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V&as in Astronomy Vol. 41, No. 4, 467492, pp. 1998 @ 1998 Elsevier Science Ltd

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LOOKING FOR THE IMPRINTS OF NONLINEAR

STRUCTURES ON THE COSMIC MICROWAVE

BACKGROUND

M.J. FULLANAa, J.V. ARNAUb, D. ShZa * Departament d' Astronomia i Astroffsica, Universitat de Valbncia, 46100 Burjassot (Valencia),

Spain b Departament de Matematica Aplicada, Universitat de Valencia, 46100 Burjassot (Valbncia),

Spain

Abstract- Many authors have estimated the anisotropies produced by one isolated cosmological non-linear inhomogeneity. This paper is an updated review about these estimates. The main methods used in order to deal with this problem are described. The limitations of these methods are analyzed. Results appear to be particularly interesting in the open non-linear case, in which a general treatment of the anisotropies produced by inhomogeneity distributions is very troublesome. The effects produced by very big structures such as the Great Attractor and the Boijtes Void are studied in detail. Some generalities about the origin, detection and features of the Cosmic Microwave Background anisotropies are also presented for the sake of completeness. @ 1998 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Since the discovery of the Cosmic Microwave Background (CMB) radiation in 1965 [83], Cosmology has undergone a great advancement. The study of the CMB radiation has played a very significant role in many achievements of Cosmology. Some basic questions about the nature and relevance of the CMB arise. What is this radiation, which were the processes generating the CMB, what are the reasons making this radiation so significant? Let us present some comments about these questions. ’

The standard theory about the primeval history of the Universe establishes that matter and radiation were initially coupled. This means that there existed a primeval plasma in which protons and electrons were not confined in atoms. Photons could not travel freely as a result of frequent interactions with the neighbouring free electrons and, consequently, these photons were not able to carry information from a point of the plasma to another one, namely, the Universe was opaque. As a result of the expansion, it cooled and be&me transparent.

* These questions and other basic ones are addressed in many books as, for example, Ref. [82].

468 M.J. Fullarm et al.

Let us make some comments about expansion. The Universe was (and is) expanding following Hubble’s Law; this means that the proper physical distance between a pair of well-separated galaxies (or particles) is an increasing function of time, that is, any galaxy is receding from any other. The recession produces a cosmological redshift of the CMB waves. The rate of recession is given by Hubble’s Law: d(t)/& = Hi(t), where l(t) measures the spatial distance between two galaxies and H is Hubble’s parameter. The redshift, 2, of a galaxy satisfies the relation 1 + Z = &a(t), where ao is the present value of the scale factor a(t). As a consequence of this relation between Z and t, the redshift can be used to measure time (the greater the redshift, the further in the past the source of the information).

The temperature decreased as a result of the expansion; thus, at a certain time, the temperature was low enough and the protons captured the free electrons to form neutral hydrogen atoms. After this process, the Universe became transparent; photons could travel almost freely because protons and electrons were confined in atoms (recombination). The mean free path of the CMB photons became huge. Recombination led to the decoupling between matter and radiation; after decoupling, CMB photons travelled almost freely in space. The so-called CMB radiation is formed by these decoupled photons. This radiation departed from all the points of the Universe in all directions during a short period of time. It is received in all directions at any point of the universe. As time goes on, any observer receives the CMB radiation from further times and distances.

The decoupling redsbift has been estimated to be z - 103. The recombination-decoupling process lasted for a certain time corresponding to a redshift increment AZ - 80 [50]. The short duration of this process allows us to consider a Last Scattering Surface (LSS) separating an ion- ized and opaque universe from a neutral and transparent one. It may happen that after this first decoupling, matter and radiation couple again (at a redshift smaller than 1000). Astrophysical phenomena related to galaxy evolution could lead to a reiunization. There must be a new decoupling after each reionization.

After discussing the origin and nature of the CMB radiation, let us analyze the reasons making its analysis one of the most important topics in Cosmology. This radiation is found to be very isotropic (see for instance Ref. [68]). Nevertheless, small CMB temperature fluctuations (anisotropies) have been found. The existence of these anisotropies strongly enhances the impor- tance of the CMB. The anisotropies are due to the density inhomogeneities present in Universe. For this reason, the detection and analysis of the anisotropies supply substantial information about inhomogeneities; that is to say, about the large-scale structure of the Universe.

The redshift undergone by a CMB photon is the effect produced by all the cosmological inhomogeneities located near the photon at some time. Each inhomogeneity acts on the CMB photons through different physical phenomena (see Section 3.2). The estimation of the total anisotropy requires a statistical treatment. In the linear regime, such a treatment is well established. How- ever, in the case of the anisotropy produced by non-linear inhomogeneities evolving in open universes, a general treatment becomes troublesome (non-linear gravity, non-Gaussian statistics, etc.). Hence, approximations leading to indications about the anisotropy corresponding to this troublesome case are worthwhile. One of these approaches estimates the anisotropy produced by spherical isolated non-linear structures. In order to do this estimation, an idealized universe composed just of one large spherical cosmological inhomogeneity is considered. If the chosen structure is similar to some observable structures and the resulting anisotropies are great enough, useful information about the anisotropy produced by a distribution of these structures can be in- ferred. This information could be useful in order to select appropriate general methods for more accurate quantitative computations.

The study of the CMB anisotropies generated by one spherical pressureless inhomogeneity can

Looking for the imprints of non-linear structures on CMB 469

be done by using two main models. One of them is the so-called Swiss-Cheese Model (SCM) and the other one is based on the Tolman-Bondi solution (TBS) of Einstein’s equations.

The aim of this review is to offer a detailed description of the CMB anisotropies generated by isolated cosmological inhomogeneities. In this section, a general introduction has been presented. In the next one, a brief description of the CMB radiation is given. Section 3 deals with the CMB anisotropies. Some observational data about anisotropies are given, a general review of the most important anisotropy sources is presented and the meaning of the angular scales of the CMB anisotropies is analyzed. Section 4 is concerned with the main goal of this paper: the estimation and interpretation of the anisotropy created by a single isolated cosmological inhomogeneity. Several models for describing these inhomogeneities are presented. Particular attention is focused on the SCM and the TB models applying in the pressureless case. The main predictions about CMB anisotropies based on the selected theoretical models are presented. The meaning and consequences of these predictions are discussed in Section 5.

2. THE COSMIC MICROWAVE BACKGROUND RADIATION

The discovery of the CMB radiation is a curious example of the intricate paths that science can take.

Lemaitre [59] was the first scientist to speculate about the possible observable remnants of the very early stages of the universe. Lemtitre imagined a hot beginning of the expansion. A relic radiation would appear in it. He proposed the cosmic rays to be the result of this radiation. Afterwards, this assumption was proved to be wrong.

Tolman [ 1 IO] introduced the idea of the thermal history of an expanding universe. He showed that expansion cools the black-body radiation while keeping a thermal spectrum.

In the late 194Os, Gamow, Alpher and Herman predicted a universal radiation background, remnant of the hot Big Bang. These authors [l-3,39,40] estimated the order of magnitude of the present background radiation temperature. A temperature of a few degrees Kelvin was obtained. Nevertheless, this prediction was forgotten for a long time.

