The punctuated Zel’dovich approximation

IL NUOVO CIMENTO VOL. 18 C, N. 5 Settembre-0ttobre 1995

The Punctuated Zel'dovich Approximation.

L. FONTANA, M. MmELLI, G. MURANTE and A. PROVENZALE

Isti tuto di Cosmogeofisica del CNR - Corso F iume 4, 1-10133 Torino, I taly

(ricevuto il 9 Giugno 1995; approvato il 24 Luglio 1995)

Summary. -- We study a phenomenological approximation to the full fluid- dynamical equations for non-linear gravitational clustering, based on an extension of the well-known Zel'dovich approximation. In this approach, called the Punctuated Zet'dovich Approximation (PZA), fluid elements move according to the Zel'dovich prescription until they reach a critical distance from each other. At this stage, particles stick with each other by conserving mass and momentum but not kinetic energy, simulating the effects of strong non-linear gravitational clustering. The PZA is then compared with pure Zel'dovich dynamics and with the adhesion approximation. PZA turns out to work at least as well as the adhesion approach, but is more flexible and physically more justified.

PACS 92.10.Lq - Turbulence and diffusion. PACS 98.65.Dx - Superclusters; large-scale structure of the Universe (including voids, pancakes, great wall, protogalaxies and primordial galaxies).

1. - I n t r o d u c t i o n .

Understanding the origin and dynamical evolution of the large-scale structure of the Universe is one of the main open issues of modern cosmology. Most theories on large-scale structure formation rely on the assumption that the structures observed today (galaxies, galaxy clusters, superclusters, voids) represent the result of the gravitational evolution of initially small-density perturbations in an otherwise homogeneous Universe, see, e.g., [1]. This picture has recently received observational support from the detection of small temperature anisotropies in the Cosmic Microwave Background Radiation (CMBR), as provided by COBE satellite[2].

For a cold, collisionless medium, the evolution of density perturbations before orbit crossing in an almost homogeneous Universe can be studied using the fluid-dynamical approximation, which represents a good approximation to the kinetic theory based on the Vlasov equation. When the amplitude of density perturbations is small compared to the average density, the growth of fluctuations can be handled analytically using linear perturbation theory, see, e.g., [3]. Galaxies and clusters of galaxies, however, are very non-linear objects. For this reason, one needs to go beyond linear theory in order to properly understand the process of structure

531

532 L. FONTANA~ M. MILELLI, G. MURANTE and A. PROVENZALE

formation. Although the full non-linear dynamics can be presently followed only by resorting to N-body simulations, various analytical approximations have been introduced to provide analytic insight, at least in the mildly non-linear gravitational regime. In general, these approximations are easier to implement and are often computationally less expensive than the full N-body simulations. For instance, in 1970 Zel'dovich [4] introduced a kinematic approximation to the fluid-dynamical equations in Lagrangian coordinates which uses the gravity-induced velocity field at early times to move the fluid elements (see the review[5]). Zel'dovich showed that gravitational instability generically leads to the formation of two-dimensional sheets, called pancakes. The Zel'dovich approximation breaks down after the first orbit crossing because it predicts that particles should continue unperturbed through the pancakes. In an attempt to solve this problem, Gurbatov and Saichev[6] and Gurbatov et al. [7] introduced the adhesion approximation which extends the useful range of the Zel'dovich approximation by using artificial viscosity to mimic some of the effects of non-linear gravity.

In this paper we discuss a different phenomenological approach, which we nick- name ((Punctuated Zel'dovich Approximatiom) (PZA). Along these lines, we extend the work on merging phenomena and ballistic agglomeration, considered by Pietronero and Kupers [8] and Carnevale, Pomeau and Young [9], in order to link the mildly non-linear dynamics of the Zel'dovich approximation with the highly non- linear one, parametrizing the formation of bound gravitational states through a radius of ,,strong)~ interaction and simulating gravity through an aggregation probability.

In the following, we mainly concentrate on the one-dimensional case, so to have a larger resolution in order to better describe the general properties of the clustering process. Furthermore, there are strong indications that one-dimensional collapse may be predominant on large scales, at least in the initial stages of galaxy formation. In fact, even though in most works spherical symmetrY is imposed, this turns out to be a non-generic case as strongly anisotropic collapse is generally expected at large scales, see, e.g., Doroshkevich[10] and Doroshkevich et al. [11]. Numerical studies confirm that anisotropic collapse at large scales seems to be a general consequence of gravitational instability.

