal spectrum with max119891 = 1 is shownin Figure 3 (a full explanation of this figure is given below)

Themaximumof119891(120572) as the dimension of the support ofthe mass distribution provides important information about

8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

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[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

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Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

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[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

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[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

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24 Advances in Astronomy

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[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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8 Advances in Astronomy

05 10 15 20 25 30 35

0204060810f()

Figure 3 Several coarse multifractal spectra for the random massdistribution of Figure 2 showing very good convergence The limitfunction 119891(120572) is a typical example of multifractal spectrum of self-similar mass distributions

voids [53] If max119891 equals the dimension of the ambientspace then either voids are not totally empty or there isa sequence of totally empty voids with sizes that decreaseso rapidly that the sequence does not constitute a fractalhierarchy of voids This type of sequence is characterizedby the Zipf law or Pareto distribution [3 53 54] It seemsthat max119891 = 3 in cosmology and furthermore that thereare actually no totally empty voids as discussed below Inmathematical terms one says that the mass distributionhas full support and we can also say that it is nonlacunarWe have already seen examples of nonlacunar distributionsin Section 22 which are further explained below Let uspoint out a general property of nonlacunar distributionsWhen max119891 equals the dimension of the ambient space thecorresponding120572 is larger because120572 ge 119891(120572) (it could be equalfor a nonfractal set of points but we discard this possibility)The points with such values of 120572 are most abundant [119891(120572)is maximum] and also are mass depletions that belong tononempty voids The full set of points in these voids ishighly complex so it is not easy to define individual voidswith simple shapes [53] and may be better to speak of ldquothestructure of voidsrdquo [52]

As simple examples ofmultifractal spectrum calculationswe consider the calculations for the one-dimensional massdistributions studied in Section 22The case of inverse devilrsquosstaircases is special because the local dimension of zero-size mass condensations is 120572 = 0 Furthermore the set oflocations of mass condensations is countable and hence haszero Hausdorff dimension that is to say 119891(0) = 0 (in accordwith the general condition 119891(120572) le 120572) The complementaryset namely the void region has Hausdorff dimension119891(120572) =1 and local dimension 120572 = 1ℎ gt 1 [deduced from the steplength scaling (2)] Since the set ofmass concentrations fulfills119891(120572) = 120572 for 120572 = 0 it is in fact the ldquomass concentraterdquo Theclosure of this set is the full interval which is the support ofthe distribution and is dominated by voids that are not totallyempty a characteristic property of nonlacunar distributionsThis type of multifractal spectrum with just one point onthe diagonal segment from (0 0) to (1 1) and another onthe horizontal half line from (1 1) to the right has beencalled bifractal [55] However the multifractal spectrum thatis calculated through the 120591 function involves the Legendretransform and therefore is the convex hull of those twopoints namely the full segment from (0 0) to (1ℎ 1) The

extra points find their meaning in a coarse-grained approachto the bifractal

Let us comment further on the notion of bifractal distri-bution We first recall that a bifractal distribution of galaxieswas proposed byBalian and Schaeffer [20] with the argumentthat there is one fractal dimension for clusters of galaxiesand another for void regions In fact the bifractal proposedby Balian and Schaeffer [20] is more general than the onejust studied because they assume that the clusters that is tosay the mass concentrations do not have 119891(120572) = 120572 = 0but a positive value Such a bifractal can be considered asjust a simplification of a standard multifractal because anymultifractal spectrum has a bifractal approximation Indeedany multifractal spectrum contains two distinguished pointsas seen above namely the point where 119891(120572) = 120572 whichcorresponds to the mass concentrate and the point where119891(120572) is maximum The former is more important regardingthe mass and the second is more important regarding thesupporting set of themass distribution (the set of ldquopositionsrdquo)whose dimension is max119891 However this supporting set isriddled by voids in nonlacunar multifractals

The second example of multifractal spectrum calculationcorresponds to the one-dimensional random mass distri-bution generated for Figure 2 This mass distribution isjust a mathematical construction that is not connected withphysics but is a typical example of continuous strictly singularmass distribution The multifractal spectrum is calculableanalytically but we have just employed coarse multifractalanalysis of the particular realization in Figure 2 This is agood exercise to assess the convergence of the method for amass distribution with arbitrary resolution In Figure 3 aredisplayed four coarse spectra for 119897 = 2minus13 2minus14 2minus15 and 2minus16which show perfect convergence in the full range of 120572 exceptat the ends especially at the large-120572 end correspondingto the emptiest voids The limit is a standard spectrum ofa self-similar multifractal unlike the bifractal spectrum ofthe inverse devilrsquos staircase Since the mass distribution ofFigure 2 is nonlacunar max119891 reaches its highest possiblevaluemax119891 = 1The long range of valueswith120572 gt 1 suggestsa rich structure of voids

Finally let us generalize the multifractal spectrum of theinverse devilrsquos staircase generated by the one-dimensionaladhesionmodel to the three-dimensional case for a later useIn three dimensions in addition to point-like masses thereare filaments and sheets Therefore the (concave envelopeof the) graph of 119891(120572) is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with some 120572max gt 3 for voids as mass depletions (theconjecture that along a given line there is essentially aone-dimensional situation [46] suggests that 120572max = 2 +1ℎ) There are three distinguished points of the multifractalspectrum in the diagonal segment namely (0 0) (1 1) and(2 2) corresponding to respectively nodes filaments andsheets where mass concentrates Each of these entities corre-sponds to a Dirac-delta singularity in themass density nodesare simple three-dimensional Dirac-delta distributions whilefilaments or sheets are Dirac-delta distributions on one- ortwo-dimensional topological manifolds There is an infinite

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Submit your manuscripts atwwwhindawicom

Advances in Astronomy 9

but countable number of singularities of each typeMoreoverthe set of locations of each type of singularities is dense whichis equivalent to saying that every ball however small containssingularities of each type The mathematical description isperhaps clumsy but the intuitive geometrical notion canbe grasped just by observing Figure 1 (it actually plots atwo-dimensional example but the three-dimensional case isanalogous)

24 Multifractal Analysis of the Large-Scale Structure Ofcourse the resolutions of the available data of large-scalestructure are not as good as in the above algorithmicexamples but we can employ as well the method of coarsemultifractal analysis with shrinking coarse-graining lengthAs regards quality we have to distinguish two types of datadata from galaxy surveys and data from cosmological 119873-body simulations

The first multifractal analyses of large-scale structureemployed old galaxy catalogues These catalogues con-tained galaxy positions but no galaxy masses NeverthelessPietronero and collaborators [6 8] managed to assign massesheuristically and carry out a proper multifractal analysiswith (4) and (5) The results were not very reliable becausethe range of scales available was insufficient This problemalso affected contemporary multifractal analyses that onlyemployed galaxy positions Recent catalogues contain morefaint galaxies and therefore longer scale ranges Further-more it is now possible to make reasonable estimates of themasses of galaxies In this new situation we have employedthe rich Sloan Digital Sky Survey and carried out a multi-fractal analysis of the galaxy distribution in the data release7 taking into account the (stellar) masses of galaxies [56]The results are reasonable and are discussed below At anyrate galaxy survey data are still poor for a really thoroughmultifractal analysis

Fortunately the data from cosmological 119873-body simu-lations are much better On the one hand the state-of-the-art simulations handle billions of particles so they affordexcellent mass resolution Several recent simulations providea relatively good scaling range namely more than twodecades (in length factor) whereas it is hard to get evenone decade in galaxy redshift surveys Of course the scalerange is still small if compared to the range in our exampleof random mass distribution in one dimension but it issufficient as will be shown momentarily On the other handthe119873 bodies simulate the full darkmatter dynamics whereasgalaxy surveys are restricted to stellar matter which onlygives the distribution of baryonic matter (at best) Of coursethe distribution of baryonic matter can be studied in its ownright A comparison between multifractal spectra of large-scale structure in dark matter 119873-body simulations and ingalaxy surveys is made in [56] using recent data (the Bolshoisimulation Figure 4 and the Sloan Digital Sky Survey) Weshow in Figure 5 the shape of the respective multifractalspectra

Both the spectra of Figure 5 are computed with coarsemultifractal analysis so each plot displays several coarsespectra The plot in the left-hand side corresponds to the

Figure 4 Slice of 125 ℎminus1Mpc side length of the cosmic webstructure formed in the Bolshoi cosmological119873-body simulation

Bolshoi simulation which has very good mass resolution itcontains119873 = 20483 particles in a volume of (250 Mpc)3ℎ3The coarse spectra in the left-hand side of Figure 5 cor-respond to coarse-graining length 119897 = 39 Mpcℎ andseven subsequently halved scales Other cosmological 119873-body simulations yield a similar result and in particularone always finds convergence of several coarse spectra toa limit function Since the coarse spectra converge to acommon spectrum in different simulations we can considerthe found multifractal spectrum of dark matter reliableNotice however that the convergence is less compelling forlarger 120572 corresponding to voids and specifically to emptiervoids

In the right-hand side of Figure 5 is the plot of four coarsespectra of a volume-limited sample (VLS1) from the 7th datarelease of the Sloan Digital Sky Survey with 1765 galaxies ina volume of 368 sdot 104 Mpc3ℎ3 for coarse-graining volumesV = 153 sdot 103 191 239 299 Mpc3ℎ3 (the coarse-graininglength can be estimated as 119897 = V13) Four coarse spectra arethe maximum number available for any sample that we haveanalyzed [56]These four coarse spectra converge in the zone120572 lt 2 of relatively strong mass concentrations but there arefewer converging spectra for larger 120572 and the shape of thespectrum of mass depletions (120572 gt 3) is questionable While itseems that the multifractal spectrum reaches its maximumvalue possible max119891 = 3 so that the distribution can benonlacunar the values 119891(120572) for larger 120572 are quite uncertain[56] In particular it is unreliable that 120572max ≳ 6 a quite largevalue to be comparedwith the considerably smaller andmorereliable value of 120572max derived from119873-body simulations

At any rate we can assert a good concordance betweenthe multifractal geometry of the cosmic structure in cosmo-logical119873-body simulations and galaxy surveys to the extentthat the available data allow us to test it that is to say inthe important part of the multifractal spectrum up to itsmaximum (the part such that 119902 gt 0) The common featuresfound in [56] and visible in Figure 5 are as follows (i) aminimum singularity strength 120572min = 1 (ii) a ldquosupercluster

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Submit your manuscripts atwwwhindawicom

10 Advances in Astronomy

1 2 3 4

05

10

15

20

25

30

Bolshoi simulation

1 2 3 4 5 6

05

10

15

20

25

30

SDSS samplef()

f()

Figure 5 The multifractal spectra of the dark matter distribution in the Bolshoi simulation (coarse-graining lengths 119897 =391 195 098 049 024 0122 0061 0031 Mpcℎ) and of the stellarmass in a volume-limited galaxy sample of the SloanDigital Sky Survey(coarse-graining lengths 119897 = 115 576 288 144 Mpcℎ)

setrdquo of dimension 120572 = 119891(120572) ≃ 25 where the massconcentrates and (iii) a nonlacunar structure (without totallyempty voids) It is to be remarked that 120572min = 1 correspondsto the edge of diverging gravitational potential

As regards the visual morphological features a webstructure can be observed in both the 119873-body and galaxydistributions In particular the slice of the Bolshoi simulationdisplayed in Figure 4 definitely shows this structure andfurthermore it looks like the two-dimensional distributionfrom the adhesion model displayed in the right-hand side ofFigure 1 However when we compare the real multifractalspectrum of the large-scale structure with the multifrac-tal spectrum of the three-dimensional mass distributionformed according to the adhesion model as seen at the endof Section 23 we can appreciate considerable differencesTherefore in spite of the fact that the real cosmic web lookslike the web structure generated by the adhesion modelthe differences in their respective multifractal spectra revealdifferences in the distributions Indeed the cosmic webmultifractal spectrum well approximated by the left-handside graph in Figure 5 contains no point-like or filamentaryDirac-delta distributions and it does not seem to containsheet distributions although the graph passes close to thepoint (2 2) The differences can actually be perceived bycarefully comparing the right-hand side of Figure 1 withFigure 4 Notice that the features in the adhesion modelcosmic web appear to be sharper because they correspondto zero-size singularities namely nodes having zero area andfilaments zero transversal length In Figure 4 point-like orfilamentary mass concentrations are not so sharp becausethey correspond to power-law concentrations