In the years 1964 and 1965, Arno Penzias and Bob Wilson, two researchers of the Bell Labo- ratories at Holmdel, were making some measurements with a radiotelescope. They did not understand the nature of a certain noise excess appearing in their measurements. A meeting attended by the Holmdel group and a Princeton University team reached the conclusion that the detected noise could be the CMB radiation. The high isotropy of this noise suggested its interpretation as the CMB remnant. The almost isotropic antenna temperature excess was found to be To = 3.5 f 1 .O K at a wavelength 2 of 7.35 cm. No fluctuations appeared during observations (about one year). These results were communicated to the Astrophysical Journal [83] and interpreted - in an associated letter by Dicke, Peebles, Roll and Wilkinson [25] - as the detection of the b&k-body cosmic radiation.

After the first observation at 7.35 cm, new measurements at different wavelengths were necessary in order to verify that the detected signal had a Planck spectrum. Some of the first observations are due to: Roll and Wilkinson [91], Field and Hitchcock [32], Thaddeus and Clauser [ 1071 and Penzias and Wilson [84] 3 . From the resulting observational data, a Planck spectrum

* The radiation tcmpcrature at a certain wavelength is defined as the black-body temperature with the same inten- sity as the one detected at such wavelength. The temperature would be the same for all wavelengths in a properly thermalized radiation.

3 See Raychaudhuri [88] for a more detailed list of these first observations.

470 M. J. Fullana et al.

with a 3 K temperature was obtained in agreement with Gamow’s [39,40] predictions. After- wards, many measurements have been done in order to test the features of the CMB spectrum. Wavelengths, J., satisfying the inequality 75 cm 1 A. z 0.05 cm were considered by the nineties

[801. The most accurate observations have been performed during the nineties by using the FIRAS

(Far InfraRed Absolute Spectrophotometer) device on board of COBE (Cosmic Background Explorer) satellite [67,68]. The data from FIRAS can be summarized as follows [105]:

(a) The best-fit black-body temperature is [34]

T,, = 2.728 f 0.002 K (1)

with a 95% confidence level (CL). At this temperature, the number of CMB photons per unit of volume, n,, , and the energy density of the CMB photons, pY, are:

ny 21413 cmm3

p ~4 68 x 1O-34 gcme3, Y . (2)

(b) A least-squares fit to all CMB measurements yields

IyI < 1.5 x 1o-5 (95% CL), (3)

l/&)1 < 9 x 10-5 (95% CL),

where y is the standard distortion parameter [ 1061 and /.Q-J is the dimensionless chemical potential [57]. The limits here [34] set limits [33,34,67,68,11] on the contributions to the CMB of processes occurring between redshifts lo3 and 5 x 106. The inequality A E/ ECMB < 2 x 10B4 is satisfied, where ECMB is the energy density of the CMB and A E is the contribution to the CMB energy density due to one of the mentioned processes.

The analysis of the CMB spectrum and anisotropies is the best source of information about the features of the Universe at large redshifr. The black-body spectrum and the high degree of isotropy of the CMB are the main observational data justifying the standard model, in which small linear perturbations evolve in a homogeneous and isotropic background. This is the so- called Friedmann-Lema?tre model. The study of the relationship between density perturbations and anisotropies in the CMB, as well as the study of black-body spectrum distortions, give important information about the history of the Universe. Observational constraints on the deviations with respect to the perfect black-body spectrum set constraints on physical processes generating this type of deviations. Physical processes producing too great deviations must be rejected. (See for example Ref. [53], or [120,121]). Observational constraints on the anisotropies set relevant constraints to the theories of large scale structure formation. Theories producing too large anisotropies would be ruled out. Up to date, there are various models for structure formation compatible with the observed anisotropies. The values of several parameters involved in these models are unknown. Further observations should contribute to fix a model and the values of the unknown parameters.

The study of the spectrum and the anisotropies of the CMB can be done either from an observational point of view - designing experiments and performing them - or from a theoretical one - estimating the CMB anisotropies corresponding to different physical processes. This paper is a theoretical one, concerned with the estimation of the CMB anisotropy produced by non-linear gravity.


3. CMB ANISOTROPIES

3.1. Some observational data

Despite the high degree of isotropy of the CMB, the observed temperatures appear to have a small dependence on the direction of observation. The amplitude of the temperature fluctuations has been measured on various angular scales 4 . Since the anisotropies of the CMB are very small, the design of experiments becomes expensive and difficult.

The lirst observed anisotropy was the dipolar component [2 1,103]. The peculiar motion - with respect to CMB - of the Local Group creates this component (see Appendix A). COBE measurements [56,33,61] have given the dipole of the CMB with high accuracy. The most accurate value given by the COBE team has been obtained from the analysis of the COBE Differential Microwave Radiometers (DMR) four-year data. They obtain a best-fit dipole amplitude [61]

D &s = 3.357 f 0.001 f 0.023 mK, (4)

where the first uncertainties are statistical and the second are estimations of the combined sys- tematics. 5

Detection of anisotropy at angular scales smaller than that of the dipole was a difficult task. Until 1992, only the dipole had been detected.

Fortunately, the experimental situation changed in 1992 with the detection of non-dipolar cosmological anisotropies with the DMR device on board the COBE satellite [lOO-102,9]. In April of that year, the DMR team announced the discovery of anisotropy in the CMB temperature on angular scales greater than 10”. Excepting the dipole (amplitude - 10d3), the amplitude of the detected anisotropies is at the level of about one part in 10’. The features of the DMR device are not compatible with the measurement of temperature variation on angular scales below 10”. In particular, the COBE team claimed that [lo], in the case of a flat power spectrum of primordial energy density fluctuations, the quadrupole of the CMB anisotropy is (see Appendix A)

Qrrns-PS = 18f 1.6pK.

The quadrupolar anisotropy is the effect of very large density fluctuations of small amplitude located near LSS.

The detection of local anisotropies produced by the hot gas of some clusters of galaxies was first claimed by Gull and Northover [44]. Other observers have detected these anisotropies in a few rich clusters (e.g. Birkinshaw et al. [ 14,151, Uson [ 1121, Birkinshaw [ 161, Klein et al. [55], Herbig et al. [46], Wilbanks et al. [ 1191, Jones et al. [51]). These anisotropies can be explained as a result of the interaction between the CMB photons and the hot electrons located inside rich clusters of galaxies. This phenomenon produces a spectral distortion in the Rayleigh-Jeans part of the black-body spectrum. This distortion is equivalent to a temperature decrease. This effect can be distinguished from other ones because it depends on frequency. The first researchers who predicted it were Sunyaev and Zel’dovich (see Ref. [ 1061, for a review); for this reason it is known as the Sunyaev-Zel’dovich effect. The amplitude of the Sunyaev-Zel’dovich effect in the case of some rich clusters is AT/ T - low4 on scales of arcminutes. predictions and observations seem to agree.

4 See Subsection 3.3 for a description of angular scales. 5 This dipole is directed - in galactic coordinates - towards (I, b) = (264O.33 f OO.04 f 0°.13,4S0.05 f OO.02 f

00.09).