The paper is organized as follows. In sect. 2 we briefly discuss the Zel'dovich and adhesion approximations, and in sect. 3 we introduce the Punctuated Zel'dovich Ap- proximation. In sect. 4 we discuss the results of numerical simulations with Gaussian initial conditions and scale-free power spectra. In sect. 5 we provide summary and conclusions.

2. - T h e Z e l ' d o v i c h and a d h e s i o n a p p r o x i m a t i o n s .

Here we concentrate on the one-dimensional problem, which can be considered equivalent to a plane-symmetric three-dimensional collapse, and stay in the framework of a non-baryonic dark-matter-dominated Universe.

Given the essentially collisionless nature of non-baryonic dark matter, the appropriate mathematical model for the evolution of density and velocity perturbations are the kinetic Vlasov-Poisson equations. For particles with small thermal velocities (approximation of cold fluid), the pressureless Euler-Poisson equations may be used to approximately describe the dynamics of a self-gravitating cosmic fluid for times shorter than the first orbit crossing.

THE PUNCTUATED ZEL'DOVICH APPROXIMATION 533

In the Newtonian context, valid for length scales small compared to the horizon and locally non-relativistic densities and velocities, the Euler-Poisson equations are

(2.1) --35 + _1 ~ u + _1 ~ ( S u ) = 0, 3t a a

au (2.2) - - + H u + ( u ~ ) u = - - - ,

~t a

(2.3) ~ ~b = 4 z G - ~ a 2 5 ,

having introduced the comoving coordinates x = r / a ( t ) and the peculiar velocity u = a ( t )& . Here 5(x) indicates the density contrast, ~ is the peculiar gravitational potential, H = d / a is the Hubble constant, a( t ) is the cosmic expansion factor, G is the gravitational constant and ~ is the background density. For simplicity, we consider a matter-dominated Einstein-de Sitter Universe, such that a( t ) ~ t 2/3, -~ ~ a - 8 and the linear growth rate b(t) of density perturbations is proportional to the scale factor a( t ) . Choosing a( t ) as the new time variable and defining a new comoving velocity

d x u V - -

d a a d '

we obtain the following form for Euler's equation:

3v 3 (2.4) - - + ( v ~ ) v = - - - (v + A ~ b ) ,

3a 2a

where A = 2 / ( 3 H 2 a 8) is a constant for a flat Universe with dust-like matter. At early times, when inhomogeneities are of very small amplitude, the solutions to

eqs. (2.1)-(2.4) may be obtained by linearization. Later, in the non-linear regime, there is no explicit general solution to the basic equations. The Euler-Poisson equations cease to be a valid approximation after the first orbit crossing, due to the development of singularities. Consequently, in the strongly non-linear regime one is forced to use the full Vlasov-Poisson equations, whose dynamical and mathematical properties remain largely to be properly understood.

At the intermediate stage of dynamical evolution, after the breakdown of linear approximation but before generalized orbit crossing, various perturbative approaches and approximations to the primitive equations (2.1)-(2.4) have been developed. Along these lines, in the '70s Zel'dovich realized the limitations of the linear approach and proposed an extrapolation of the Eulerian linear solutions into the mildly non-linear regime by employing the Lagrangian picture of continuum mechanics. The Zel'dovich approximation may be obtained from (2.4) by setting its right-hand side to zero,

(2.5) Dv _ 3v + (v~) v = 0, Da 3a

534 L. FONTANA, M. MILELLI, G. MURANTE and h. PROVENZALE

where D / D a is the convective derivative. This equation has an immediate solution in terms of the displacement of the fluid elements from their initial Lagrangian (unperturbed) position q with constant velocity [4]

(2.6) x = q + a ( t ) v ( q ) .

The Zel'dovich approximation is exact in 1D because the gravitational force is a Lagrangian (material) invariant, until the first orbit crossingi which coincides with the first structure formation. This corresponds to the formation of a pancake (sheet-like structure) by contraction along one of the principal axes. However, multi- stream flows invariably form at the location of pancakes, which grow progressively thicker leading to the ultimate breakdown of the Zel'dovich approximation. Vice versa, in the framework of a realistic description of gravitational dynamics, we expect that the potential wells associated with the non-linear structures should be able to retain and accrete from surrounding regions.