In fact the notion of gravitational sticking in the adhesionmodel is a rough approximation that can only be valid forsheets because the accretion of matter to a point (120572 = 0) or asharp filament (120572 = 1) involves the dissipation of an infiniteamount of energy (in Newtonian gravity) It can be provedthat any mass concentration with 120572 lt 1 generates a diverginggravitational potential Therefore it is natural that the realcosmic web multifractal spectrum starts at 120572 = 1 (Figure 5)

Moreover Figure 5 shows that 119891(1) = 0 that is to say theset of locations of singularities with 120572 = 1 has dimensionzero which suggests that the gravitational potential is finiteeverywhere

The above considerations are based on the assumptionof continuous matter and even the adhesion model initiallyassumes a fluid model for the dynamics even though it givesrise to point-likemass condensations (at nodes)However119873-body dynamics is intrinsically discrete and this can modifythe type of mass distribution on small scales In fact cosmo-logical119873-body simulations have popularized halo models inwhich the matter distribution on small scales is constructedon a different basis

25 Halo Models The idea of galactic dark matter halos hasits roots in the study of the dynamics in the outskirts ofgalaxies which actually motivated the introduction of theconcept of dark matter These dark halos are to be relatedto the visible shapes of galaxies but should be in generalmore spherical and extend to large radii partially fillingthe intergalactic space and even including several galaxiesTherefore the large-scale structure of dark matter should bethe combination of quasi-spherical distributions in haloswiththe distribution of halo centers This picture is analogous toa statistical model of the galaxy distribution introduced byNeyman and Scott in 1952 [57] in which the full distributionof galaxies is defined in terms of clusters of points aroundseparate centers combined with a distribution of clustercenters Neyman and Scott considered a ldquoquasi-uniformrdquodistribution of cluster centers and referred to the Poissondistribution [57] Of course modern halo models considerthe clustering of halo centers [24]

The halo model is apparently very different from themodels that we have seen above but it is actually connectedwith the basic fractal model and its cluster hierarchy Indeedwe can consider Neyman and Scottrsquos original model asconsisting of a hierarchy with just two levels namely point-like galaxies and their clusters with little correlation betweencluster centers A specific formulation of such model is Fryrsquos

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Submit your manuscripts atwwwhindawicom

Advances in Astronomy 11

hierarchical Poisson model which he employed to study thestatistics of voids [58] This two-level hierarchy can be gener-alized by adding more levels (as already discussed [53]) Aninfinite cluster hierarchy satisfying Mandelbrotrsquos conditionalcosmological principle constitutes the basic fractal model Ofcourse halo models are normally defined so that they havepatterns of clustering of halos that are independent of thedistributions inside halos departing from self-similarity

In fact the distribution of halo centers is often assumed tobe weakly clustered and is sometimes treated perturbativelyHowever it tends to be recognized that halos must prefer-entially lie in sheets or filaments and actually be ellipsoidal[25 31] Neyman and Scott required the distribution ofpoints around the center to be spherical without furtherspecification Modern halo models study in detail the halodensity profiles in addition to deviations from sphericalsymmetry For theoretical reasons and also relying on theresults of cosmological 119873-body simulation density profilesare assumed to have power-law singularities at their centersThere are a number of analytic forms of singular profiles withparameters that can be fixed by theory or fits [24] A detailedaccount of this subject is beyond the scope of this work In ourcontext the power-law singularity and its exponent whichnormally has values in the range [minus15 minus1] are especiallyimportant

We can compare the density singularities in haloswith thesingular mass concentrations already studied in multifrac-tal distributions A coarse-grained multifractal distributionindeed consists of a collection of discretemass concentrationsand can be formulated as a fractal distribution of halos [2728] Models based on this idea constitute a synthesis of fractaland halo models Furthermore the idea naturally agrees withthe current tendency to place halos along filaments or sheetsand hence assume strong correlations between halo centersThe range [minus15 minus1] of the density power-law exponent isequivalent to the range 120572 isin [15 2] of the mass concentrationdimension defined by119898 sim 119903120572 In view of Figure 5 this rangeis below the value for the mass concentration set [fulfilling119891(120572con) = 120572con] which is 120572con ≃ 25 This difference canbe attributed to a bias towards quasi-spherical rather thanellipsoidal halos because the former are more concentratedIn the web structure generated by the adhesion model nodesare more concentrated than filaments or sheets

It is to be remarked that the theories of halo densityprofiles are consistent with a broad range of power-lawexponents and even with no singularity at all (Gurevich-Zybinrsquos theory [41] is an exception) In fact the quoteddensity-singularity range ismainly obtained from the analysisof 119873-body simulations For this and other reasons it isnecessary to judge the reliability of 119873-body simulations inthe range of scales where halos appear [32] Notice that thereliability of 119873-body simulations on small scales has beenquestioned for a long time [59ndash61] The problem is that two-body scattering spoils the collisionless dynamics altering theclustering properties Although the mass distribution insidean 119873-body halo is smooth (except at the center) this canbe a consequence of discreteness effects an 119873-body haloexperiences a transition from a smooth distribution to a very

anisotropic and nonsmooth web structure over a scale thatshould vanish in the continuum limit119873 997888rarr infin [31]

To conclude this section let us try to present an overviewof the prospects of the halo model The original idea of thehalo model namely the description of the cosmic structurein terms of smooth halos with centers distributed in a simplemanner and preferably almost uniformly seems somewhatnaive inasmuch as it is an ad hoc combination of two simpletypes of distributions which can only be valid on either verysmall or very large scales Moreover smooth halos seem tobe the result of 119873-body simulations in an unreliable rangeof scales In spite of these problems the notion of galacticdark matter halos that is to say of small-scale dark matterdistributions that control the dynamics of baryonic matterand the formation of galaxies is surely productive Howeverthe cold dark matter model has problems of its own becausethe collisionless dynamics tends anyway to generate toomuch structure on small scales and it is expected that thebaryonic physics will change the dynamics on those scales[62] The halo model provides a useful framework to studythis question [26] The ongoing efforts to unify the halo andmultifractal models [27ndash31] will hopefully help to give rise toa better model of the large-scale structure of the universe

3 Formation of the Cosmic Web

The halo or fractal models are of descriptive nature althoughthey can be employed as frameworks for theories of structureformation In contrast the Zeldovich approximation and theadhesion model constitute a theory of structure formationUnfortunately this theory is quite simplistic insofar as thenonlinear gravitational dynamics is very simplified to theextent of being apparently absent At any rate in the correctequations the equation for the gravitational field is linear andthe nonlinearity is of hydrodynamic type (see the equationsin Section 32)This nonlinearity gives rise to turbulence andin the adhesion model in particular to Burgers turbulencewhich is highly nontrivial

So we place our focus on the properties of Burgersturbulence However we begin by surveying other modelsof the formation of the cosmic web that are not directlybased on approximations of the cosmological equations ofmotion (Section 31) Next we formulate these equations inSection 32 and proceed to the study of turbulence in Burgersdynamics first by itself (Section 33) and second in the settingof stochastic dynamics (Section 34) Finally we consider thefull nonlinear gravitational dynamics (Section 35)

31 Models of Formation of Cosmic Voids Let us recall twomodels of the formation of the cosmic web namely theVoronoi foammodel [33 34] and Sheth and van deWeygaertrsquosmodel of the hierarchical evolution of voids [63] Both thesemodels focus on the formation and evolution of cosmic voidsbut the Voronoi foam model is of geometric nature whereasthe hierarchical void model is of statistical nature Theyare both heuristic and are not derived from the dynamicalequations of structure formation but the Voronoi foammodel can be naturally connected with the adhesion model

12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

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[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

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[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

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[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

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[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

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24 Advances in Astronomy

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[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

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[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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12 Advances in Astronomy

The Voronoi foam model focuses on the underdensezones of the primordial distribution of small density per-turbations The minima of the density field are the peaksof the primordial gravitational potential and therefore theseeds of voids Indeed voids form as matter flows away fromthose points The basic model assumes that the centers ofvoid expansion are random namely they form a Poissonpoint field and it also assumes that every center generates aVoronoi cell The formation of these cells can be explained bya simple kinematicmodel that prescribes that matter expandsat the same velocity from adjacent centers and therefore aldquopancakerdquo forms at the middle that is to say a condensationforms in a region of the plane perpendicular to the joiningsegment through its midpoint [34 Fig 40] The Voronoifoam results from the simultaneous condensation on all theseregions

Let us describe inmore detail the formation of the cosmicweb according to this kinematic model [33 34] After thematter condenses on a wall between two Voronoi cells itcontinues to move within it and away from the two nucleiuntil it encounters in a given direction the matter movingalong the other two walls that belong to the cell of a thirdnuclei The intersection of the three walls is a Voronoi edgeon which the matter from the three walls condenses andcontinues its motion away from the three nuclei Of coursethe motion along the edge eventually leads to an encounterwith the matter moving along three other edges that belongto the cell of a fourth nuclei and it condenses on thecorresponding node All this process indeed follows the rulesof the adhesion model but the Zeldovich approximation ofthe dynamics is replaced by a simpler kinematicmodel whichproduces a particular cosmic web namely a Voronoi foam Itis easy to simulate matter distributions that form in this wayand those distributions are suitable for a correlation analysis[34] This analysis reveals that the two-point correlationfunction is a power-law with an exponent 120574 that grows withtime as matter condenses on lower dimensional elements ofthe Voronoi foam Naturally 120574 is close to two in some epoch(or close to the standard value 120574 = 18)

The only parameter of the Voronoi foam model withrandom Voronoi nuclei is the number density of thesenuclei In fact this construction is well known in stochasticgeometry and is known as the Poisson-Voronoi tessellationThe number density of nuclei determines the average sizeof Voronoi cells that is to say the average size of cosmicvoids The exact form of the distribution function of Voronoicell sizes is not known but it is well approximated by agamma distribution that is peaked about its average [64]Therefore the Voronoi foam model reproduces early ideasabout cosmic voids namely the existence of a ldquocellularstructurerdquo with a limited range of cell sizes [37] This breaksthe self-similarity of the cosmic web found with the adhesionmodel by Vergassola et al [46] (see Section 21) The self-similarity of cosmic voids in particular is demonstratedby the analysis of galaxy samples or cosmological 119873-bodysimulations [53] Notice that gamma distributions of sizes aretypical of voids of various shapes in a Poisson point field [53]In a Voronoi foam with random nuclei the matter pointswould be the nodes of condensation namely the Voronoi

vertices and these do not constitute a Poisson point field butare somehow clustered At any rate a distribution of sizes ofcosmic voids that is peaked about its average is not realistic

Moreover a realistic foam model must account for theclustering of minima of the primordial density field and forthe different rates of void expansion giving rise to othertypes of tessellations In fact such models seem to convergetowards the adhesion model [33] The adhesion modelactually produces a Voronoi-like tessellation characterizedby the properties of the initial velocity field (or the initialdensity field) [47] In cosmology the primordial density fieldis not smooth and hence has no isolated minima Althougha smoothened or coarse-grained field can be suitable forstudying the expansion of cosmic voids the distribution ofvoids sizes is very different from the one predicted by thebasic Voronoi foam model [53]

Let us now turn to Sheth and van de Weygaertrsquos hier-archical model of evolution of voids [63] It is based on aningenious reversal of the Press-Schechter and excursion setapproaches to hierarchical gravitational clustering In factthe idea of the Press-Schechter approach is better suited forthe evolution of voids than for the evolution of mass concen-trations (a comment on the original approach can be foundat the end of Section 35)The reason is the ldquobubble theoremrdquowhich shows that aspherical underdense regions tend tobecome spherical in time unlike overdense regions [33 34]

Sheth and van de Weygaertrsquos hierarchical model of voidsis more elaborate than the Voronoi foam model but it alsopredicts a characteristic void size To be precise it predictsthat the size distribution of voids at any given time is peakedabout a characteristic void size although the evolution of thedistribution is self-similar Actually the lower limit to therange of void sizes is due to a small-scale cut-off which is putto prevent the presence of voids inside overdense regions thatare supposed to collapse (the ldquovoid-in-cloud processrdquo) [63]The resulting distribution of void sizes looks like the gammadistribution of the Voronoi foam model and Sheth and vande Weygaert [63] indeed argue that the hierarchical modelwhich is more elaborate somehow justifies the Voronoi foammodel which is more heuristic