Successive summaries of experiments for the measurement of CMB anisotropies can be found in Weiss [117], Partridge [80], Readhead and Lawrence [89], White et al. [118]. Nowadays, there exists about a dozen experiments (in realization or in project) in which measurements on scales greater than 5’ with sensitivity between - lo-’ and 10e6 are being or will be performed. Such observations will help us to better understand the structure, formation and history of the Universe. It seems that we are in a critical moment - of technological development - which can lead to accurate detections of anisotropies on small angular scales. This possible detection would allow us to reject some theoretical models and to reinforce others.

3.2. Anisotropy sources

Density fluctuations break the homogeneity of the Universe. After an intricate evolution depending much on the spatial scales, the primordial linear fluctuations produce CMB anisotropies through different physical processes. In this subsection, the processes producing CMB anisotropies are described.

Anisotropies are usually classified as primary or secondary. The anisotropies created by physical processes near decoupling are called primary, while those produced long after decoupling are called seconuizly.

Let us describe some physical phenomena producing primary and secondary anisotropies. The bulk peculiar velocity of the plasma (free protons and electrons) with respect to the CMB

- at decoupling time - produces a Doppler effect. This effect depends on the radial component of the peculiar velocity at decoupling time: ~(0, (p) .

The existence of photon temperature fluctuations - associated with density fluctuations - on the LSS is another anisotropy source. Hereafter, this anisotropy is called thermal. It is roughly proportional to the baryonic density fluctuations at decoupling time (AP/P)b(td) (see Ref. [57]). The following relation is satisfied:

(6)

Another anisotropy appears as a result of the difference between the gravitational potential created by the inhomogeneities on the LSS and that created at the observation point. This is the so-called Sachs-Wolfe effect [93]. The corresponding temperature fluctuations obey the equation

where 4 is the gravitational potential and subscripts E and 0 stand for emission and observation, respectively.

In addition, in the case of a linear inhomogeneity evolving in an open universe, there appears an integrated Sachs-Wolfe effect which is associated to the time variation of the peculiar gravitational potential created by the inhomogeneities. In the flat case, such a linear effect is negligible.

The anisotropies described above are essentially produced by linear inhomogeneities located near LSS; therefore, they are primary anisotropies. A description of some anisotropies generated by inhomogeneities placed far from LSS - some of which are non-linear - is given below. These anisotropies are secondary.

The relative motion between the observer and the CMB produces a Doppler effect. The corresponding anisotropy can be expanded in spherical harmonics (see Appendix A). The dipole amplitude depends on the peculiar velocity of our Local Group of Galaxies, VLG, as follows:

Looking for the imprints of non-linear structures on CMB

AT VLG -N-* T c

The quadrupole amplitude of such a Doppler effect is

473

(8)

(9)

This small quadrupole is just a relativistic effect. The time variation of the gravitational potential created by a non-linear inhomogeneity pro-

duces secondary anisotropies when the CMB photons pass near it. This is an integrated effect over the photon null geodesic. Various authors [4,19,65,66,86,1 I 1] have used different methods in order to study such an effect.

The motion of CMB photons across an overdensity is slower than in the homogeneous background and, consequently, a time delay appears. This delay produces anisotropy. This is the Rees-Sciama effect [go]. The resulting anisotropy is negligible in linear cases; hence, it is a purely non-linear gravitational effect.

The Sunyaev-Zel’dovich effect described in Section 3.1 is another source of secondary anisotropy. We can distinguish two problems: The estimation and detection of the effect produced by single Abel1 clusters located at low redshifts and the estimation and detection of the net effect produced by far unresolved clusters. The second problem becomes particularly difficult. Problems with its estimation appear as a result of current uncertainties about cluster evolution. It is believed that the net effect is small and, consequently, detection is expected to be difficult. This subject is being studied at present.

Reionizations would erase primary anisotropies on some scales, but would produce new secondary anisotropies on other scales. The so-called Vishniac effect [76,114] plays an important role in reionizations. The study of this kind of phenomena has been recently improved by Hu and Silk [48], Hu et al. [47], Dodelson and Jubas [26] and Persi [85].

Recently, the anisotropies generated by the peculiar bulk motion of highly non-linear structures - such as clusters - have been estimated. This effect also generates a secondary anisotropy [ 1111.

Although the physical phenomena listed above are the most accepted explanations for the origin of CMB anisotropies, other anisotropy sources have been proposed. Cosmic strings and other topological defects would produce non-Gaussian anisotropies with special features. This possibility is being studied in detail. Other possible causes of anisotropies such as the rotation of the Universe or a possible anisotropic expansion have been also considered (see Ref. 1571).

In short, it can be stated that the main sources of primary anisotropies are: (a) the Doppler effect produced by peculiar velocities on the LSS, (b) initial fluctuations on the photon gas near the LSS (thermal anisotropies), and (c) the Sachs-Wolfe effect. Secondary anisotropies are orig- inated by (i) the motion of the observer (Doppler effect); (ii) the time variation of the gravitational potential along the world line of the CMB photons; (iii) the Rees-Sciama effect; (iv) the Sunyaev-Zel’dovich effect; (v) the Vishniac effect, and (vi) the bulk motion of non-linear structures.

3.3. Angular scales

Since recombination is not an instantaneous process, we should speak about a recombination

shell rather than about a LSS. The thickness of this shell is - 7h-‘52,1’2 Mpc. Such a distance

474 M. J. Fulha et al.

is seen, at present, to subtend an angle 0d ‘v (452d,‘2)‘6 [53]. A detailed study of the interaction between electrons and photons shows that anisotropies on scales less than & are essentially erased during the recombination-decoupling process; so there should not exist significant primary anisotropy on scales less than 6Jd. In other words, the possible anisotropies measured on these angular scales would be secondary.

The particle horizon of a certain observer, at a given time, is the maximum distance that a massless particle can travel from Big Bang to the assumed time. l%o points located at a distance larger than the particle horizon are not causally connected. In the standard cosmological model (without inflation), with decoupling at z m 103, the angle subtended by the particle horizon at decoupling is &..r 21 (2ad/2)o. So, in the absence of inflation, anisotropies subtending angular scales greater than 01-l cannot be causally generated. Inflation solves this problem, allowing us to understand the presence of anisotropies on scales greater than 0~. These scales were inside the particle horizon during inflation and correlations on these scales were causally produced by the inflationary field.

The eflective horizon, at a certain time, is the maximum distance at which physical interactions can produce net energy-momentum exchanges. The angle subtended by the effective horizon at decoupling (z - 103) is 6%~ - (ls2i’2)0. Density fluctuations subtending angular scales less than &r are very sensitive to local physical interactions, while fluctuations subtending greater angles evolve essentially governed by gravitational instability laws.

We have just defined three significant angular scales related to CMB anisotropies: these scales are @d, &, and f&R. According to the relationship between the angular scale corresponding to a sort of anisotropy and the angles 0t-l and &.n, three kinds of anisotropies can be distinguished. Anisotropies on angular scales 6J < &r are named smull angular scale anisotropies, those on scales 8 > OH are called large angular scale anisotmpies and, finally, the ones on scales between &.ff and 01.l are named intermediate angular scale anisotmpies. Roughly, it may be said that small and large angular scales correspond to angles smaller and greater than l”, respectively 7 , while intermediate angular scales correspond to angles of the order of lo.