In order to overcome the smearing of pancakes arising in the framework of the Zel'dovich approach, Gurbatov and Saichev[6] and Gurbatov et a/.[7] have introduced the adhesion approximation, based on the idea of gravitational sticking of particles. In this approximation the right-hand side of (2.4) is replaced by an artificial viscosity term to mimic the effect of non-linear gravity on small scales and to stabilize the thickness of pancakes. The resulting equation is known as the Burgers equation, which is well known in hydrodynamics. This has the form

~v (2.7) - - + ( v ~ ) v = u g l y ,

~a

where v is a coefficient of (artificial) viscosity, parametrizing small-scale strong gravitational interactions, in a way which is conceptually similar to the idea of dynamical friction. It is interesting to note that in the limit u-~ 0 the right-hand side of (2.7) remains finite only in those regions where large gradients in the velocity field exist (i.e. inside the pancakes) and vanishes elsewhere. For any non-vanishing value of v, the artificial viscosity prevents the penetration of one particle stream into another and avoids the orbit crossing occurring in the Zel'dovich approximation. In the framework of Burgers dynamics, the structures which are forming are thin sheets, whose thickness depends on the viscosity parameter v. Since viscosity is

relevant on a scale length a V r ~ v (below which dissipation occurs and velocity gradients are erased), we expect that this should also be the typical thickness of the non-linear structures.

Despite the fact that eq. (2.7) is manifestly non-linear, it can be linearized in a straighforward way and its analytical solution can be explicitly written. First we introduce the velocity potential r defined by $~ ~b = v, and then use the Hopf-Cole transformation

(2.8) U = exp [ - q)/2u].

After substituting into the Burgers equation, we get

(2.9) 3U - v 3 2U 3a 3x 2 '

THE PUNCTUATED ZELPDOVICH APPROXIMATION 535

which is the usual linear diffusion (heat) equation, having a well-known analytic solution. In terms of the initial potential function r (q), the velocity field at any later time is thus given by

~r SIn U (2.10) v = - 2 v - ,

~x Sx

(2.11) U(x, a)= (4~va) "1/2 exp -~u ] L 4--ua "

The reason for the great deal of attention paled to the adhesion approach (e.g.,[12-14]) lies essentially in the fact that the availability of a solution for the velocity field v provides more insight into the details of the non-linear dynamics, while avoiding some of the limitations of the Zel'dovich approach. See, in particular, [14] for a physical and mathematical study of the adhesion approximation in the limit of vanishing viscosity, a case which allows for the development of a fast simulation code. Clearly, the adhesion approximation has several drawbacks as well. First, the fluid elements are passively transported by the velocity field and the density behaves as a passive scalar, while in real gravitational clustering the density perturbations play a crucial role. Second, the adhesion approximation does not conserve the linear momentum of the fluid elements. There are various ways to overcome this latter problem; in the next section we discuss a solution based on what we call the Punctuated Zel'dovich Approximation.

3. - The Punc tua ted Zel 'dovich Approximat ion.

This phenomenological approach is rooted into the understanding that the Zel'dovich approximation represents a significant step beyond linear theory for the description of gravitational evolution on large, weakly non-linear scales. On the other hand, we know that on small scales it becomes less accurate, essentially because it ignores the variations of the gravity exerted on a particle as it moves near high- density concentrations.

To avoid orbit crossing, we consider a different ~ approximation where particles move according to the Zel'dovich prescription (2.6) until they are far apart from each other, but when they get closer than some pre-selected critical distance de, then

1) the Zel'dovich dynamics is interrupted,

2) the particles stick inelastically and are replaced by a new single particle according to some interaction rules and

3) the Zel'dovich dynamics continues with this new set of particles and velocities.

A similar sticking scenario has been discussed in [8] for the clustering of galaxies, without explicit link to the Zet'dovich approximation, and it has been studied in detail in [9] in a fluid-dynamical context. The idea of interrupting an HamiItonian dynamics by localized, strong dissipation events has been used also in the study of turbulence, see, e.g., [15]. This type of approach fits into the general framework of punctuated Hamilt0nian dynamics, where Hamiltonian evolution is occasionally interrupted by

536 L. FONTANA, M. MILELLI, G. MURANTE and A. PROVENZALE

some abrupt instantaneous transformations, which simulate dissipative events. This is why we have called the approximation used here the Punctuated Zel'dovich Approximation (PZA).