If the small-scale cut-off is ignored by admitting thatvoids can form inside overdense regions then the distribu-tion of void sizes in Sheth and van de Weygaertrsquos modelhas no characteristic void size and for small sizes it isproportional to the power minus12 of the void volume [63]Power-law distributions of void sizes characterize lacunarfractal distributions which follow the Zipf law for voidsbut the exponent is not universal and depends on thefractal dimension [53] At any rate the results of analysesof cosmic voids and the cosmic web itself presented in [53]and Section 24 show that the matter distribution is surelymultifractal and nonlacunar This implies that cosmic voidsare not totally empty but have structure inside so that the self-similar structure does not manifest itself in the distributionsof void sizes but in the combination of matter concentrationsand voids (as discussed in [53]) In fact the cosmic webgenerated by the adhesion model with scale-invariant initialconditions is a good example of this type of self-similarity(Section 21)

Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 13

Therefore the conclusion of our brief study of models offormation of cosmic voids is that just the adhesion model ismore adequate than these other models for describing theformation of the real cosmic web structure The adhesionmodel derives from the Zeldovich approximation to thecosmological dynamics which we now review

32 Cosmological Dynamics We now recall the cosmologicalequations of motion in the Newtonian limit applicable onscales small compared to the Hubble length and away fromstrong gravitational fields [4 5 33] The Newtonian equationofmotion of a test particle in an expanding background is bestexpressed in terms of the comoving coordinate 119909 = 119903119886(119905)and the peculiar velocity 119906 = = 119907 minus 119867119903 where 119867 = 119886119886 isthe Hubble constant (for simplicity we can take 119886 = 1 in thepresent time) So the Newton equation of motion 119889119907119889119905 =119892T with total gravitational field 119892T can be written as

119889119906119889119905 + 119867119906 = 119892 (9)

where the net gravitational field 119892 = 119892T minus 119892b is definedby subtracting the background field 119892b = ( + 1198672)119903 Theequation for this background field is

3 ( + 1198672) = nabla sdot 119892b = minus4120587119866984858b + Λ1198882 (10)

that is to say the dynamical FLRW equation for pressurelessmatter (with a cosmological constant) This equation gives119867(119905) and hence 119886(119905) To have a closed system of equationswe must add to (9) the equation that gives 119892 in terms of thedensity fluctuations

nabla sdot 119892 = minus4120587119866 (984858 minus 984858b) (11)

and the continuity equation

120597984858120597119905 + 3119867984858 + nabla sdot (984858119906) = 0 (12)

These three equations form a closed nonlinear system ofequations

The equations can be linearized when and where thedensity fluctuations are small Within this approximationone actually needs only (9) which is nonlinear because of theconvective term 119906 sdot nabla119906 in 119889119906119889119905 but becomes linear whenthe density fluctuations and hence 119892 and 119906 are small Thelinearized motion is

119909 = 1199090 + 119887 (119905)119892 (1199090) (13)

where 119887(119905) is the growth rate of linear density fluctuationsRedefining time as = 119887(119905) we have a simple motion alongstraight lines with constant velocities given by the initial netgravitational field

33 Burgers Turbulence The Zeldovich approximation pro-longs the linear motion (13) into the nonlinear regime where119906 is not small To prevent multistreaming the adhesion

model supplements the linear motion with a viscosity termas explained in Section 21 resulting in the equation

119889119889 equiv 120597

120597 + sdot nabla = ]nabla2 (14)

where is the peculiar velocity in -time To this equationit must be added the condition of no vorticity (potentialflow) nabla times = 0 implied by nabla times 119892(1199090) = 0 Itis to be remarked that the viscosity term is in no wayfundamental and the same job is done by any dissipationterm in particular by a higher-order hyperviscosity term[65] Eq (14) is the (three-dimensional) Burgers equation forpressureless fluids (tildes are suppressed henceforth) Any] gt 0 prevents multistreaming and the limit ] 997888rarr 0 isthe high Reynolds-number limit which gives rise to Burgersturbulence Whereas incompressible turbulence is associatedwith the development of vorticity Burgers turbulence isassociated with the development of shock fronts namelydiscontinuities of the velocity These discontinuities arise atcaustics and give rise to matter accumulation by inelasticcollision of fluid elements The viscosity ] measures thethickness of shock fronts which become true singularities inthe limit ] 997888rarr 0

A fundamental property of the Burgers equation is that itis integrable namely it becomes a linear equation by applyingto it the Hopf-Cole transformation [1 35 36] This propertyshows that the Burgers equation is a very special case of theNavier-Stokes equation In spite of it Burgers turbulence is auseful model of turbulence in the irrotational motion of verycompressible fluids At any rate the existence of an explicitsolution of the Burgers equation makes the development ofBurgers turbulence totally dependent on the properties of theinitial velocity distribution in an explicit form The simplestsolutions to study describe the formation of isolated shocksin an initially smooth velocity field Fully turbulent solutionsare provided by a self-similar initial velocity field such that119906(120582119909) = 120582ℎ119906(119909) ℎ isin (0 1) as advanced in Section 21Indeed (14) is scale invariant in the limit ] 997888rarr 0 namely itis invariant under simultaneous space and time scalings 1205821199091205821minusℎ119905 for arbitrary ℎ if the velocity is scaled as 120582ℎ119906Therefore(14) has self-similar solutions such that

119906 (119909 119905) = 119905ℎ(1minusℎ)119906( 1199091199051(1minusℎ) 1) (15)

A self-similar velocity field that is fractional Brownian onscales larger than the ldquocoalescence lengthrdquo 119871 prop 1199051(1minusℎ)contains a distribution of shocks that is self-similar on scalesbelow 119871 Such velocity field constitutes a state of decayingturbulence since kinetic energy is continuously dissipated inshocks [1 35 36] In the full space R3 as is adequate forthe cosmological setting the transfer of energy from largescales where the initial conditions hold to the nonlinearsmall scales proceeds indefinitely

The simplicity of this type of self-similar turbulenceallows one to analytically prove some properties and makereasonable conjectures about others [35 36 46] A case thatis especially suitable for a full analytical treatment is the one-dimensional dynamics with ℎ = 12 namely with Brownian

14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

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[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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14 Advances in Astronomy

initial velocity As regards structure formation the dynamicsconsists in the merging of smaller structures to form largerones in such a way that the size of the largest structures isdetermined by the ldquocoalescence lengthrdquo and therefore theyhave masses that grow as a power of 119905 [1 46] The mergingprocess agrees with the ldquobottom-uprdquo picture for the growthof cosmological structure [5] This picture is appropriate forthe standard cold dark matter cosmology

From the point of view of the theory of turbulence cen-tered on the properties of the velocity field the formed struc-ture consists of a dense distribution of shock fronts with veryvariable magnitude Therefore the kinetic energy dissipationtakes place in an extremely nonuniform manner giving riseto intermittency [1] A nice introduction to the phenomenonof intermittency as a further development of Kolmogorovrsquostheory of turbulence can be found in Frischrsquos book [66 ch8] Kolmogorovrsquos theory assumes that the rate of specificdissipation of energy is constant Intermittency is analyzed bymodeling how kinetic energy is distributed among structureseither vortices in incompressible turbulence or shocks inirrotational compressible turbulence Mandelbrot proposedthat intermittency is the consequence of dissipation in afractal set rather than uniformly [3 sect10] Mandelbrotrsquos ideais embodied in the 120573-model of the energy cascade [66]However this model is monofractal and a bifractal model ismore adequate especially for Burgers turbulence Althoughthis bifractality refers to the velocity field [66] it correspondsin fact to the bifractality of the mass distribution in the one-dimensional adhesion model already seen in Section 23 Letus briefly study the geometry of intermittency in Burgersturbulence

Kolmogorovrsquos theory of turbulence builds on theRichard-son cascade model and is based on universality assumptionsvalid in the limit of infinite Reynolds number (] 997888rarr 0) andaway from flow boundaries As explained by Frisch [66] itis possible to deduce by assuming homogeneity isotropyand finiteness of energy dissipation in the limit of infiniteReynolds number the following scaling law for the momentsof longitudinal velocity increments

⟨(120575119906 sdot 119903119903)119896⟩ prop (120576119903)1198963 (16)

where

120575119906 (119909) = 119906(119909 + 1199032) minus 119906(119909 minus 1199032) (17)

120576 is the specific dissipation rate and 119896 isin N To introducethe effect of intermittency the Kolmogorov scaling law canbe generalized to

⟨|120575119906|119902⟩ = 119860119902 119903120577(119902) (18)

where 119902 isin R and 119860119902 does not depend on 119903 (and does nothave to be related to 120576) Eq (18) expresses that the fractionalstatistical moments of |120575119906| have a power-law behavior givenby the exponent 120577(119902) Therefore the equation is analogousto the combination of (4) and (5) for multifractal behaviorThe functions 120577(119902) and 120591(119902) are indeed connected in Burgersturbulence (see below)

If 120577(119902) prop 119902 in (18) as occurs in (16) then the probability119875(120575119906) is Gaussian The initial velocity field that we havechosen for Burgers turbulence is Gaussian namely fractionalBrownian with ⟨|120575119906|2⟩ prop 1199032ℎ (Section 21) Therefore 120577(119902) =ℎ119902 if 119903 ≫ 119871(119905) (which would fit (16) for ℎ = 13) In generalintermittency manifests itself as concavity of 120577(119902) [66] In theBurgers turbulence generated by the given initial conditionsa type of especially strong intermittency develops for any119905 gt 0 and 119903 ≪ 119871(119905) This intermittency is such that 120577(119902) inaddition to being concave has amaximumvalue equal to oneThis can be proved by generalizing the known argument forisolated shocks [35] to a dense distribution of shocks as isnow explained

Let us replace in (18) the ensemble average with a spatialaverage (notice that fractional statistical moments of 120575119898119872in (4) are defined in terms of a spatial average) The averagein (18) understood as a spatial average can be split intoregular and singular points (with shocks) At regular pointsthe inverse Lagrangian map 1199090(119909 119905) is well defined and from(13)

119906 (119909 119905) = 1199060 (1199090) = 1198920 (1199090) = 119909 minus 1199090 (119909 119905)119905 (19)

Self-similarity in time allows one to set 119905 = 1 without loss ofgenerality Hence we deduce the form of the spatial velocityincrement over 119903 at a regular point 119909

120575119906 (119909) = 119903 minus 1205751199090 (119909) (20)

Since the properties of the Lagrangian map are known onecan deduce properties of the velocity increment For exampleat a regular point 119909 the term 1205751199090(119909) is subleading for 119903 ≪ 119871because the derivatives of the inverse Lagrangianmap vanish

To proceed we restrict ourselves to one dimension inwhich the analysis is simpler and we have for 119903 ≪ 119871

⟨|120575119906|119902⟩ sim 119862119902 119903119902 +sum119899

10038161003816100381610038161205751199061198991003816100381610038161003816119902 119903 (21)

where the first term is due to the regular points and the secondterm is due to the set of 119906-discontinuities (shocks) such that|120575119906119899| ≫ 119903 Clearly putting 119905 back 120575119906 asymp 119903119905 gt 0 correspondsto the self-similar expansion of voids whereas 120575119906 lt 0 and|120575119906| ≫ 119903119905 corresponds to shocks If 119902 lt 1 the first termdominates as 119903 997888rarr 0 and vice versa Therefore 120577(119902) = 119902 for119902 le 1 and 120577(119902) = 1 for 119902 gt 1 Notice that the number of termsin the sum increases as 119903 997888rarr 0 but the series is convergentfor 119902 ge 1 due to elementary properties of devilrsquos staircasesif the spatial average is calculated over the interval Δ119909 thatcorresponds to an initial interval Δ1199090 ≫ L thensuminfin119899=1 |120575119906119899| =Δ1199090 and suminfin119899=1(|120575119906119899|Δ1199090)119902 lt 1 when 119902 gt 1 (actually theseries converges for 119902 gt 119863BT where 119863BT is the Besicovitch-Taylor exponent of the sequence of gaps which is in this caseequal to ℎ because of the gap law 119873(119872 gt 119898) prop 119898minusℎ [3 p359])

The form of 120577(119902) with its maximum value of one impliesthat the probability 119875(120575119906) is very non-Gaussian for 119903 ≪ 119871 asis well studied [35 36] and we now show In one dimensionthe ratio 120575119906119903 is a sort of ldquocoarse-grained derivativerdquo related

Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 15

by (20) to the coarse-grained density 1205751199090119903 (the length of theinitial interval 1205751199090 that ismapped to the length 119903 is the coarse-grained density normalized to the initial uniform density)Hence we derive a simple relation between the density andthe derivative of the velocity namely 984858 = 1minus120597119909119906 We can alsoobtain the probability of velocity gradients from119875(120575119906) in thelimit 119903 997888rarr 0 However this limit is singular and indeedthe distribution of velocity gradients can be called strictlysingular like the distribution of densities because at regularpoints where 984858 = 0 then 120597119909119906 = 1 and at shock positionswhere 984858 = infin then 120597119909119906 = minusinfin Naturally it is the shocks thatmake 119875(120575119906) very non-Gaussian and forbid that 120577(119902) growsbeyond 120577 = 1 The intermittency is so strong that the formof 120577(119902) is very different from Kolmogorovrsquos Eq (16) and theaverage dissipation rate 120576 seems to play no role

As the mass distribution can be expressed in terms of thevelocity field the intermittency of the latter is equivalent tothe multifractality of the former Indeed the bifractal natureof 120591(119902) in one dimension is equivalent to the dual form of120577(119902) for 119902 gt 1 or 119902 le 1 Let us notice however that 120577(119902) isuniversal whereas the bifractal 119891(120572) studied in Section 23depends on the value of ℎ (at the point for voids) and so does120591(119902) Actually the full form of 119875(120575119906) also depends on ℎ

For the application to cosmology we must generalizethese results to three dimensions Although this general-ization is conceptually straightforward it brings in somecomplications in addition to point-like mass condensationsthere appear another two types of mass condensationsnamely filaments and sheets the latter actually correspond-ing to primary shocks (furthermore there is mass flow insidethe higher-dimensional singularities and this flow must bedetermined [47]) The form of 120577(119902) is still determined by theprimary shocks and is therefore unaltered [35 36] Nev-ertheless the mass distribution itself is no longer bifractalas is obvious The form of 119891(120572) has already been seen inSection 23 namely it is the union of the diagonal segmentfrom (0 0) to (2 2) and the segment from (2 2) to (120572max 3)with 120572max gt 3

In summary the adhesion model with initial velocityand density fields having Gaussian power-law fluctuationsnamely being fractional Brownian fields leads to a self-similar and strongly intermittent Burgers turbulence suchthat various quantities of interest can be calculated analyti-cally This dynamics gives rise to a self-similar cosmic webthat is quite realistic Therefore it constitutes an appealingmodel of structure formation in cosmology [1 46] Howeverit has obvious shortcomings (i) in cosmology the initial butldquoprocessedrdquo power spectrum is not a power-law [5 33] (ii)the Burgers equation is integrable and therefore the adhesionmodel may be too simplified model of the gravitationaldynamics which actually is chaotic and (iii) the condensationofmatter in shock fronts is considered as an inelastic collisionbut the dissipated kinetic energy is assumed to disappearwithout further effect In general chaos and dissipationare connected chaotic dynamics erases memory of initialconditions giving rise to an irreversible process in whichentropy grows To deal with these problems we need a moregeneral framework A first step is taken by connecting theBurgers equation with a well-studied stochastic equation

namely the KardarndashParisindashZhang (KPZ) equation as westudy next

34 The Stochastic Burgers Equation After studying freelydecaying Burgers turbulence in the adhesion model weproceed to the study of forced Burgers turbulence [35 36] It isunderstood that the forcing in cosmology is not external butis just the feedback of dissipated energy Indeed we intendto achieve a more complete understanding of the dynamicsthat is based on a general relation between dissipation andfluctuating forces such as the relation embodied in theclassical fluctuation-dissipation theorem [67] So we rely onthe theory of stochastic dynamics with origin in the Langevinequation for Brownian motion The Langevin equation isderived by simply adding a fluctuating force 119865(119905) to Newtonrsquossecond law This force is due to the interaction with theenvironment and must contain a slowly varying dissipativepart and a rapidly varying part with null average (a ldquonoiserdquo)

The Langevin equation has a limited scope A qualitativedevelopment in stochastic dynamics consisted in consideringpartial differential equations with fluctuating forces thatdepend on space and time A simple example is the Edwards-Wilkinson equation which results from adding noise to thediffusion equation [68] Assuming that the scalar field ofthe Edwards-Wilkinson equation is the ldquoheightrdquo of somesubstrate the equation can be employed in the study ofstochastic growthThe Edwards-Wilkinson equation is linearbut a more suitable equation for stochastic growth namelythe KardarndashParisindashZhang (KPZ) equation was obtained byadding to it a nonlinear term For several reasons thisequation has become a paradigm in the theory of stochasticnonlinear partial differential equations In fact by takingthe scalar field in the KPZ equation as a velocity potentialthe equation becomes equivalent to the stochastic Burgersequation in which a noise term that is to say a random forceis added to the right-hand side of (14) so that it containsdissipation plus noise

The presence of noise in the stochastic Burgers equationalters the evolution of the velocity and density fields Forexamplemass condensations can gain energy andhence frag-ment so adhesion is no longer irreversible It may even nothappen at allThe different kind of evolution induced by noiseis best appreciated if the spatial region is bounded with size119871 (typically the use of periodic boundary conditions makesspace toroidal) Then the total mass and kinetic energy arebounded and the free decay of Burgers turbulence eventuallyleads to a simple state For example in one dimension allthe kinetic energy is dissipated and all the mass condensateson a single point (for generic initial conditions) [35 36] Asnoise injects energy the dissipative dynamics cannot relax tothe state of minimum energy but to a fluctuating state whichmay or may not have mass condensations according to thecharacteristics of the noise

In the Langevin dynamics the particle motion relaxesuntil it is in thermal equilibriumwith themediumand has theMaxwell velocity distribution [67] The strength of the noiseis given by the temperature 119879 and the velocity fluctuationsfulfill

16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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16 Advances in Astronomy

⟨[119907 (119905 + Δ119905) minus 119907 (119905)]2⟩ = 2119879119898

Δ119905119905rel (22)

where 119898 is the mass of the particle and 119905rel is the relax-ation time This equation says that the variance of ΔV isindependent of the absolute time 119905 and this holds for all119905 provided that Δ119905 ≪ 119905rel In contrast the value of ⟨119907(119905)⟩explicitly depends on 119905 and on the initial condition 119907(0) Theproportionality of the variance of ΔV to Δ119905 is characteristic ofBrownian motion

The thermal noise in the Langevin equation is Gaussianand macroscopically uncorrelated [67] that is to say it haswhite frequency spectrum However there is no reason toassume thermal equilibrium in general Indeed various ldquocol-orsrdquo are considered in stochastic dynamics In the theory ofstochastic partial differential equations the noise is Gaussianand has a general power spectrum 119863(119896 120596) In particular anoise with power-law spectrumof spatial correlations but stilluncorrelated in time is favored namely with 119863(119896) prop 119896minus2120588(see below)

The Edwards-Wilkinson equation for a scalar field 120601(119909 119905)is equivalent to the stochastic Burgers equation for 119906 =minusnabla120601 with the convective term suppressed The solutionof the Edwards-Wilkinson equation is more difficult thanthe solution of the Langevin equation but it is a linearequation and hence soluble by decoupling of the Fourierspatial modes The dynamics also consists in a relaxation toa stationary state [68] However the relaxation takes place ata rate that depends on the ldquoroughnessrdquo of 120601(119909 119905) at 119905 = 0Indeed Fourier spatial modes have relaxation times inverselyproportional to 1198962 so relaxation is faster on the smaller scales

TheKPZdynamics also begins to relax on the small scalesOn these scales the nonlinear term which correspondsto the convective term in the Burgers equation becomesimportant As said above the combination of nonlinearityand dissipation tends to build structure whereas the noisetends to destroy it In fact the KPZ dynamics is ruled bythe competition between nonlinearity and noise To proceedlet us introduce the dynamic scaling hypothesis [68] as ageneralization of (22)

Given a general stochastic partial differential equation fora scalar field 120601(119909 119905) it is reasonable to assume that

⟨[120601 (119909 + Δ119909 119905 + Δ119905) minus 120601 (119909 119905)]2⟩= |Δ119909|2120594 119891( Δ119905

|Δ119909|119911) (23)

where 120594 and 119911 are critical exponents and 119891 is a scalingfunction This function expresses the crossover between thepower laws of purely spatial and purely temporal correlations119891(119904) approaches a constant for 119904 997888rarr 0 and behaves as 1199042120594119911for 119904 997888rarr infin Eq (23) generalizes (22) to both space andtime and hence is analogous to the space-time similarityalready considered in (15) [the redefinition 119891(119904) = 1199042120594119911119892(119904)makes the connection with (15) or (22) more apparent] (Letus notice that the averages considered in this section are takenwith respect to realizations of the random noise unlike inthe preceding section in which they are taken with respect to

realizations of the random initial conditions) Eq (15) impliesthat the initial conditions are modified below a scale thatgrows as a power of time [with exponent 1(1 minus ℎ)] andlikewise (23) implies that relaxation to the noise-inducedasymptotic distribution takes place below a scale that growsas a power of Δ119905 with exponent 1119911

In the case of the Edwards-Wilkinson equation in 119889dimensions for a field 120601(119909 119905) dynamic scaling is borne outby the explicit solution In the solution the Fourier mode120601k(119905) is a linear combination of the noise Fourier modesso a Gaussian noise produces a Gaussian 120601 [68] This fieldfulfills (23) so 120601 and 119906 = minusnabla120601 for fixed 119905 are actuallyfractional Brownian fields with Hurst exponents 120594 and 120594 minus1 respectively With noise spectrum 119863(119896) prop 119896minus2120588 theexplicit solution yields 120594 = 1 minus 1198892 + 120588 and 119911 = 2as can also be deduced from the scaling properties of theequation Therefore 120601 and 119906 at fixed 119905 become less roughas 120588 grows For example in one dimension and with whitenoise 120594 = 12 so 120601(119909 119905) describes for fixed 119905 a Browniancurve whereas 119906 = minus120597119909120601 is uncorrelated (white noiselike) Moreover the dynamics consists in a relaxation tothermal equilibrium in which fluid elements acquire theMaxwell velocity distribution in accord with the fluctuation-dissipation theorem

The KPZ equation is nonlinear and its solution is anon-Gaussian field Therefore (23) is just an assumptionsupported by theoretical arguments and computer simula-tions The one-dimensional case with white noise is specialinsofar as the effect of the nonlinear term is limited thefluctuation-dissipation theorem holds and 120594 = 12 likein the Edwards-Wilkinson equation For general dimensionand noise uncorrelated in time the Burgers equation hasGalilean invariance (invariance under addition of a constantvector to 119907 plus the consequent change of 119909) which enforcesthe relation 120594 + 119911 = 2 [68] In one dimension and withwhite noise this relation implies that 119911 = 32 which showsthat the KPZ and Edwards-Wilkinson equations differ intemporal correlations that is to say the convective termof thestochastic Burgers equation does have an effect Neverthelessthe dynamics also consists in a relaxation to the Maxwellvelocity distribution In the state of thermal equilibriumthere is no structure

At any rate for cosmology we must consider the three-dimensional KPZ equation The study of the KPZ equationwith ldquocoloredrdquo noise has been carried out by Medina et al[69] employing renormalized perturbation theory namelyperturbation of the linear equation by the nonlinear term andsubsequent renormalization Within this approach the typesof noise that give rise to dynamical scaling are found as fixedpoints of the dynamical renormalization group For small 119896the noise power spectrum assumes the universal form

119863 (119896) asymp 1198630 + 119863119896minus2120588 120588 gt 0 (24)

that is to say the noise consists of a white noise plus apower-law long-range noise [69] Unfortunately in threedimensions renormalized perturbation theory is of littleconsequence the only stable renormalization group fixedpoint is the trivial one (vanishing nonlinear convective

Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 17

term) and it is stable only if 120588 lt 12 [69] This meansthat a noise with sufficient power on large scales inevitablyleads to a strong-coupling stationary state and questions theapplication of perturbation theory to the stochastic Burgersequation with spatially correlated noise in three dimensionsIn spite of it the stochastic adhesionmodel has been long stud-ied in cosmology and mostly with perturbative treatmentswhich are still being applied [70] Certainly nonperturbativemethods seem more sensible [71]