Excluding the dipolar and quadrupolar anisotropies produced by the peculiar motion of the observer, it is usually accepted that the main anisotropy source on large angular scales is the Sachs-Wolfe effect produced by primeval inhomogeneities placed near LSS. Intermediate angular scales are dominated by the Doppler effect produced by peculiar motions on the LSS. Finally, in the simplest models, small angular scales are dominated by thermal fluctuations; but effects produced by reionizations, non-linear gravity and electron-photon interactions (Sunyaev- Zel’dovich) could contribute to these scales.

4. NONLINEAR COSMOLOGICAL INHOMOGENEITIES. MODELS AND ANISOTROPIRS

On large spatial scales, the Universe seems to be formed by great voids of galaxies surrounded by walls. The number of galaxies inside a void (wall) is smaller (greater) than that corresponding to the same volume of the background Universe. The surrounding walls are not homogeneous. Groups of galaxies, clusters of galaxies and flatter structures such as superclusters are inside these sharpened walls (see for instance Refs. [23,75,24,92,18,78]).

6 C2 is the density parameter. The subindex 0 stands for present time throughout the paper. 7 In Partridge’s report [EO] one can find a detailed description of observations at different angular scales.

L.ooking for the imprints of non-linear structures on CMB 475

The spatial distribution of galaxies can be directly observed, but there is an important energy component - collisionless dark matter- which cannot be directly observed. This component produces observable dynamical effects on the baryonic matter through gravitational interaction. It is not known whether galaxies trace the dark matter distribution.

Voids of galaxies tend to sphericity [49] while galaxy excesses tend to diverge from sphericity. The flatness of the cosmological structures strongly depends on the spatial size. Large overdensities such as superclusters are strongly flattened structures, while clusters have ellipsoidal shapes. Some clusters - such as the Coma Cluster - are quasispherical. These considerations strongly suggest the interest of spherically symmetric models of cosmological structures. The fact that collisional baryonic matter seems to be subdominant points out the interest of pressureless models.

In order to study the spatial distribution of galaxies, various methods have been used; among them let us mention the classical statistical studies based on the N-point correlation functions [81], modem developments based on fractals and multifractals [62], and studies about the topology of the galaxy distributions based on the “genus” concept [43]. Classical statistical approaches have the advantage that they can be extended to early linear periods and applied to study the CMB. In general, it is not always easy to study the CMB anisotropies by using general methods describing a statistical distribution of density fluctuations. This is specially true in the case of open universes and non-linear structures. In this case, the study of the effect produced by an unique inhomogeneity becomes particularly significant. Once this study has been done, one tries to infer the maximum information about the anisotropies which should appear in the case of realistic distributions of the chosen inhomogeneities. According to previous comments, spherical pressureless models can play an important role in the estimation of the anisotropies produced by isolated cosmologically significant non-linear structures.

With the essential aim of summarizing the main investigations about the anisotropy produced by a unique isolated non-linear structure, this section is organized as follows: The methods used to describe an universe containing an unique inhomogeneity are presented in Section 4.1, and the anisotropies produced by one of these isolated inhomogeneities are studied in Sections 4.2 and 4.3.

4.1. Pressureless spherical models for isolated inhomogeneities

Basically, there are two approaches describing an isolated pressureless spherical structure. One of them consists of describing the inhomogeneity with the help of various solutions of Einstein’s equations, which are appropriately matched. The second approach uses an unique pressureless spherically symmetric solution of the same equations.

Lemtitre [60] considered an inhomogeneity as a spherical homogeneous condensation described by a time-dependent spherically symmetric metric. All the physical quantities describing the condensation were appropriately matched with those of the outer region, which evolves as a homogeneous expanding universe. Actually, this author introduced many spherical overdensities with the essential aim of studying the effect of many condensations on the evolution of the background universe. At that time, a possible causal relation was suggested between the condensation of overdensities and the expansion of the background. He ruled out these ideas by proving that, in the absence of pressure, the motion of the background universe (regions located outside the condensations) is independent of the condensation evolution.

Einstein [29] proposed a spherically symmetric model for star clusters and, as a limiting case, he considered a cavity surrounded by a material shell which matches to an outer Schwarzschild’s space-time. In that paper, Einstein’s goal was to prove that Schwarzschild’s singularities are

476 M. J. Fulluna et al.

not physically significant * . Afterwards, Einstein and Straus [30,3 l] used another model in order to investigate the influence of the cosmological expansion on the evolution of individual stars. These authors seeded many vacuoles into a space-time involving cosmological expansion. Schwarzschild’s solution of Einstein’s equations describes the space-time inside the vacuole. This solution was matched with a cosmological outer one.

Lemtitre [60], Einstein [29], and Einstein and Straus [30,31] set the basis for the development of the SCM g . In this model, the solution describing the inner region is fitted to another solution corresponding to the intermediate region, which is matched to a homogeneous outer region expanding as the Universe (see Appendix B). The mass of the intermediate region equals that of the central part (compensation) and, consequently, the homogeneity at large spatial scales required by the Cosmological Principle is achieved. The SCM has been used for different purposes (see the references given so far or, for instance, Ref. [54]). Nevertheless, we are only concerned with applications to the estimation of CMB anisotropies (see Subsection 4.2). The first researchers using a certain SCM in the study of CMB anisotropies were Rees and Sciama [90]; nevertheless, these authors used a Newtonian treatment of the problem, which should be considered as a qualitative approach.

Another model for describing pressureless spherical non-linear inhomogeneities - a good alternative to SCM- is based on a single solution of Einstein’s equations. This solution describes the spherically symmetric space-time created by a pressureless perfect fluid in a Friedmann- Robertson-Walker background. It tends to this homogeneous background at very large distances from the symmetry centre. Such a solution is a certain particularization of the general time dependent line element with spherical symmetry. It seems that this line element was first studied by Lema?%e [60], who calculated the components of the energy-momentum tensor and solved the field equations in some particular cases lo . It seems that Tolman [ 109,l lo] derived Lemaitre’s results independently, but he referred to Lemahre’s prior discovery. Tolman discussed some interesting consequences of this solution. On the other hand, Bondi [ 171 made a systematical study of the pressureless case. He published an extension of the work developed by the preceding authors. This author obtained the equations of motion and compared them with the Newtonian equations. He found that the total energy is an important parameter, which determines the geometry of 3-spaces. Despite Lemaitre’s previous work, the cosmological pressureless and spherically symmetric solution of the Einstein equations is often called the TBS. This solution is only useful before shell-crossing [45,69]. TBS has been used in the literature to deal with various cosmological problems (see, for instance, Refs. [7 1,99,74,73], and references therein). A description of the Tolman-Bondi solution of Einstein’s equations is given in Appendix C. The use of this solution in the investigation of CMB anisotropies is analyzed in detail in Subsections 4.2.2 and 4.3.