With respect to previous studies, we remark that the moving particles considered here are not individual galaxies but rather elements of the cosmic fluid, and that the PZA is used as a natural extension of the Zel'dovich Approximation, which has a rigorous mathematical justification. By contrast, the sticking episodes are a heuristic parametrization of strongly non-linear gravitational clustering, in the same spirit of the adhesion approximation.

In general, we may define a probability Pa that particles aggregate irreversibly and a probability Ps = 1 - Pa that they scatter, as in Pietronero and Kupers [8], who discussed various functional forms for Pa. In particular, Sylos Labini et al. [16] consider an environment-dependent form of Pa in an attempt at simulating strongly non-linear fractal clustering. Here, we focus on the simple case of a unitary value of Pa and perform detailed numerical simulations for the distribution of a large number of fluid elements on the line. This choice of Pa is particularly interesting, as in this case the PZA has interesting analogies (as well as differences) with the adhesion approach.

Once fLxed the functional form of P~, the following step is to describe which quantities are conserved during the sticking episodes. Here we opt for the conservation of the mass and linear momentum of the sticking particles. Consequently, in the one-dimensional case the properties of the new particle j generated by a specific sticking episode are determined by the two formulas

(3.1) m j = ~ m i ,

(3.2) my vj = ~ mi vi ,

where the index i ranges from 1 to n, and n is the number of particles which participate in the sticking episode (usually, n- -2) . Expression (3.2), which imposes the conservation of momentum, makes the conceptual difference between the present version of PZA and the adhesion approach. The position of the new particle is then fLxed as the position of the centre of mass of the sticking particles

(3.3) m j x j = ~ m i xi .

Between two successive merging episodes, the particles move according to the Zel'dovich prescription (2.6), with velocity determined by (3.1), (3.2). Particles are moved in subsequent steps of duration 5a, with the caveat that the distance between two subsequent steps, Adj = vj Aa, where vj is the velocity of the j-th particle, should not exceed the critical sticking distance de. The time step ha is thus chosen as

(3.4) A a = de I v m ~ ,

where Vm~x is the maximum (in absolute value) of the initial velocity field. After each (,Zel'dovich, step, a search is made over all particles which have come closer than de, and sticking occurs. Due to the large number of sticking episodes, this method is faster than solving the equations for the minimal approach between two straight orbits.

THE PUNCTUATED ZELtDOVICH APPROXIMATION 537

4. - N u m e r i c a l s i m u l a t i o n s .

In this section we analyse the evolution of 1D density fields, obtained by the numerical implementation of the approximation schemes discussed above. Our aim is to study the properties of the Punctuated Zel'dovich Approximation (PZA), and comparing it with the classical Zel'dovich (ZA) and adhesion (AA) approaches.

Each simulation has been evolved from a set of initial Gaussian density perturbations, whose statistical properties are completely characterized by the shape of the power spectrum. An initially Gaussian density perturbation field arise quite naturally in most inflationary models [1], and the central-limit theorem ensures that these fluctuations are the generic outcome of a wide variety of random processes. For simplicity, in the following we consider the evolution of initially scale-free power spectra

(4.1) P ( k ) = 15k 12 = A k ~

and we select the spectral indices n = -1 , 0, 1. The spatial Fourier coefficients of the density contrast, 5k, are obtained through (4.1) and we assume that their phases are randomly distributed in the range (0, 2~). We use the same sets of random phases for the different spectral indices, in order to have similar locations for the main structures.

The normalization in one spatial dimension is forcefully quite arbitrary; here we choose the normalization constant A by requiring that the linearly evolved density fluctuation field, smoothed On a box with comoving size 8h -~ Mpc, has a present r.m.s. value

(4.2) a(8h -1Mpc) = b -1 ,

where b is the bias factor, herein chosen to have the value b = 1. This normalization may be seen also as a definition of what we mean by (,today, in the simulations.

The modulus and phase of the initial peculiar velocity field are obtained in k-space with the formulas, valid in the linear regime (see [3], with a( t ) = b(t) because ~9 = 1),

1 1 (4.3) vk - 5k,

a k

(4.4) q~v = cf 6 + - - , 2

while for the velocity potential ~ we have

1 1 (4.5) O k - a k; ~ (~k,

(4.6) ~o r = ~o ~ .

Subsequently, we obtain the corresponding real quantities through a Fast Fourier Transform (FFT) algorithm in 1D.