In fact renormalized perturbation theory in any dimen-sion 119889 is valid only for 120588 lt (119889 + 1)2 [69] For larger120588 the long-range noise correlations induce analogous long-range correlations in 120601 and 119906 and give rise to intermittencyof 119906 (as studied for 119889 = 1 in [72]) In particular thefield 119906 becomes strongly non-Gaussian This is a purelynonlinear effect and may seem counterintuitive because inthe Edwards-Wilkinson equation 120601 and 119906 at fixed 119905 becomeless rough as 120588 grows The progressive smoothness of 119906 turnsit into a real hydrodynamical field (in contrast with the whitenoise like velocity in 119889 = 1 with 120588 = 0 for example)Although the nonlinearity namely the convective term of theBurgers equation does not spoil this smoothing generallyspeaking it gives rise to shocks and hence intermittencyThe combination of smoothing and formation of shocks canbe seen for example in simulations in 119889 = 1 [72 Fig 4]Therefore for 120588 gt (119889 + 1)2 we have the regime in whichstructures can form in spite of the noise that is to say theproperly turbulent regime As we focus on the field 119906 in thisregime we replace (23) with (18) [we have 120577(2) = 2(120594 minus 1)]

The turbulent regime of forced Burgers turbulence hasbeen well studied with a combination of computer simula-tions and theoretical arguments related to the Kolmogorovtheory A simple argument of Chekhlov and Yakhot [73]for 119889 = 1 shows that 120588 = 32 corresponds to anldquoalmost constantrdquo (logarithmic) energy flux in Fourier space(a balanced Richardson energy cascade) Furthermore theirnumerical simulations show that 119906 develops shocks (with thecorrespondingmass condensations)which give rise to power-law correlations with exponents 119911 = 23 and 120577(2)2 = 120594minus1 =13 The latter corresponds to the Kolmogorov scaling inaccord with (16) Let us also notice that the noise exponent120588 = 32 is such that the noise strength 119863 has the dimensionsof the dissipation rate 120576 But Chekhlov and Yakhotrsquos argumentis incomplete so that larger values of 120588 are suitable [72] Abreakthrough in the study of intermittency and Kolmogorovscaling in forced Burgers turbulence (in one dimension) hasbeen the use by Polyakov [74] of nonperturbative methodsborrowed from quantum field theory Polyakov [74] assumesthat the Burgers noise correlation function (in ordinaryspace) is twice differentiable that is to say 120588 ge 52 Boldyrev[75] has extended Polyakovrsquos methods to the range 120588 isin[32 52]

For cosmology we must consider 119889 = 3 Kolmogorovrsquosuniversality and its consequences for cosmology are consid-ered as a guiding principle in [71] The conclusion is that theinteresting range of the exponent 120588 for the cosmic structurein 119889 = 3 is 120588 isin (52 72)The lower limit 120588 = 52 is such thatthe noise strength119863 has the dimensions of 120576 and the Burgersnoise correlation function is proportional to log 119903 Moreover

120576diverges for ] 997888rarr 0 if120588 le 52The reason for the upper limit120588 = 72 is that the 119903-dependent part of the noise correlationfunction depends explicitly on the system size 119871 for 120588 gt 72The range 120588 isin (52 72) corresponds to large-scale forcingthat is such that the dissipation rate 120576 only depends on whathappens on large scales and such that the 119903-dependent partof the noise correlation function proportional to 1199032120588minus5 isuniversal

Furthermore there is intermittency for 120588 gt (119889 + 1)2 =2 in particular in the range 120588 isin (52 72) To measure itwe can use (18) where now the exponent 120577(119902) depends onthe noise exponent 120588 (and the average is with respect to therandomnoise)The intermittency increases with increasing 120588and progressively affects exponents 120577(119902) with lower values of119902 [76] The analysis of the range 120588 isin (52 72) obtains strongintermittency namely 120577(119902) = 1 for 119902 ge 3 but 23 lt 120577(2) le1 In particular 120577(2) = 4(120588 minus 2)3 while in the range 120588 isin(52 114) [76] At 120588 = 114 the exponent 120577(2) reaches itsmaximum 120577(2) = 1 and it remains fixed for 120588 gt 114 Theminimum value 120577(2) = 23 for 120588 = 52 corresponds to theKolmogorov scaling (16)

Especially interesting for cosmology is that the densitytwo-point correlation function can be obtained in terms ofcorrelation functions of the velocity field in fact it can beobtained in terms of just the two-point correlation function[71] Furthermore the density two-point correlation functionis a power-law taking the form in (3) with an exponent 120574determined by 120577(2) namely 120574 = 2 minus 120577(2) Given the range of120577(2) we deduce that

1 lt 120574 lt 43 (25)

which is a reasonable range (although the preferred value of120574 is somewhat larger)

To summarize the stochastic Burgers equation withspatially correlated noise constitutes an interesting modelof structure formation in which the cosmic web is notdetermined by ad hoc initial conditions but is the resultof the interplay of the inelastic gravitational condensationof matter with the consequent dynamical fluctuations sothat energy can be conserved on the average The relaxationto the asymptotically stable state takes place over a scalethat grows with time namely 119871 prop 1199051119911 where 119911 is deter-mined by the dynamics instead of the initial conditions Theasymptotically stable state has simple scaling properties andfurthermore it achieves Kolmogorovrsquos universality Howeverthis universality does not lead to uniqueness in the senseof total independence of the large scales indeed there is anallowed range for the noise exponent 120588 At any rate themodelpredicts a stable fractal cosmic web such that the exponentof the density two-point correlation function is within areasonable range Furthermore the formation of structuresthat are stable in the highly nonlinear regime connects inspirit with famous Peeblesrsquo stable clustering hypothesis [4]to be studied in the next section

The rationale for a fluctuating force is not restricted to thecosmological equations in the Zeldovich approximation Onecan employ the original equations namely (9) (11) and (12)This system of equations can be supplemented with viscous

18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

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[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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18 Advances in Astronomy

and random forcing terms [77] At any rate it behooves us totake a more general standpoint and study the full dynamicsof gravitational clustering

35 Nonlinearity Chaos and Turbulence in GravitationalClustering The first comprehensive attempts to treat ana-lytically the theory of large-scale structure formation aredue to Peebles and collaborators [4] They employed generalprinciples of statistical mechanics to formulate the problemand added a scaling hypothesis to obtain definite solutionsThis interesting work did not lead to a geometrical picture ofstructure formation until the connection with Mandelbrotrsquosideas [3] and the development of the Zeldovich approxima-tion and the adhesion model [1] Let us recall the basics ofPeeblesrsquo approach (a complete introduction to it with somenovelties is provided by Shani and Coles [33])

Surely the most general approach to the nonequilibriumstatistical mechanics of a system of particles is based on theLiouville equation and the derived hierarchy of equationsfor phase-space probability functions involving increasinglymore particles known as the BBGKY hierarchy (BBGKYstands for Bogoliubov-Born-Green-Kirkwood-Yvon) Everysuccessive equation of this hierarchy gives the evolution ofan 119873-particle correlation function but involves the (119873 + 1)-particle correlation functionTherefore to have amanageableset of equations it is necessary to close the hierarchy atsome level by assuming some relation between the119873-particlecorrelation function and lower order correlation functionsClosure approaches are also much employed in the statisticaltheory of turbulence and indeed the Kolmogorov theorycan be understood as a closure approach with an additionalscaling hypothesis [66] Peebles and collaborators [4] followa similar path considering density correlation functions inaddition to velocity correlation functions which are the onlyones present in incompressible turbulence

In fact the scaling hypothesis of Peebles et al focuses onthe reduced two-point density correlation function 120585 whichfulfills a simple equation that just expresses the conservationof particle pairs

120597120585120597119905 +

11199092119886

120597120597119909 [1199092 (1 + 120585) V] = 0 (26)

where V is the mean relative peculiar velocity of particlepairs Of course the presence of V makes this equation notclosed To proceed it is necessary to assume closure relationsnot only for (26) but also for the full set of equationsNevertheless it is possible to obtain information about 120585only from the conservation of particle pairs using scalingarguments [4] However these arguments are restricted to theEinstein-de Sitter cosmological model with null curvatureand therefore without characteristic scale so that 984858b(119905) and119886(119905) are power laws namely 984858b(119905) prop 119905minus2 and 119886(119905) prop 11990523Furthermore the initial spectrum of density fluctuationsmust be a power-law

Given the absence of characteristic scales Peebles andcollaborators propose similarity solutions of the equationsnamely solutions that are functions of the sole variable119904 = 119909119905120572 with an exponent 120572 that is to be determined

In particular they seek a similarity solution 120585(119904) of (26) Asthis equation is not closed one has to assume a relationbetween the expectation value of the relative velocity and 120585and furthermore that this relation adopts a simple scalingform in the linear or very nonlinear regimes namely in thecases 120585 ≪ 1 or 120585 ≫ 1 In the linear case one knowsthat 120585 prop 11990543119909minus3minus119899 which is in accord with the Zeldovichapproximation 120585 prop 2119909minus3minus119899 with prop 11990523 for the Einstein-de Sitter cosmology (119899 is the exponent of the Fourier powerspectrum of density fluctuations) Therefore one deducesthat 120572 = 4(3119899 + 9)

Of course the problematic case is the very nonlinearregime To deal with it Peebles and collaborators proposedthe stable clustering hypothesis namely that the averagerelative velocity of particle pairs vanishes in physical (notcomoving) coordinates [4]This hypothesismay seemnaturalfor tightly bound gravitational systems It plays a role that issomewhat analogous to Kolmogorovrsquos hypothesis of constantdissipation rate 120576 in turbulence Indeed the constancy of 120576provides a link between large and small scales and so doesthe stable clustering hypothesis In particular substitutingV = minus 119886119909 in (26) and furthermore taking into account thefact that 120585 ≫ 1 it is easy to solve the equation and obtain120585(119904) prop 119904minus120574 where 120574 = 2(120572 + 23) In terms of variables (119905 119909)or (119886 119903) where 119903 = 119886119909 is the physical distance

120585 prop 1199052120572(120572+23)119909minus2(120572+23) prop 1198863119903minus2(120572+23) (27)

Naturally as regards the 119903-dependence (27) is just a particu-lar case of (3) in which 120574 is expressed in terms of 120572 which canbe derived from the initial power spectrum exponent 119899 (inthe Einstein-de Sitter cosmology) giving 120574 = 2(120572 + 23) =3(3 + 119899)(5 + 119899) As regards the dependence on 119886 that is tosay on time (27) has new content which can be generalized

First let us remark that (27) can be deduced withoutinvoking the equation for conservation of particle pairs orclosure relations bymeans of a heuristic argument also due toPeebles [5 sect22] This argument is based on the estimation ofthe density of a bound system which can be assumed to givethe average conditional density namely the average density atdistance 119903 from an occupied point This statistic is expressedas

⟨984858 (0) 984858 (119903)⟩⟨984858⟩ = ⟨984858⟩ [1 + 120585 (119903)] (28)

where ⟨984858⟩ = 984858b is the background cosmic density For a boundsystem 120585 ≫ 1 and the density of a system of size 119903 is theconditional density 984858b 120585(119903) The average conditional densityis especially useful to define the extremely nonlinear limit inwhich 120585 997888rarr infin and 984858b 997888rarr 0 while its product stays finiteThis limit actually allows one to dispense with the transitionto homogeneity and study directly a fractal mass distribution[3 18]

To estimate the density of a bound system one can followthe evolution of an overdensity 120575984858 = 984858 minus 984858b of comovingsize 119909 which initially grows as 120575984858 prop 984858b 11990523119909minus(3+119899)2 Overa time 119905(119909) prop 1199093(3+119899)4 the overdensity is of the order ofmagnitude of the background density and the linear theory

Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Advances in Astronomy 19

is no longer valid [notice that this time is the ldquocoalescencetimerdquo that can be deduced in the adhesionmodel by invertingthe ldquocoalescence lengthrdquo expression 119871 prop 1(1minusℎ)] In termsof the physical size 119903 the time for nonlinearity fulfills therelation 119905 prop [119903119886(119905)]3(3+119899)4 which solving for 119905 gives 119905(119903) prop1199033(3+119899)(10+2119899) At this time the background density is 984858b prop119905minus2 prop 119903minus3(3+119899)(5+119899) and our overdensity as approximatelytwice the background density follows the same rule In fact afactor of twice the background density seems hardly sufficientfor the formation of a bound system for example thespherical collapse model favors a factor that is about 180 [33]The exact value of this factor is irrelevant and the importantpoint is that the self-similar evolution of an overdense blobof size 119903 leads to a bound system with densityprop 119903minus3(3+119899)(5+119899)Therefore interpreting this density as a conditional densityone obtains