One of the main differences between the SCM and the TB model is that, in the SCM, the density and the velocity profiles are very particular (each region is homogeneous), while TBS allows us the use of arbitrary profiles see Ref. [6]. This is an important advantage of the TB model.

*Einstein’s conclusion is that Schwarzschild’s singularity does not appear in physical reality because matter cannot be concentrated arbitrarily. Einstein claims that, as the constituting particles reach the singularity, its speed tends to that of light.

g The name indicates that a distribution of vacuoles (or overdensities) in the universe recalls the features of a Swiss cheese. to Lcmake only considered small deviations from an Einstein universe.

Laoking for the imprints of non-linear structures on CMB 477

4.2. On the anisotropy generated by one cosmological structure

Many authors have computed the anisotropies produced by one isolated cosmological inhomogeneity. These authors have integrated the differential equations of the photon world lines in an ideal universe containing only one inhomogeneity. The differences among the methods used by different authors appear either as a result of using different solutions of Einstein’s equations to describe the inhomogeneity (see the preceding subsection), or as a consequence of using different approximating conditions, to integrate the differential equations satisfied by the null geodesics.

In this section, some of the most important contributions to this field are presented. Papers using the SCM, the TB model and other models are considered. In each of the selected cases, the most important aspects of the model and the main conclusions are briefly presented.

4.2.1. Estimates based on the SCM The SCM has been used to compute the anisotropy produced by regions with uniform density

excesses and deficits. In both cases the central region is surrounded by a spherical compensating shell. Such a model has been used to study the anisotropies generated by big spherical structures such as the Great Attractor and great voids of galaxies. As stated before, the first authors who used the SCM in the study of CMB anisotropies were Rees and Sciama [go]. They were inter- ested in the CMB anisotropy produced by quasar clusters. These cluster were described by using a Newtonian version of the SCM, which should be only used in the case of irregularities much smaller than the observable universe. These authors separated and estimated various contributions to the CMB temperature fluctuations tl and discussed the sign of each contribution and that of the net fluctuations.

Dyer [27] used a version of the SCM improving on the Newtonian one due to Rees and Sciama [90]. The resulting temperature profile - across a hole - is different from that obtained by Rees and Sciama [90]. In Dyer’s model, comoving spheres of dust are removed from the background universe and, then, clumps of the same mass are placed at the hole centres. The resulting gravitational effect appeared to be of the order of 10m4 in the case of very big masses of 10’9Mo. In terms of the mass M, this effect approximately scales as M2j3. Nottale [72] used a model similar to Dyer’s one in order to study contracting clusters. He stated that, for @I 21 0.1-0.2, the richest clusters should yield an observable effect of the order of the Sunyaev-Zel’dovich one; nevertheless, it is nowadays known that this author overestimated the masses of the rich clusters.

Kaiser [52] calculated - to order of magnitude - the CMB temperature fluctuations in an universe populated by Swiss-cheese lumps having various sizes. Combining this model with the available observational data on CMB anisotropies (1982), he concluded that the values of the spectral index - corresponding to the power-law spectrum of density fluctuations - must be greater than - 1.

Zel’dovich and Sazhin [123] used the SCM to compute the CMB anisotropies induced by stationary objects in an expanding universe. They found a fluctuation of 6T/ T - 2 x 10m7 for a distant stationary cluster of galaxies with mass M - 3 x 1O”M~. For clusters placed at large redshifts - the authors mention the redshift z = 3 - the effect rises up to AT/T - 2 x 10m6. The typical angular scale of the effect is estimated to be some tens of arcminutes. All the computations were done in a flat background.

Dyer and Ip [28] calculated the CMB anisotropies appearing in a lumpy Universe. The SCM was a generalization of that used by Dyer in 1976. Depending on the initial conditions for lumps, these authors distinguished synchronous and asynchronous models. The authors pointed out two

” These contributions correspond to various term involved in Eq. (6) of Rees and Sciama [go].

478 M.J. Fullana et al.

main differences between their calculation and those of Kaiser [52]. One of these differences is that Kaiser only considered the CMB photons passing by the centre of the spherical structures, while Dyer and Ip [28] also considered other directions, and the second difference appears as a result of the fact that Kaiser neglected the effect produced by structures located between the observer and his LSS. According to Dyer and Ip [28], this effect can be actually important, even dominant and, consequently, the CMB anisotropies need not be dominated by perturbations at the LSS; nevertheless, the masses of the objects producing such a dominant effect are not realistic (see Ref. [7]).

Estimates of the CMB anisotropies produced by voids of galaxies based on the SCM were also done. Opposite to the clusters, the voids are underdense inner regions surrounded by overdense compensating shells. Thompson and Vishniac [108] considered SCM voids in a flat universe. They estimated the anisotropy produced by one void and by a universe filled with voids. In the case of a Bootes-like void, the anisotropy amplitude was found to be AT/T - 10m6.

The integration of the differential equations corresponding to the null geodesics simplifies when the potential approximation due to Martinez-Gonzalez, Sanz and Silk [64] is used. In this approximation, the anisotropy in a certain direction is given by a formula involving three terms I2 : the Doppler term due to the infall toward the symmetry centre of the observer and the emitter, the Sachs-Wolfe effect, and an integrated effect along the null geodesics. The integral giving this last effect depends on the partial time derivative of the gravitational potential along the null geodesics. For compensated structures located at low redshifts, only the integral (depending on the gravitational potential generated by the structure itself) contributes to the CMB anisotropy. The potential approach was used by Martinez-Gonzalez and Sanz [63] to estimate the CMB anisotropies produced by different SCM realizations of Great Attractors and voids. The greatest amplitude, AT/T 2 lo-‘, was obtained in the case of the Super Great Attrac- tor [98]. When comparisons are possible, the results obtained by these authors agree with those obtained by Thompson and Vishniac [ 1081 and Dyer [27].

4.2.2. Estimates based on the TB model Let us now describe some estimations of the anisotropies produced by pressureless spherical

structures based on the use of the TBS. The first authors doing this type of estimate were Raine and Thomas [87]. These authors calculated the anisotropy produced by a large spherical linear overdensity having a present density contrast of 0.015 and a present size of lOOOh-’ Mpc. This overdensity was centred at a redshift Z w 3 and, then, the dipolar anisotropy produced by the motion of the Local Group towards the symmetry centre was computed. Raine and Thomas [87] concluded that the total motion of the Local Group with respect to the CMB only can be explained as the effect of one large linear overdensity in the case of very open universes. These authors did not consider non-linear structures. Paczyfiski and Piran [77], Wu and Fang [122], and MCszSuos and Molnar [70] also used the TBS in order to discuss the origin of the CMB dipole.

Until the early nineties, the exact differential equations of the null geodesics of a TB space- time were not numerically integrated. Independently, Panek [79] and Arnau et al. [6] built up two numerical codes for the integration of these equations. Neither linearizations nor approximating conditions are used in these codes. The difference between both integrations are due to distinct methods for introducing initial conditions and different numerical treatments of the TBS. Recently, it has been verified that, in a selected case, both codes lead to very similar results [36].