For all cases, we evolve Np = 105 particles, each of which represents a single fluid element, in a one-dimensional ambient space with comoving size Lbox =

= 100h-lMpc. Periodic boundary conditions are imposed. Using a( t ) as the time

5 3 8 L. FONTANA, M. MILELLI , G. MURANTE and A. PROVENZALE

variable, we normalize such that a = 1 at the present epoch. Finally, we fix the artificial viscosity and the critical distance such that v = 1.0 and d c= 0.1 h - lMpc .

Once obtained the particle distribution at a given time, we evaluate the fluctuation density field with the Cloud-In-Cell (CIC) scheme. Here we use Nmd = 2048, with a consequent spatial resolution A x = L b o x / N ~ d = 50kpc. Figures la)-c) show the density field (~(x~d) obtained by the PZA for the three spectral indices considered. Note the ,,spiky, appearance of the density field, due to the presence of single particles which have been produced by the sticking of several initial fluid elements.

300

200

100

0 500

400-

3 o 0 - g

200-

~D

100-

0

300

200

L,~.L.~ i.,..l.l J~l .i,,.a~J ......

a)

iL ........... ~ .......... ],, .A ... . . . . . ~,

b)

' "" ~ ' l l ' " t " " " I L ' ~ ,i,J ,Ltl . tJ .... , J ! 1 , , l l

c)

100

0 400 1600 2000 ,I I

i

800 1200

x (grid points)

Fig. 1. - Density fluctuation field ~p/p generated by the PZA for scale-free initial conditions with spectral index n = - 1 (panel a)), n = 0 (panel b)) and n = 1 (panel c)). Abscissae are given in units of grid points.

THE PUNCTUATED ZEL~DOVICH APPROXIMATION 539

It is worth recalling that the PZA, like the AA, is expected to produce good results on intermediate scales, being presumably unable to properly reproduce the strong non-linear clustering on small scales. In order to obtain a smoother density field, we thus consider the results obtained by convolving the density fluctuation field attained by the CIC method with a Gaussian window

1 (4.7) W ( x - ~j) = --V2~-x~ e x p [ - ( x - y)2 / 2 x 2 . ]

with width R = 2 x , = i h- lMpc. This corresponds to the thickness of pancakes at

present time, a V r ~ v , in the case a = 1 and u = 1.0.

0

0

30-

20-

i0-

O-

a)

30-

20-

b)

10-

I ~ I I I ]

Fig. 2. - Smoothed density fluctuation field generated by PZA (panel a)), AA (panel b)) and ZA (panel c)), for scale-free initial conditions with spectral index n = - 1 .

30 c) 20

0 400 800 1200 16'00 2000 x (grid points)

~4O L. FONTANA, M. MILELLI, G. MURANTE and ~ PROVENZALE

Figures 2a-c) show the behaviour of 6(xg~d) at the epoch a = I for the case n = - 1, as obtained from the PZA, the AA and the ZA, respectively. In this case we have much power on large scales and, consequently, there are only a few large peaks, while underdense regions are very extended. In the case of the ZA, pancakes have thickened too much; this had to be expected as particles were continuing the unperturbed trajectories beyond orbit crossing. Panels a) and b) show an impressive similarity between the position and amplitude of the peaks calculated by PZA and AA. While ZA allows pancakes to re-expand into the surrounding low-density regions, both PZA and AA keep the peaks thin and coherent. Of course, in real gravitational clustering the peaks tend to fragment into a series of smaller structures. But either PZA or AA do not capture the degree of this fragmentation. Consequently, what appears as a chain of groups and clusters in N-body simulations, appears in PZA and AA as a single thin structure.

�9 o

O

O �9

2~ I 10

01 &_

20-

10

0 I

a)

i i i i

b)

I i I I ~ I

20

10

0

c)

i I i , r i i i i

400 800 1200 1600 2000 x (grid points)

Fig. 3. - Same as fig. 2 but for n = 0.

THE PUNCTUATED ZELtDOVICH APPROXIMATION 541

Figures 3a)-c) and 4a)-c) refer to the cases with n = 0 and n = 1, respectively. As expected, for larger spectral indices (i.e. less power on large scales) peaks are more frequent and underdense regions are less extended. It is clear that the failure of ZA is more evident in the situations characterized by a large amount of power on small scales, where truly non,linear structures rapidly develop. Consequently, for n = 0 and n = 1 the ZA shows an almost homogeneous density distribution, resulting from generalized orbit crossing. Again, PZA is able to go beyond ZA and it originates a density field which is very similar to that of AA. It is interesting to note that on small scales PZA shows a larger number of small peaks with respect to AA, where the viscous term forces a dissipative smoothing process. The conservation of momentum in PZA allows for a matter distribution which is definitely more structured than that obtained with the AA.