120585 prop 984858minus1b 119903minus3(3+119899)(5+119899) prop 1198863 119903minus3(3+119899)(5+119899) (29)

in accord with (27)We can drawmore general conclusions by only appealing

to the stable clustering hypothesis and without assuming anyscaling in the cosmological evolution Indeed if particle pairskeep on the average their relative physical distances thenthe average conditional density is a function 119891(119903) that doesnot depend on time Therefore 120585 = 1198863119891(119903) where 119891 is anarbitrary function This form of 120585 is also obtained as thegeneral solution of (26) under stable clustering conditionsnamely V = minus 119886119909 and 120585 ≫ 1 which allow us to write (26)as

119886120597120585120597119886 minus 11199092

120597120597119909 (1199093120585) = 0 (30)

One can check that 120585 = 1198863119891(119886119909) solves the equation by directsubstitution

Of course the stable clustering hypothesis can be ques-tioned Peebles says that ldquomerging might be expected todissipate the clustering hierarchy on small scalesrdquo [5 sect22]and indeed the effect of merging is often presented as anargument against the hypothesis However this argumentdisregards that stable clustering occurs only on the averagethat is to say that merging can be accompanied by split-ting keeping the average physical distance of particle pairsconstant Actually the conservation of energy in the localinertial frame demands it so We have used this argument forintroducing the stochastic adhesion model in Section 34 amodel in which the merging and subsequent fragmentationof mass condensations take place in a stable manner Thestable clustering hypothesis in cosmology is actually relatedto the possibility of studying in the local inertial frameand without a cosmological model the gravity laws andspecifically the laws of bound systems a possibility ofwhich Newton certainly took advantage (The problem ofthe influence of the global cosmological expansion on localdynamics is still being discussed [78] and is connected withanother polemic issue namely back-reaction [79])

Accepting the stable clustering hypothesis we can furtherargue that 119891(119903) is singular as 119903 997888rarr 0 and that it must have

in particular a power-law form 119891(119903) prop 119903minus120574 But this formof 119891 is independent of the hypothesis Notice that the averageconditional density has dimensions of mass density so thata finite limit of 119891(119903) in the limit 119903 997888rarr 0 would define auniversal mass density However there is no universal massdensity in the theory of general relativity (Adding quantummechanics we have the universal mass density 1198885(ℎ1198662) ≃1097 kgm3 (ℎ is the Planck constant) In this regard a largebut reasonable universal mass density would constitute anaturalness problem analogous to but less serious than theproblem of the cosmological constant (or vacuum energydensity) [80] Of course reasonable universal mass densitiescan be obtained by adding other universal constants egthe nucleon mass the electron charge) On the other handa universal mass density could also be given by the initialconditions but the only density in them is 984858b which is timedependent and too small it is actually the limit of the averageconditional density for 119903 997888rarr infin Therefore it is natural toassume that 119891(119903) is singular as 119903 997888rarr 0

A power-law singularity is not the only possibility but isthemost natural possibility In fact the power-law formof 120585 isrequired by the notion of hierarchical clustering (Section 22)which is related to the full form of Peeblesrsquo closure approachIf we dispense with the full form and just keep the stableclustering with a power-law form of 120585 there is no way toobtain the value of the exponent 120574 because there is no reasonto relate it to the initial conditions which do not need to beself-similar The evolution of the FLRW cosmological modeldoes not need to be self-similar either Therefore if 120574 couldbe derived from the initial conditions this derivation wouldpresent an awfully difficult problem A possible option thatis conceptually different is that 120574 should be determined bya nonlinear eigenvalue problem [81] One example of thesolution of a problem of this type is provided by Polyakovrsquosnonperturbative theory of forced Burgers turbulence [74]Another example which is even more relevant is providedby Gurevich and Zybinrsquos theory [41] described below

Let us remark that the presence of length scales in theFLRW cosmological model or in the initial spectrum ofdensity fluctuations does not rule out a sort of space-timesimilarity in the strongly nonlinear regime and let us remarkthat such a situation has already been shown to arise inthe stochastic adhesion model in Section 34 Naturally thezero-size mass condensations of the adhesion model are notrealistic so it is necessary to analyze what type of massconcentrations can form under the effect of Newtonrsquos gravity

The formation of power-law density singularities in colli-sionless gravitational dynamics has been proved by Gurevichand Zybinrsquos analytic theory [41]This study is especially inter-esting since it brings to the fore the role of nonlinear dynam-ics and chaos in the formation of gravitational structureThey take an isolated overdensity of generic type namely ageneric local maximum of the density field (assumed to bedifferentiable) and study its evolutionThis evolution leads toa density singularity in a dynamical time and after that itconsists in the subsequent development of a multistreamingflowwith an increasing number of streams that oscillate aboutthe origin within a region defined by the first caustic In this

20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

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[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

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[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

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[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

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[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

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[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

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[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

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[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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High Energy PhysicsAdvances in

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The Scientific World Journal

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Acoustics and VibrationAdvances in

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20 Advances in Astronomy

region themultiplicity of streams in opposite directions givesrise to strongmixing which leads to a steady state distributionwith an average density This average density has a power-law singularity at the origin with an exponent that can becalculated (under reasonable assumptions) This exponent isindependent of the particular form of the initial overdensity

In the case of spherical symmetry Gurevich and Zybinfind

984858 (119903) prop 119903minus2 [ln(1119903)]minus13 (31)

It corresponds to a diverging gravitational potential but thedivergence is weak and does not induce large velocities Letus recall from Section 24 that the local mass concentrationexponent 120572min = 1 is found in cosmological 119873-bodysimulations and the SDSS stellar mass distribution and noticethat it corresponds to 984858(119903) prop 119903minus2 in the case of sphericalsymmetry Arguably the strongest mass concentrations havespherical symmetry The density 984858(119903) prop 119903minus2 is also the sin-gular isothermal density profile where thermal equilibriumis represented by an 119903-independent average velocity and theMaxwell distribution of velocities An 119903-independent circularvelocity of stars is observed in the outskirts of spiral galaxies(this observation is of course one of the motivations for theexistence of dark matter) so the isothermal density profilemay have a role on these small scales [5 sect3] However thecollisionless gravitational dynamics does not lead to thermalequilibrium In particular the distributions of velocities inthe steady states found by Gurevich and Zybin are veryanisotropic [41]

For the sake of generality Gurevich and Zybin treat thecase without full spherical symmetry that is to say with lowellipticity so that transverse velocities are small Remarkablythis case leads to a singularity that is not exactly a power-lawbut can be well approximated by one namely 984858(119903) prop 119903minus120573with 120573 ≃ 17ndash19 Furthermore this form is not very sensitiveto the degree of ellipticity The opposite case of very largeellipticity can be approximated by one-dimensional or two-dimensional solutions Gurevich and Zybin only considerone-dimensional flows and derive 984858(119909) prop 119909minus47 In this caseit is possible to study in some detail the nonlinear causticoscillations and the role of transfer of energy successivelyto higher harmonics [41] The picture is analogous to theRichardson energy cascade in the theory of incompressibleturbulence [66] As regards the one-dimensional power-lawexponent 47 it has to be noticed that Aurell et al [49] analyzea soluble example of one-dimensional collapse with differentinitial conditions and obtain a less singular value

In general that is to say in the collapse in dimensionshigher than one and without symmetry it is commonlyaccepted that power-law singularities must appear but thereis no agreement about the exponent This question is relatedof course to the problem of halo density profiles in thehalo model examined in Section 25 We have seen that arange of exponents has been considered relying on variousarguments and on the results of simulations Let us notice thathalo models usually assume a bias towards quasi-sphericalmass concentrations and therefore towards more singular

values of the mass concentration exponent 120572 Here we haveto consider all singularities on the same footing

If Gurevich and Zybinrsquos theory were to be completed byadding the case of two-dimensional collapse then one couldthink of obtaining a universal exponent by averaging theexponents over suitable positions of singularities at nodesfilaments and sheets of the cosmic web Gurevich and Zybinindeed consider the problem of how to integrate their resultsfor isolated singularities in a hierarchical structure Theydivide this structure into an intermediate nonlinear range setto 50ndash100 Mpc where the cosmic web (ldquocellularrdquo) structureappears and a lower-scale range where steady nondissipativegravitational singularities appear [41] However this is anartificial restriction of the cosmic web structure amountingto a separation into a regular ldquocellularrdquo structure and a sort ofhalos which are smooth except for their central singularitiesIn fact a distribution of isolated and smooth halos is only anapproximation of the real cosmic web [27]The halos are partof the cosmic web structure rather than separate entities andthey may not be smooth [31]

In particular Gurevich and Zybinrsquos treatment of thecollapse of isolated overdensities ignores that maxima of theinitial density field are not isolated in the ldquobottom-uprdquomodelof growth of cosmological structure isolated density maximaof the initial density field only exist provided that this field iscoarse grained which introduces an artificial scale [82] Theresulting halos with central singularities but smooth profilesbelong in the model of fractal distribution of halos which isjust a coarse-grained approximation of a multifractal model[27 31]

In fractal geometry the density is not a suitable variablebecause it only takes the values zero or infinity as explainedin Section 22 Besides in a multifractal model with a rangeof local exponents 120572 the average of local exponents overspatial positions takes a special form for global magnitudessuch as the moments M119902 each is dominated by a specificset of singularities with local dimension 120572(119902) = 1205911015840(119902)The bulk of the mass belongs to the mass concentrate withexponent 120572con = 120572(1) but the singularities of this set are notparticularly strong in the real cosmic web (120572con ≃ 25 seeFigure 5) According to the argument given above that thedensity of bound systems is to be identified with the averageconditional density and hence is connected with M2 wemay think that the important exponent is 120572(2) lt 120572(1)Certainly the focus is usually placed on the exponent 120574 in(3) which is the one most accessible to measures and the onethat Peeblesrsquo closure approach tries to determine Howeverthe correlation dimension 120591(2) = 3 minus 120574 does not directlydetermine the corresponding local dimension 120572(2) unlessthe multifractal spectrum is known [in general 120572(2) lt 120591(2)]

These difficulties do not arise if we have only one fractaldimension In fact Peeblesrsquo closure approach only employsthe three-point correlation function in addition to the two-point correlation function and assumes the hierarchical formfor it [4 33] which is valid for a monofractal but not for ageneral multifractal However the present data support fullmultifractality (Figure 5) and so does Gurevich and Zybinrsquosanalytic theory unless it is restricted to quasi-spherical mass

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

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[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

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[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

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[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

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[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

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[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

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[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

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[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

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[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

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[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

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Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

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[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in Astronomy 21

concentrations Moreover even the intuitive image of thecosmic web consisting of sheets filaments and knots showsthat it could hardly involve only one dimension But thethree-dimensional morphology of the cosmic web is by nomeans the cause of multifractality one-dimensional cosmo-logical119873-body simulations already showmultifractality [43]It seems that we are obliged to accept that we have to deal witha range of fractal dimensions namely with a full multifractalspectrum that we cannot quite model yet

Multifractality of the mass distribution formed by gravi-tational clustering is surely a universal phenomenon linkedto the intermittency of highly nonlinear processes Nonlineargravitational clustering is indeed a phenomenon very relatedto turbulence which began to be studied somewhat earlierbut is in a similar state of development The specific natureof gravity leads to some helpful constraints chiefly thestable clustering hypothesis but this is not sufficient for asolution and specific nonlinear methods are needed [33]The guiding idea for studying nonlinear phenomena liketurbulence or gravitational clustering is surely the scalingsymmetry which is naturally motivated by the observedmultifractality The renormalization group prominently fea-tures among the analytic methods based on scaling as hasbeen advocated before [83] The efficacy of this method isattested by its results for the stochastic adhesion modelspecifically the results obtained employing nonperturbativeformulations of the renormalization group [84] Stochasticgravitational clustering demands a random noise with anarbitrary exponent as proposed by Antonov [77] who onlyapplies perturbation theory One could try to determinethe arbitrary exponent from universality constraints likein [71] The stochastic approach deserves to be exploredfurther