Panek [79] estimated the anisotropies produced by the Great Attractor, big voids, and rich clusters of galaxies. Various models for these structures and the background were considered.

l2 See formula (7) in Ref. [64].


This author concluded that the amplitudes of the gravitational anisotropy produced by the Great Attractor, Boiites-like voids, and rich clusters of galaxies are smaller than m 2 x 10e6, m 4 x 10A7, and m 6 x IO=+, respectively. In the case of rich clusters, this gravitational anisotropy was comparable with that appearing as a result of the Zel’dovich-Sunyaev effect.

Tbe main results obtained by the authors of this paper - by using the TBS - are summarized in a separate section.

4.2.3. Estimates based on other models We begin with some non-linear spherical models including pressure. Bertschinger [ 121 used self-similar spherically symmetric solutions of Einstein equations in

order to model cosmological structures. Self-similarity and spherical symmetry are the main limitations of these models, which admit a collisional baryonic component.

The so-called thin-shell void approximation was used by Sato and Maeda [96] to describe the evolution of voids with walls in an Einstein-de Sitter Universe. These walls involve a baryonic component. Sato [97] applied this model to estimate the anisotropies produced by voids. This author also studied the anisotropy produced by a distribution of non-linear observable voids located near the line of sight. The amplitude of the resulting anisotropy appeared to be a few times 10e7.

Finally, pressure is also considered in the numerical spherically symmetric models studied by Quilis et al. [86]. These authors used high resolution shock capturing techniques in order to solve the differential equations describing the evolution of spherical rich clusters with a baryonic component. They verified that the effect of pressure is not important and pointed out that the most important part of the gravitational anisotropy produced by a rich cluster should be generated during the brief period of cluster relaxation. The consequences of these facts were discussed. The amplitude of the gravitational anisotropy appeared to lie in the interval (10m6, 1O-5).

Non-spherical structures have been also considered in the literature. Atrio-Barandela and Kashlinsky [8] estimated the CMB anisotropy produced by the Great Wall. This structure was modelled by using the so-called Zel’dovich approximation. The potential approximation was used in order to estimate the anisotropy produced by the Great Wall. The amplitude of this anisotropy depends on both L2c and the bias parameter. Its order of magnitude is 10M6. This anisotropy is an integrated effect on the null geodesics due to the time variation of the gravitational potential created by the Great Wall. CMB anisotropies produced by similar structures located at higher redshifts were found to be smaller than those produced by the Great Wall itself.

On the other hand, Chodorowski [20] also used the potential approximation to compute the CMB anisotropies produced by flattened cosmic structures as the Great Wall. He did the calculations in a flat universe and obtained values of AT/T smaller or equal to 1 0m6 for the Great Wall.

Deviations from sphericity have been also considered. Argueso and Martinez-Gonzalez [5] obtained analytical formulae for the CMB dipole and quadrupole created by a linear spheroid. They concluded that spheroids centred very close to the observer’s position produce a much higher quadrupole than that of spheres with the same density and volume. Beyond a certain distance from the observer, spheroids and spheres behave in a similar way.

Non-symmetric isolated structures can be modelled by using N-body simulations. Van Kam- pen and Martinez-Gonzalez [ 1131 used this type of simulation to compute the gravitational potential corresponding to a distribution of collisionless particles. The amplitudes of the anisotropies produced by these distributions of matter were found to be a few times 10v6 (10d7) for Great Attractor-like objects (rich clusters). The background was assumed to be flat.


Bertschinger et al. [ 131 estimated the gravitational anisotropy (Sachs-Wolfe effect) produced by a Great Attractor progenitor located near the LSS. They used linear methods to describe the structure, which was normalized by using current data about the Great Attractor velocity field. In spite of the fact that the Great Attractor is a non-linear inhomogeneity, a linear normalization based on the velocity field of the surrounding galaxies is expected to be appropriate. The universe was assumed to be flat. The amplitude of the resulting gravitational anisotropies was found to be AT/T = (1.7 f 0.3) x 10m5 on an angular scale of lo and, consequently, the authors concluded that, if Great Attractor-like fluctuations of the gravitational potential are present elsewhere in the Universe, the resulting anisotropy should be detectable in current experiments.

4.3. Our applications of the TBS

The authors of this review have applied the TBS to estimate the CMB anisotropies produced by isolated cosmologically significant objects. In this section, we are going to describe: (i) the main features of the codes used in our computations, and (ii) the main results obtained with them. In a paper published by Amau et al. [6], a code for the integration of the differential equations satisfied by the null geodesics of the TB space-time was described. No approximations were done to facilitate the integration. In that preliminary paper, the code was tested and the amplitude of the anisotropy produced by voids without walls was estimated to be of the order of 10e7. In other papers, the same code was used to estimate the anisotropies produced by cosmological objects as the Great Attractor, the Bootes Void, and the progenitors of these big structures (see below).

The TBS depends on two arbitrary functions of the comoving radial coordinate M. These functions are tl(M) and f(M) (see Appendix C). Once these functions are fixed, the TBS is fully determined. ‘Iwo initial profiles are necessary in order to fix these functions. The initial density and velocity profiles were used in our calculations. Great Attractor-like and Boijtes Void- like objects were modelled. In all cases, the following initial energy density profile was assumed:

4 2 c Ei -= PBd i=l 1 + (RI&Y ’ (10)

where R is a radial coordinate. The values of the parameters Ei, Ri and a! must be appropriately chosen in each case (see Ref. [94] for the case of Great Attractor-like structures, and Ref. [36], for the case of voids). The subscripts B and d stand for background and decoupling, respectively. The redshift at decoupling time was assumed to be Zd = 103.

A second initial profile is necessary. The initial profile of the peculiar velocity, VP, is always used in our papers. This velocity is that corresponding to the chosen energy density profile in the case of vanishing non-growing modes [S 11. It has the following form:

VPd = (11)

where (. . .) denotes the mean value between R = 0 and R. After fixing the TB geometry using the above profiles, the differential equations of the null

geodesics can be solved. These equations were written in an appropriate form to facilitate numerical integration. In the final form, anisotropy production in a certain direction is described by a system of three differential equations (see Eqs. (3.14)-(3.16) in Ref. [6]).

Once the temperature corresponding to many directions is computed, the mean temperature and the contrasts with respect to this mean can be easily obtained see Ref. [6]; then the temperature contrast can be expanded in spherical harmonics (see Appendix A), and the dipole and the


relativistic Doppler quadrupole produced by the infalling motion of the observer can be removed to get the gravitational anisotropy.

For several admissible realizations of the Great Attractor (Bootes Void), the CMB gravitational anisotropy was found to be of the order of 10m6 (lo-?) (see Ref. [94,35]). The CMB gravitational anisotropy for Great Attractor-like structures placed at various distances from the Local Group were computed in Refs. [7,37,95]. The greatest CMB gravitational anisotropies appeared in the case of Great Attractor-like objects evolving in open universes with Qa 5 0.4 and located at redshifts z_ E [2,30]. The angular scales of the resulting anisotropies were found to be of a few degrees and the amplitudes reached the order - 10d5.