We now turn to the statistical analysis of the density fields shown in fig. 2)-4).

0

�9

2O

20

10

0 J ~ ~ ~ . ~ ~.~

20

a)

b)

10

c)

0 ~ o ' 4 6 o ' 8 6 o ' 12'oo

x (grid points)

Fig. 4. - Same as fig. 2 but for n = 1.

' 1600 ' 20'00

5~2 L. FONTANA, M. MILELLI, G. MURANTE and A. PROVENZALE

Figures 5a)-c) show the behaviour of the power spectrum of the density fluctuation field. Again, there is a very good agreement between AA and PZA, especially on large scales. On very large scales, the spectrum keeps the memory of its initial shape, but on intermediate scales there is an asymptotic evolution towards a spectrum characterized by a spectral index n --0, independent of initial conditions. Note also the lack of power in the spectra generated by ZA, due to orbit crossing and to its failure in describing the strong clustering regime.

@

0

i0 4

10 2

i0 ~

10 -2

i0 - 4 _ _

10 4

10 2

10 0

10 -2

i0 -4__ 10 4

I 0 2

i 0 ~

i 0 -2

% ) " ! { ' ! : ; . ,

ii ! "~'li.!

I I i l I I i l

b) ~

I

loll i0 -2

, . ,

c)

-=, " I ]11'%~ \ . . . ,,

I i I I I I I

10 -1 10 ~

wave number

Fig. 5. - Power spectra of the density fluctuation fields generated by the PZA (dot-dashed line), AA (solid line) and ZA (dotted line). Panel a) refers to n -~ - 1, panel b) to n = 0 and panel c) to n = l .


Another important statistics is the one-point probability distribution (PDF) of the density fluctuations, P(~). In the linear regime at the beginning of the simulation (for a = 4.10 -2) ~ is normally distributed. Figures 6a)-c) show how the influence of the non-linear gravitational evolution, as modelled by the AA and PZA, modifies the initial shape of P(5). Underdense regions become even more underdense (although not indefinitely, since the density is defined as a positive quantity), while positive density enhancements tend to grow without bound. Clearly, the symmetry between positive and negative fluctuations cannot be maintained and the PDF becomes skewed. Note that for this statistics there are some differences between PZA and AA.

o

10 7

10 ~

10 -1

10 -2

101

a)

I I - - 7 - I

lo 0 ~/ : '~ ~

b)

V ' ~,,' A i 1 ! / i ~.

10 -2

10 7

I - I I I I ~ I - - I - -

c)

i0 ~

-1 10 4 ~!~ !' ,t

10-2~ - , , - 1 0 - 0 8 - 0 : 6 - 0 4 - 0 : 2 o o 0:2 04 0:6 0:8 10

density contrast

Fig. 6. - PDFs for the PZA (solid line) and AA (dotted line). Panel a) is for n = - 1, panel b) for n = 0 and panel c) for n = 1.

544 L. FONTANA, M. MILELLI~ G. MURANTE and A. PROVENZALE

I n f a e t , t h e r e i s a l a r g e r n u m b e r o f v o i d s i n P Z A w i t h r e s p e c t t o A A , w h i c h s h o w s a

m o r e g r a d u a l d e c r e a s e o f P ( 5 ) a s 6 i n c r e a s e s . T h i s m a y b e a t t r i b u t e d t o t h e d i f f e r e n c e

b e t w e e n t h e d i s s i p a t i v e s m o o t h i n g p r o c e s s u s e d i n A A a n d t h e p u n c t u a t e d ,

m o m e n t u m - c o n s e r v i n g m e r g i n g p h e n o m e n a i n P Z A .

6. - C o n c l u s i o n s a n d p e r s p e c t i v e s .