After considering sophisticated methods of nonlineardynamics let us briefly comment on a simple method ofobtaining the mass function of objects formed by gravita-tional collapse namely the Press-Schechtermethod [33]Thismethod is based on the spherical collapse model which isnot realistic in three dimensions In fact Vergassola et al[46] find that the Press-Schechter mass function does notagree with the mass function derived from the adhesionmodel in more than one dimension They define this massfunction referring it to mass condensations in balls of fixedsize given that there are extended objects of arbitrary sizesuch as filaments in more than one dimension With asimilar definition of the mass function the form of thisfunction has been studied in the distributions generated bycosmological 119873-body simulations [28 29] The result is amass function of Press-Schechter type but with a fixed power-law exponent that is independent of 119899 (the initial powerspectrum exponent) This is contrary to the Press-Schechterapproach

One can see that the Press-Schechter approach is in awayanalogous to the Peebles approach in the sense that the latteralso finds under questionable assumptions a simple relationin this case a relation between 119899 and another importantexponent namely the 120585-exponent 120574 However it seems thatsimple approaches to the nonlinear gravitational dynamicsthat attempt to bypass its true complexity have limited scope

4 Discussion

We have reviewed the main ideas and theories that have ledto the current understanding of the geometry of the cosmicstructure The appealing denomination of it as a ldquocosmicwebrdquo is most appropriate in regard to its three-dimensionalappearance but some important properties related to itsfractal geometry are independent of the web morphologyand already appear in one-dimensional cosmologicalmodelsin which the geometry is much simpler

We have also considered halo models very briefly Halomodels are based on the statistical properties of discretepoint distributions rather than on the geometrical analysisof continuous mass distributions However we have shownthat the correlations between points cannot be too simpleand must consist of a hierarchy that allows us to naturallyconnect themwith fractal models Of course this connectioncan actually take place only in the limit of an infinite numberdensity of points in which we obtain a continuous massdistribution

The abstract study of the geometry of mass distributionshas been essentially a mathematical subject Since it is welldeveloped we propose to take advantage of this body ofknowledge and we argue that the cosmic web structuremust be studied as a strictly singular but continuous massdistribution The geometry of these distributions is notsimple even in one dimension because they have nonisolatedsingularities However in the one-dimensional case everymass distribution can be described in terms of the massdistribution function which is just a monotonic functionthat gives the mass distribution by differentiation (it mustbe differentiable almost everywhere) If the function iscontinuous so is the mass distribution It is not easy togeneralize this construction to higher dimensions wherevarious geometrical and topological issues arise At any ratea study of the type of singularities in a continuous massdistribution is always possible This is the goal of multifractalanalysis which is based on the analysis of the behaviorof fractional statistical moments M119902(119897) when 119897 997888rarr 0 Incosmology only the integral moments M119899 are normallyconsidered but they do not provide sufficient information

The simplest strictly singular and continuous mass distri-bution consists of a uniformmass distribution on a fractal setnamely on a self-similar set of the type of the Cantor setThismass distribution has just one kind of singularitiesThereforeit is a monofractal described by just one dimension theHausdorff dimension of the fractal setThis type of fractal hasa sequence of individual empty voids that is characterized bya particular form of the Zipf law

Next in complexity we consider a bifractal distributionspecifically the type of distribution generated by the one-dimensional adhesion model It is actually a noncontinuousdistribution with some interesting properties namely it doesnot contain any properly fractal set with nontrivialHausdorffdimension but it is obviously self-similar If we disregardits void structure it just consists of a collection of point-like masses whose magnitudes follow a power-law which isone cause of its self-similarity Another cause is the spatialdistribution of the masses which gives rise to a self-similar

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

22 Advances in Astronomy

void structure In fact this structure is crucial for the fractalnature or rather bifractal of this mass distribution becausethe set of point-like masses with 119891(120572) = 120572 = 0 has nothingfractal to it and the fractality is given by the other point ofthe bifractal spectrum at (1ℎ 0)

The transition from the preceding bifractal to a full multi-fractal is best perceived by focusing on the mass distributionfunction let us substitute the points of discontinuity thatis to say the points with vertical segments in the graphwhich correspond to 120572 = 0 by points of nondifferentiabilitysuch that the derivative grows without limit We so obtain acontinuous monotone function that has weaker singularitieswith 0 lt 120572 lt 1 This may seem an artificial procedurethat is difficult to be realized in practice On the contraryany generic mass distribution is of this type Moreoverthese distributions normally have a nontrivial multifractalspectrum If the graph of the function besides not havingvertical segments does not have horizontal segments thenthere are no individual empty voids and the multifractal isnonlacunar In this case the voids look like the voids of theadhesion model distribution but they have a more complexstructure because the multifractal spectrum contains a fullinterval with 120572 gt 1

Naturally the large-scale mass distribution is three-dimensional and therefore more complex with notablemorphological features These features in particular theshape of voids inspired the early models of the cosmicweb eg the Voronoi foam model Nowadays importantinformation on the structures of matter clusters and voidscan be obtained from the multifractal analysis of 119873-bodysimulations of the dynamics of cold dark matter alone orwith baryonic matter in combination with the multifractalanalysis of galaxy catalogues The most obvious facts arefirst the self-similarity of the structures and second that themultifractal spectra are incompatible with monofractal dis-tributionsWe also notice the concordance of themultifractalspectra of cold dark matter and baryonic matter to the extentallowed by the quality of the dataThis concordance supportsthe hypothesis of a universal multifractal spectrum for thestructure generated by gravitational clustering of both darkand baryonic matter

Of course this hypothesis challenges us to explain howsuch universality arises Some features of the multifractalspectrum can be easily related to natural properties of themass distribution For example max119891 = 3 is related to thenonlacunar structure or full support of the mass distributionIt can be explained by the adhesion model which is a reliableapproximation for the rough structure of voids (althoughnot good enough for predicting the detailed multifractalspectrum for 120572 gt 3) On the opposite side of the multifractalspectrum we have that 120572min = 1 This value disagrees withthe adhesion model prediction (120572min = 0) and indeed theadhesion model is a poor approximation for the formation ofstrong mass concentrations However the basic condition ofboundedness of the gravitational potential imposes 120572min = 1In fact amonofractal of dimension equal to onewas proposedby Mandelbrot for a similar reason namely the moderatevalue of cosmic velocities [3 sect9] But certainly the large-scale mass distribution is not monofractal and the argument

properly considered only sets a lower bound to the localfractal dimension

To go beyond these two properties of the mass dis-tribution we need to understand better the dynamics ofstructure formation in cosmology It is useful to begin witha further analysis of the adhesion model that is to say witha study of Burgers turbulence because the hydrodynamictype of nonlinearity of the dynamical equations proves thatthe methods of the theory of turbulence are appropriate Weconsider the Richardson energy cascade the Kolmogorovhypothesis and the intermittency of the velocity field TheBurgers turbulence with scale-invariant initial conditionslends itself to a fairly complete analysis and one can derivethe form of the function 120577(119902) that encodes the type ofintermittency

However a treatment of the dynamics of structure forma-tion that only involves the velocity field (because it is basedon the Zeldovich approximation) is bound to be insufficientA crucial problem of the adhesion model is that decayingBurgers turbulence does not conserve energy This can beamended by studying forced Burgers turbulence in the con-text of stochastic dynamics The stochastic Burgers (or KPZ)equation gives rise to a scale-invariant asymptotic state that isindependent of the initial conditions The techniques basedon dynamical scaling are powerful but perturbation theorycannot deal with the three-dimensional problem and onehas to apply nonperturbative methods Simple universalityarguments lead to a constraint on 120577(119902) and hence to areasonable range for the two-point correlation exponent 120574But it is doubtful that these arguments can be pushed muchfurther while keeping within the scope of Burgers turbulence

Therefore we must keep in mind the methods of turbu-lence but deal with the full gravitational dynamics A lot ofgood work has been done on nonlinear gravitational dynam-ics and we cannot evenmentionmost of it so we have limitedourselves to two relevant approaches the classic Peebles clo-sure approach and Gurevich-Zybinrsquos theory of nondissipativegravitational singularities The full form of Peeblesrsquo closureapproach involves somequestionable hypotheses butwe havefocused precisely on the stable clustering hypothesis for thehighly nonlinear regimeThismost natural hypothesis gives aprominent role to the conditional mass density and hence tofractal geometry For the calculation of power-law exponentsnamely fractal dimensions the stable clustering hypothesisfalls short and additional assumptions are necessary

Definite power-law exponents of gravitational singular-ities are provided by the Gurevich-Zybin theory whichconsists in an appropriate treatment of multistreaming incollisionless gravitational collapse However this treatmentcan only obtain exponents for singularities that form inisolation We find no simple way to relate the exponentsobtained by this theory with the dimensions in the multi-fractal spectrum of the real cosmic web At any rate thederivation of power-law singularities from the full dynamicsof gravitational collapse is certainly an advance over the zero-size singularities of the adhesion model

In conclusion the problem of explaining the geometry ofthe cosmic web on a dynamical basis has only been solvedpartially This problem seems to have a similar status to the

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

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Acoustics and VibrationAdvances in

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Submit your manuscripts atwwwhindawicom

Advances in Astronomy 23

problem of the geometry of the flow in fluid turbulenceHopefully the current developments in the study of nonlineardynamics will bring further progress

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S F Shandarin and Ya B Zeldovich ldquoThe large-scale structureof the universe turbulence intermittency structures in a self-gravitating mediumrdquo Reviews of Modern Physics vol 61 no 2pp 185ndash220 1989

[2] S N Gurbatov A I Saichev and S F Shandarin ldquoLarge-scalestructure of theUniverseTheZeldovich approximation and theadhesion modelrdquo Physics-Uspekhi vol 55 no 3 pp 223ndash2492012

[3] B B Mandelbrot The Fractal Geometry of Nature (rev ed ofFractals 1977) W H Freeman and Company New York NYUSA 1983

[4] P J PeeblesTheLarge-Scale Structure of the Universe PrincetonUniversity Press Princeton NJ USA 1980

[5] P J E Peebles Principles of Physical Cosmology PrincetonUniversity Press Princeton NJ USA 1993

[6] P H Coleman and L Pietronero ldquoThe fractal structure of theuniverserdquo Physics Reports vol 213 no 6 pp 311ndash389 1992

[7] S Borgani ldquoScaling in the Universerdquo Physics Reports vol 251no 1-2 pp 1ndash152 1995

[8] F Sylos Labini M Montuori and L Pietronero ldquoScale-invariance of galaxy clusteringrdquo Physics Reports vol 293 no2-4 pp 61ndash226 1998

[9] J Gaite A Domınguez and J Perez-Mercader ldquoThe fractaldistribution of galaxies and the transition to homogeneityrdquoTheAstrophysical Journal vol 522 no 1 pp L5ndashL8 1999

[10] B J T Jones V J Martınez E Saar and V Trimble ldquoScalinglaws in the distribution of galaxiesrdquo Reviews of Modern Physicsvol 76 no 4 pp 1211ndash1266 2004

[11] P J E Peebles ldquoThe fractal galaxy distributionrdquo Physica DNonlinear Phenomena vol 38 no 1-3 pp 273ndash278 1989

[12] J Einasto M Joeveer and E Saar ldquoStructure of superclustersand supercluster formationrdquo Monthly Notices of the RoyalAstronomical Society vol 193 no 2 pp 353ndash375 1980

[13] Y B Zeldovich J Einasto and S F Shandarin ldquoGiant voids inthe universerdquo Nature vol 300 no 5891 pp 407ndash413 1982

[14] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[15] A A Klypin and S F Shandarin ldquoThree-dimensional numer-ical model of the formation of large-scale structure in theuniverserdquoMonthlyNotices of the Royal Astronomical Society vol204 no 3 pp 891ndash907 1983

[16] D H Weinberg and J E Gunn ldquoLarge-scale structure andthe adhesion approximationrdquo Monthly Notices of the RoyalAstronomical Society vol 247 pp 260ndash286 1990

[17] L Kofman D Pogosyan S F Shandarin and A L MelottldquoCoherent structures in the universe and the adhesion modelrdquoThe Astrophysical Journal vol 393 no 2 pp 437ndash449 1992

[18] L Pietronero ldquoThe fractal structure of the universe Correla-tions of galaxies and clusters and the average mass densityrdquo

Physica A StatisticalMechanics and its Applications vol 144 no2-3 pp 257ndash284 1987