For Bootes Void-like objects located at z E [ 1, lo], the CMB gravitational anisotropies [36] reached amplitudes of a few times 10W6 in the case of open enough backgrounds (fia ( 0.4). The angular scales appeared to be of a few degrees as in the case of Great Attractor-like objects.

It was verified that the Great Attractor-like and Bootes-like structures producing the greatest CMB anisotropies were evolving in the mildly non-linear regime (contrasts near one) when the CMB photons passed near them [35]. Therefore, the time variation of the gravitational potential seems to be responsible for the resulting anisotropies.

In order to discuss our results, it should be considered that, at scales of several degrees, the amplitude of the observed anisotropies is - 10m5 (see Refs. [22,116,115,102,120]). Hence, the greatest anisotropies obtained in our calculations become of the same order or a little smaller than the observed ones. These results strongly suggest that, in open enough universes, structures placed at redshifts between 1 and 30 could give an important contribution to the total large scale anisotropy and, consequently, the idea that this total anisotropy is produced only by linear perturbations placed near the last scattering surface must be revised, at least, in the case of open enough universes (tic < 0.4). In this case, the estimation of the CMB anisotropy created by a distribution of structures - including the big ones we have studied - seems to be a very promising project.

5. DISCUSSION

In this section, we discuss the potential interest of the investigations of the CMB anisotropies produced by isolated cosmological structures.

The anisotropy produced by a certain inhomogeneity is superimposed on the effects of all the objects located near the solid angle subtended by the chosen inhomogeneity. These objects should produce anisotropies as a result of various effects and in a wide range of scales (see Section 3). Observations measure the total anisotropy and it is not easy to separate the contribution of a single structure. The angular scales of the anisotropies produced by an inhomogeneity range between the scale subtended by the inhomogeneity itself and a few times this scale, see Ref. [37]. For these scales, the anisotropies produced by the inhomogeneity can be compared with the total observed anisotropy. Three cases are possible: (ar) the inhomogeneity produces CMB anisotropies greater than that measured in current observations. In this case, the inhomogeneity should not exist in the Universe, except in the unlikely case of a strong cancellation of the anisotropies produced by this type of inhomogeneities, (B) the structure produces anisotropies of the same order of - or a little smaller than - the observed one. In this case, the contribution of this type of structure to the total observed effect is significant and, consequently, general statistical techniques should be used in order to estimate the total effect of a realistic distribution of inhomogeneities and, (y) the anisotropy produced by the structure becomes negligible and no conclusions can be reached at all.


In case (a), predictions about the anisotropy produced by an unique inhomogeneity could be useful in order to get some bounds on the parameters defining both the structures and the background. This is possible when the anisotropy produced by a certain type of inhomogeneity is too great to be compatible with current observations for some values of the mentioned parameters. Then these values can be ruled out and, consequently, some bounds arise. For instance, Kaiser [53] obtained an upper limit of -1 for the spectral index of the power law spectrum of energy density fluctuations. Goicoechea and Sanz 1411 calculated upper limits on some relevant physical quantities by using a spherical linear model for large scale structures. The same authors (1985) established some limits on the relative mass fluctuations of large scale inhomogeneities - with radius N 103h-’ Mpc - located outside our present horizon. Atrio-Barandela and Kashlinsky [S] claimed that observations towards the Great Wall could be useful to constrain the density parameter, $20, and the bias parameter. Similar proposals were presented by Fullana [35] and Fullana et al. [38], who found that the anisotropy of Great Attractor-like objects and Bootes-like objects strongly depend on the density parameter and, consequently, they claimed that, if such an anisotropy could be detected in future in the direction of one of these objects, the $20 value could be bounded. Nevertheless, this type of conclusion should be seen with caution as a result of the fact that the anisotropy of any structure is superimposed on other anisotropies in such a way that separation is not easy (see Ref. [36]).

In case (/I), relevant effects arise. For example, Dyer and Ip [28] claimed that some structures located between the LSS and the observer could produce relevant contributions to the total CMB anisotropy. Unfortunately, the structures considered by these authors do not correspond to observed objects. Recently, the authors of this paper predicted [7,95,36] significant amplitudes for the anisotropies of Great Attractor-like and Boiites Void-like objects located at redshifts between 1 and 30 in open universes with density parameter no 1. 0.4. The angular scales of these anisotropies are of a few degrees. The main question is: What would be the effect of a distribution of such structures in an open universe? This question is being currently studied.

Various authors have performed calculations of the anisotropy generated by one cosmological inhomogeneity with the essential aim of estimating the total anisotropy produced by a distribution of these structures. Some examples have been briefly mentioned in Section 4. Sato [97] used the model proposed by Sato and Maeda [96] to estimate the anisotropy of a set of voids located near the line of sight. Thompson and Vishniac [108] first calculated the anisotropy produced by one void - with the SCM - and, afterwards, they used the results to discuss the production of CMB anisotropy in an universe filled by voids. Dyer and Ip [28] used the SCM to lump the universe. Although these efforts have given some qualitative ideas about the anisotropy produced by inhomogeneity distributions, it should be recognized that the computation of the anisotropy produced by an isolated inhomogeneity is not the best way to estimate the anisotropy produced by a realistic distribution of structures. Distributions of isolated spherical structures are not realistic. We think that the anisotropy produced by a realistic distribution of non-linear structures should be calculated by using well motivated power spectra for the distribution of non-linear density fluctuations and admissible hypothesis about the statistical features of such a distribution. In the case of flat universes and Gaussian statistics, this type of calculation has been addressed by Anninos et al. [4], Martinez-Gonzalez et al. [65,66] and Tuluie and Laguna [ill]. Since the non-Gaussian features of the distribution of non-linear structures are not well known, and there are limitations in our current knowledge of a non-linear power spectrum including all the significant scales, general statistical estimations of non-linear anisotropies are troublesome. In the case of open backgrounds, these estimations are more difficult than in the flat case. Further researches, perhaps numerical simulations, are necessary in order to rigorously treat the case of open universes with non-Gaussian statistics and realistic power spectra.

Looking for the imprints of non-linear structures on CMB

APPENDIX A. EXPANSIONS IN SPHERICAL HARMONICS

483

In this appendix, our notation in relation to the CMB dipole and quadrupole is fixed. This notation is compared with other ones used in the literature.

Any function f (8, #J> of the spherical coordinates 19 and 4 - with proper mathematical conditions accomplished - can be expanded in spherical harmonics, Y/,(8, $J), as follows:

f (e,#) = 2 f: a;“L(@, 4), 1~0 m=-1

(12)

where a;f are the numerical coefficients

n 27i

al m=

ss f(e,~)Y1,(8,~)d~sinede. (13)

0 0

In some cases, the bt,,, coefficients defined by the following relations become more appropriate:

a;” = Jj J-(bt,,, - ibt-m), al”’ (-lY

= T(btm + ibt-m). (14)

If angular celestial coordinates a! (right ascension) and S (declination) are considered, angle (Y (8) plays the role of the angle 4 ((n/2) - 0) involved in (12).

Temperature fluctuations, ST@, S), are usually defined as follows:

(15)

where To@, 6) is the present temperature in the direction (01,6) and (. . .) stands for a mean temperature over all the directions.

The function 8~ (a!!, 8) is usually expanded as follows [ 1041:

ST(a,S)=To+T,cosa!cosS+Tysincwcosg+T,sin6+~Q,(3sin2S-1)

+Q2 cos (Y sin 2S + Q3 sin Q sin 26 + Q4 cos 2o cos2 S

+ Q5 sin 2c~ cos2 6 + higher order (1 < 2), (16)

where To is the monopole, which is not physically significant. Expansions (12) and (16) are identical. The relationship between the bl, coefficients and those

of (16) is as follows: The dipolar components are

(17)

(18)

(1%

Then, the dipole, l3, of &T is

(20)


The quadrupolar components are

(21)

Q2=- J

&b2,,

Q3 = -

(22)

(23)

Q5 =

and the quadrupole Q of ST is defined by

(24)

(25)

There exists an expansion similar to (16) in terms of the galactic coordinates 1 and b or in terms of any pair of appropriate spherical coordinates.

Finally, the quadrupole used by Smoot et al. [ 1021 to present COBE data is

Q E LQ. mE Jr5

(27)

APPENDIX B. THE SWISS-CHEESE MODEL

In the SCM, Friedmann-Robertson-Walker cosmological solutions are matched to the spherically symmetric Schwarzschild’s one. An universe involving a single spherical structure is thus described. Besides, a universe populated with structures of various sizes - like a Swiss cheese- can be built up by using the SCM (these structures do not interact). In this appendix, we describe the mentioned matching of solutions, namely, the SCM. Our description is based on a more ex- haustive one due to Kaiser [52], which seems to be clear and general. This version of the SCM is appropriate in the case of overdensities. Underdensities require another parallel treatment [log].

Let us begin with the description of the background universe. The Friedmann Robertson Walker solution of the Einstein equations is assumed. This solution corresponds to an universe containing a uniform pressureless fluid. The line element in comoving spatial coordinates (taking c=G= 1)is

ds2 = -dt2 + a2(r) [ dw2 + $(w)da2 1 , (28) where t is the proper time, 0 and 4 are spherical coordinates, w is a dimensionless radial coordinate, function Sk (0) takes on the form sin o, o, sinh w in cases k = 1 , 0, - 1, respectively, and the solid angle element, da, is defined by da 2 = de2 + sin2 8d#2. The function a(t) is the well-known scale factor, whose evolution is determined by the Friedmann equation:

’ (2%


where M s $lspa3 = constant and p is the energy density. The parametric solution of this equation is

a = Mk[l - Ck(tl)l, (30)

r = MHtl - &(l~)lv (31)

where ck(r]) c cos Q, n, cash r] fork = 1 , 0, - 1, respectively. At early times, rl << 1, the relation a cx t2i3 is satisfied for any k value.

Within the homogeneous background, a sphere of dust is carved out and, then, it is replaced by a smaller expanding dust sphere having the same gravitational mass. The interior will behave like a finite sphere in another Friedmann universe (with different M). The exterior is unperturbed and, on account of Birkoff’s theorem I3 the spherically symmetric vacuum space-time in the , intermediate region is Schwarzschild.

Let us see how the fitting of the solutions is done. Let us consider two observers A and B lying on the outer and inner edges of the vacuum shell. These observers can be considered as comoving observers in their respective Friedmann universes or, equivalently, as test particles following time-like radial geodesics in Schwarzschild geometry.

Let us write the line element for Schwarzschild geometry in the usual xi = (t, r, 8,#) coordinates,

It can be seen that k’ E (1, 0, 0,O) is a Killing vector which provides us with a constant of motion k . U = U-J = -E, where I/ is the 4-velocity of the test particle. From this, and the condition U . U = - 1 for time-like geodesics, one gets the equations of motion

2m E2-V2=1--q (33)

r

where V 5 dr/dt, and

(34)

Eq. (33) is essentially the Friedmann (Newton) equation, which has the following parametric solution:

r = %[I - ck(v')l,

r=- $ [d - Sk(d)],

where S = dm. Now, the above 4-geometries are to be matched together at their intelfaces. This matching is

established by demanding that: (i) if one measures the area of an element of the interface in which A or B lies, the result obtained is independent of the metric used in the calculation. On account of (28) and (32), this condition implies the following relationship between Schwarzschild and Friedmann radial coordinates: r = B&(w). Then, comparing solution (30) and (31) with (35)

I3 See Section 99 of Ref. [l lo].


and (36) one finds MS2&(w) = m, 17’ = Q; (ii) the proper time elapsed since the Big Bang as measured by an observer at an interface (as A and B) is the same for metrics (28) and (32). Then (30), (31) and (35), (36) imply m = MS3, so S = Sk(w) and m = !npr3. In conclusion, the Schwarzschild solution for mass m matches smoothly on to a Friedmann region with parameter M at a comoving radius w via m = MS:(w). The specific energy of the test particle is E = C&w). Eqs. (35) and (36) provide F-(T) parametrically. One can also integrate (34) to get the full solution for the orbits (see Ref. [52]).

APPENDIX C. THE TOLMAN-BOND1 SOLUTION

The TBS is a particular case of the general non-stationary spherically symmetric line element

ds2 = -e2”dt2 + e2AdM2 f Y2(de2 + sin2 ed#2), (37)

where u, A and Y are arbitrary functions of t and M. Coordinates M, 8 and 4 are comoving. Kramer et al. [58] gave the solution of Einstein’s field equations for the above line element and the energy momentum tensor of a pressureless perfect fluid. These authors presented the TBS as a particular case in which the line element (38) reduces to

ds2 = _dt2 + caRlaMj2 r2(M)

dM2 + R2(t, M)(de2 + sin’ 0d$2),

where R(t , M) is a non-comoving radial coordinate and r2( M) is

r2(M)=1+f(M)=1+ 2 2GM

--I R (39)

G being the gravitational constant and f(M) a certain conserved total energy within a sphere of radius M.

The explicit form of the TBS depends on the value of f(M). In the case f(M) > 0, the solution has the following parametrical form:

R = f(M) cos GM ( hrl-11,

t+tl(M)= rf(%:3,2 tsinh v - rl).

For f(M) K 0 the TBS is

R= GM

--(1 - cosrj), f(M)

t+tl(M)=

Finally, in the case f(M) = 0, one has

[t + tdM)12’3.

(40)

(42)

At each time t, a shell is a spherical surface with constant M. Shells are centred at M = 0. The spherical symmetry implies that all physical quantities are constant on each shell.


As can be seen from (41), (42) and (43), TBS depends on two arbitrary functions f(M) and 11(M) which are to be determined from two initial profiles. A Friedmann-Robertson-Walker universe can be considered as a special Tolman-Bondi space-time, and any cosmologically admissible Tolman-Bondi space-time must tend to a Friedmann-Robertson-Walker universe as the spatial distance from the symmetry centre increases.

In spite of the fact that TBS is an exact solution of the Einstein equations, the use of this solution requires numerical treatment, at least, in the fully non-linear case (see for instance Refs.

W31).

ACKNOWLEDGEMENT

This paper has been supported by the project GV-2207194.

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