I n t h i s p a p e r w e h a v e d i s c u s s e d a p h e n o m e n o l o g i c a l m o d e l f o r n o n - l i n e a r

g r a v i t a t i o n a l c l u s t e r i n g , w h i c h w e h a v e n i c k n a m e d ~ P u n c t u a t e d Z e l ' d o v i e h

A p p r o x i m a t i o n , ( P Z A ) . T h i s a p p r o a c h i s b a s e d o n a m o d i f i c a t i o n o f t h e s t a n d a r d

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F ig . 7. - P a r t i c l e d i s t r i b u t i o n in a s l ice of a t h r e e - d i m e n s i o n a l , 64 a g r i d p o i n t s s i m u l a t i o n o f g r a v i t a t i o n a l c l u s t e r i n g o b t a i n e d b y t h e P Z A (pane l a)) , a P~ M code (pane l b)), t h e Z A (pane l c)) a n d t h e T r u n c a t e d Z A ( p a n e l d)). C o o r d i n a t e s a r e in un i t s of g r id po in t s .


Zel'dovich Approximation, based on simulating small-scale non-linear gravity through an aggregation probability of the particles passing nearby each other. We have analysed the statistical properties of the matter distribution generated by this model in a one-dimensional ambient space and compared them with the standard Zel'dovich (ZA) and adhesion (AA) approaches.

The results indicate that PZA is capable of going beyond ZA, by avoiding orbit crossing and forming bound gravitational states. There is an overall similarity between PZA and A/~ as both of them predict the formation of thin, coherent matter concentrations. We claim PZA to be even more satisfactory than AA, as the former is capable of conserving linear momentum. This leads to a larger number of voids and of small-scale structures in PZA with respect to AA, which is characterized by a smoother matter distribution due to the viscous dissipation. Nevertheless, neither PZA nor AA properly capture the degree of the fragmentation which is present in truly N-body simulations.

Work is now in progress to compare PZA to the full N-body simulations (PaM code) in a three-dimensional ambient space, using different types of initial conditions (e.g., CDM, scale-free). As an anticipation, in fig. 7a)-d) we show the matter distribution at the present non-linear epoch, for CDM initial conditions, obtained with the PZA, with a PSM code, with ZA and with the Truncated Zel'dovich Approximation (see, e.g.,[17]). The PZA shows a better agreement with the PSM results on large scales with respect to ZA and TZA, though even the PZA underestimates the presence of small clumps.

As a final comment, we want to stress the interesting potentialities of PZ/L Although extremely simple, the sticking approach seems to be able to capture some of the essential features of the clustering process on large scales.

It is a pleasure to thank Prof. C. Castagnoli, whose experience, encouragement and advice has always guided our navigation in the difficult seas of Physics.

R E F E R E N C E S

[1] COLES P. and LUCCHIN F., Cosmology (Wiley, Chichester) 1995. [2] SMOOT G. F. et al., Astrophys. J., 396 (1992) L1. [3] PEEBLES P. J., The Large-Scale Structure of the Universe (Princeton University Press,

Princeton) 1980. [4] ZEL'DOVICH YA. S., Astron. Astrophys., 4 (1970) 84. [5] SHANDARIN S. F. and ZEL'DOVICH YA. B., Rev. Mod. Phys., 61 (1989) 185. [6] GURBATOV S. N. and SAICHEV A. I., Isv. VUZ, Radiofizika, 27 (1984) 456. [7] GURBATOV S. N., SAICHEV A. I. and Sm~NDARIN S. F., Soy. Phys. Dokl., 30 (1985) 921. [8] PIETRONERO L. and KUPERS R., Fractals in Physics (Elsevier) 1986, p. 319. [9] CARNEVALE G. F., POMEAU Y. and YOUNG W. R., Phys. Rev. Lett., 64 (1990) 2913.

[10] DOROSHKEVICH A. G., Astrofiz., 6 (1970) 581. [11] DOROSrIKEVICH A. G., RYABENKII V. S. and SrIANDARIN S. F., Astrofiz., 9 (1973) 257. [12] GURBATOV S. N., SAICnEV A. I. and SttANDARiN S. F., Mora Not R. Astron~ Soa, 236 (1989) 385. [13] WEINBERG D. H. and GUNN J. E., Mon. Not. R. Astron. Soc., 247 (1990) 260. [14] VERGASSOLA M., DUBRULLE B., FRISCH U. and NOULLEZ A., Astro~ Astrophys., 289 (1994) 325. [15] WEISS J. B. and McWILL~_aMS J. C., Phys. Fluids A, 5 (1993) 3. [16] SYLOS LABINI F. and PIETRONERO L., preprint. [17] FONTANA L., MURANTE G. and PROVENZALE A., to be published in Nuovo Cimento C,

GIFCO Proceedings (1996).

Documents

The punctuated Zel’dovich approximation