[19] B J Jones V J Martinez E Saar and J Einasto ldquoMultifractaldescription of the large-scale structure of the universerdquo TheAstrophysical Journal vol 332 pp L1ndashL5 1988

[20] R Balian and R Schaeffer ldquoGalaxies fractal dimensions countsin cells and correlationsrdquoTheAstrophysical Journal vol 335 ppL43ndashL46 1988

[21] R Valdarnini S Borgani and A Provenzale ldquoMultifractalproperties of cosmological N-body simulationsrdquoTheAstrophys-ical Journal vol 394 no 2 pp 422ndash441 1992

[22] S Colombi F R Bouchet and R Schaeffer ldquoMultifractal anal-ysis of a cold dark matter universerdquo Astronomy amp Astrophysicsvol 263 no 1 1992

[23] G Yepes R Dominguez-Tenreiro and H M Couchman ldquoThescaling analysis as a tool to compare N-body simulationswith observations - application to a low-bias cold dark mattermodelrdquoThe Astrophysical Journal vol 401 pp 40ndash48 1992

[24] A Cooray and R Sheth ldquoHalo models of large scale structurerdquoPhysics Reports vol 372 no 1 pp 1ndash129 2002

[25] P Monaco ldquoApproximate methods for the generation of darkmatter halo catalogs in the age of precision cosmologyrdquo Galax-ies vol 4 no 4 p 53 2016

[26] A Del Popolo and M Le Delliou ldquoSmall scale problems of theΛCDMModel a short reviewrdquo Galaxies vol 5 no 1 p 17 2017

[27] J Gaite ldquoThe fractal distribution of haloesrdquo EPL (EurophysicsLetters) vol 71 no 2 pp 332ndash338 2005

[28] J Gaite ldquoHalos and voids in a multifractal model of cosmicstructurerdquoThe Astrophysical Journal vol 658 no 1 I pp 11ndash242007

[29] J Gaite ldquoFractal analysis of the dark matter and gas distribu-tions in theMare-Nostrum universerdquo Journal of Cosmology andAstroparticle Physics vol 2010 no 03 pp 006-006 2010

[30] C A Chacon-Cardona and R A Casas-Miranda ldquoMillenniumsimulation dark matter haloes Multifractal and lacunarityanalysis and the transition to homogeneityrdquoMonthly Notices ofthe Royal Astronomical Society vol 427 no 3 pp 2613ndash26242012

[31] J Gaite ldquoSmooth halos in the cosmicwebrdquo Journal of Cosmologyand Astroparticle Physics no 4 p 020 2015

[32] J Gaite ldquoHalo models of large scale structure and reliability ofcosmological N-body simulationsrdquoGalaxies vol 1 no 1 pp 31ndash43 2013

[33] V Sahni and P Coles ldquoApproximation methods for non-lineargravitational clusteringrdquo Physics Reports vol 262 no 1-2 pp1ndash135 1995

[34] R van de Weygaert ldquoFroth across the universe dynamics andstochastic geometry of the cosmic foamrdquo inModernTheoreticaland Observational Cosmology M Plionis S Cotsakis and IGeorgantopoulos Eds vol 276 p 119 Kluwer 2002

[35] U Frisch and J Bec ldquoBurgulencerdquo in Les Houches 2000 NewTrends in Turbulence M Lesieur A Yaglom and F David Edspp 341ndash383 Springer EDP-Sciences 2001

[36] J Bec and K Khanin ldquoBurgers turbulencerdquo Physics Reports vol447 no 1-2 pp 1ndash66 2007

[37] S F Shandarin and Y B Zeldovich ldquoTopology of the large-scalestructure of the universerdquoComments onAstrophysics vol 10 pp33ndash45 1983

[38] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press Oxford UK 1975

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

24 Advances in Astronomy

[39] V I Arnold Catastrophe theory Springer Berlin Germany1984

[40] J Hidding S F Shandarin and R van de Weygaert ldquoTheZelrsquodovich approximation Key to understanding cosmic webcomplexityrdquoMonthly Notices of the Royal Astronomical Societyvol 437 no 4 pp 3442ndash3472 2014

[41] A V Gurevich and K P Zybin ldquoLarge-scale structure of theUniverse Analytic theoryrdquo Physics-Uspekhi vol 38 no 7 pp687ndash722 1995

[42] E Aurell D Fanelli and P Muratore-Ginanneschi ldquoOn thedynamics of a self-gravitating medium with random and non-random initial conditionsrdquo Physica D Nonlinear Phenomenavol 148 no 3-4 pp 272ndash288 2001

[43] B N Miller J Rouet and E Le Guirriec ldquoFractal geometry inan expanding one-dimensional Newtonian universerdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 76no 3 pp 0367051ndash03670514 2007

[44] M Joyce and F Sicard ldquoNon-linear gravitational clustering ofcold matter in an expanding universe Indications from 1D toymodelsrdquoMonthly Notices of the Royal Astronomical Society vol413 no 2 pp 1439ndash1446 2011

[45] T Buchert and A Domınguez ldquoAdhesive gravitational cluster-ingrdquo Astronomy amp Astrophysics vol 438 no 2 pp 443ndash4602005

[46] M Vergassola B Dubrulle U Frisch and A Noullez ldquoBurgersrsquoequation Devils staircases and the mass distribution for largescale structuresrdquo Astronomy amp Astrophysics vol 289 pp 325ndash356 1994

[47] F Bernardeau and P Valageas ldquoMerging and fragmentation inthe Burgers dynamicsrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 016311 2010

[48] T Buchert ldquoLagrangian theory of gravitational instability ofFriedman-Lematre cosmologies - a generic third-order modelfor non-linear clusteringrdquo Monthly Notices of the Royal Astro-nomical Society vol 267 no 4 pp 811ndash820 1994

[49] E Aurell D Fanelli S N Gurbatov and A Y Moshkov ldquoTheinner structure of Zeldovich pancakesrdquo Physica D NonlinearPhenomena vol 186 no 3-4 pp 171ndash184 2003

[50] K Falconer Fractal Geometry JohnWiley and Sons ChichesterUK 2003

[51] M Monticino ldquoHow to construct a random probability mea-surerdquo International Statistical Review vol 69 no 1 pp 153ndash1672001

[52] S Gottlober E L Łokas A Klypin and Y Hoffman ldquoThestructure of voidsrdquo Monthly Notices of the Royal AstronomicalSociety vol 344 no 3 pp 715ndash724 2003

[53] J Gaite ldquoStatistics and geometry of cosmic voidsrdquo Journal ofCosmology and Astroparticle Physics vol 2009 no 11 pp 004-004 2009

[54] J Gaite and S C Manrubia ldquoScaling of voids and fractality inthe galaxy distributionrdquoMonthly Notices of the Royal Astronom-ical Society vol 335 no 4 pp 977ndash983 2002

[55] E Aurell U Frisch A Noullez and M Blank ldquoBifractality ofthe Devilrsquos staircase appearing in the Burgers equation withBrownian initial velocityrdquo Journal of Statistical Physics vol 88no 5-6 pp 1151ndash1164 1997

[56] J Gaite ldquoFractal analysis of the large-scale stellar mass distribu-tion in the Sloan digital sky surveyrdquo Journal of Cosmology andAstroparticle Physics vol 2018 no 07 pp 010-010 2018

[57] J Neyman and E L Scott ldquoA theory of the spatial distributionof galaxiesrdquoThe Astrophysical Journal vol 116 p 144 1952

[58] J N Fry ldquoStatistics of voids in hierarchical universesrdquo TheAstrophysical Journal vol 306 pp 358ndash365 1986

[59] A L Melott ldquoMore resolution isnrsquot always better resolutionrdquoComments on Astrophysics and Space Physics vol 15 pp 1ndash81990

[60] R J Splinter A L Melott S F Shandarin and Y SutoldquoFundamental discreteness limitations of cosmological N-bodyclustering simulationsrdquo The Astrophysical Journal vol 497 no1 pp 38ndash61 1998

[61] M Joyce B Marcos and T Baertschiger ldquoTowards quantitativecontrol on discreteness error in the non-linear regime ofcosmological N-body simulationsrdquoMonthly Notices of the RoyalAstronomical Society vol 394 no 2 pp 751ndash773 2009

[62] D H Weinberg J S Bullock F Governato R Kuzio de Narayand A H Peter ldquoCold dark matter controversies on smallscalesrdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 112 no 40 pp 12249ndash12255 2015

[63] R K Sheth and R Van De Weygaert ldquoA hierarchy of voidsMuch ado about nothingrdquo Monthly Notices of the Royal Astro-nomical Society vol 350 no 2 pp 517ndash538 2004

[64] T Kiang ldquoRandom fragmentation in two and three dimen-sionsrdquo Zeitschrift fur Astrophysik vol 64 pp 433ndash439 1966

[65] J P Boyd ldquoHyperviscous shock layers and diffusion zonesmonotonicity spectral viscosity and pseudospectral methodsfor very high order differential equationsrdquo Journal of ScientificComputing vol 9 no 1 pp 81ndash106 1994

[66] U Frisch Turbulence The Legacy of A N Kolmogorov Cam-bridge University Press Cambridge UK 1995

[67] F Reif Fundamentals of Statistical and Thermal Physics ch15McGraw-Hill New York NY USA 1965

[68] T Halpin-Healy and Y-C Zhang ldquoKinetic roughening phe-nomena stochastic growth directed polymers and all thatAspects of multidisciplinary statistical mechanicsrdquo PhysicsReports vol 254 no 4-6 pp 215ndash414 1995

[69] E Medina T Hwa M Kardar and Y C Zhang ldquoBurgersrsquoequation with correlated noise renormalization-group analysisand applications to directed polymers and interface growthrdquoPhysical Review A Atomic Molecular and Optical Physics vol39 no 6 pp 3053ndash3075 1989

[70] G Rigopoulos ldquoThe adhesion model as a field theory for cos-mological clusteringrdquo Journal of Cosmology and AstroparticlePhysics no 1 p 014 2015

[71] J Gaite ldquoA non-perturbative Kolmogorov turbulence approachto the cosmic web structurerdquo EPL (Europhysics Letters) vol 98no 4 p 49002 2012

[72] MK Verma ldquoIntermittency exponents and energy spectrumofthe Burgers and KPZ equations with correlated noiserdquo PhysicaA Statistical Mechanics and its Applications vol 277 no 3 pp359ndash388 2000

[73] A Chekhlov and V Yakhot ldquoKolmogorov turbulence in arandom-force-driven Burgers equationrdquo Physical Review EStatistical Nonlinear and Soft Matter Physics vol 51 no 4 ppR2739ndashR2742 1995

[74] AM Polyakov ldquoTurbulence without pressurerdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 52 no 6part A pp 6183ndash6188 1995

[75] S A Boldyrev ldquoVelocity-difference probability density func-tions for Burgers turbulencerdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 55 no 6 part A pp6907ndash6910 1997

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in Astronomy 25

[76] F Hayot and C Jayaprakash ldquoAspects of the stochastic Burgersequation and their connection with turbulencerdquo InternationalJournal of Modern Physics B vol 14 no 17 pp 1781ndash1800 2000

[77] N V Antonov ldquoScaling behavior in a stochastic self-gravitatingsystemrdquo Physical Review Letters vol 92 no 16 p 161101 2004

[78] M Carrera and D Giulini ldquoInfluence of global cosmologicalexpansion on local dynamics and kinematicsrdquo Reviews ofModern Physics vol 82 no 1 pp 169ndash208 2010

[79] T Buchert ldquoOn backreaction in Newtonian cosmologyrdquoMonthly Notices of the Royal Astronomical Society vol 473 no1 pp L46ndashL49 2018

[80] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[81] G I Barenblatt Scaling Self-Similarity and IntermediateAsymptotics Cambridge University Press Cambridge UK1996

[82] J M Bardeen J R Bond N Kaiser and A S Szalay ldquoThestatistics of peaks of Gaussian random fieldsrdquoThe AstrophysicalJournal vol 304 pp 15ndash61 1986

[83] J Gaite and A Domınguez ldquoScaling laws in the cosmicstructure and renormalization grouprdquo Journal of Physics AMathematical and Theoretical vol 40 no 25 pp 6849ndash68572007

[84] T Kloss L Canet B Delamotte and N Wschebor ldquoKardar-Parisi-Zhang equation with spatially correlated noise a unifiedpicture from nonperturbative renormalization grouprdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 89no 2 article 022108 2014

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom