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
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),
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.
Since the discovery of the Cosmic Microwave Background (CMB) radiation in 1965 , 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. .
468 M.J. Fullarm et al.
Let us make some comments about expansion. The Universe was (and is) expanding following Hubbles 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 Hubbles Law: d(t)/& = Hi(t), where l(t) measures the spatial distance between two galaxies and H is Hubbles 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; af- ter 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 . 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 decou- pling after each reionization.
After discussing the origin and nature of the CMB radiation, let us analyze the reasons mak- ing its analysis one of the most important topics in Cosmology. This radiation is found to be very isotropic (see for instance Ref. ). 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 inho- mogeneities 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 Einsteins 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  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 under- stand 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  and interpreted - in an associated letter by Dicke, Peebles, Roll and Wilkinson  - as the detection of the b&k-body cosmic radiation.
After the first observation at 7.35 cm, new measurements at different wavelengths were nec- essary in order to verify that the detected signal had a Planck spectrum. Some of the first ob- servations are due to: Roll and Wilkinson , Field and Hitchcock , Thaddeus and Clauser [ 1071 and Penzias and Wilson  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  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 Gamows [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 :
(a) The best-fit black-body temperature is 
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 poten- tial . The limits here  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 impor- tant 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 generat- ing this type of deviations. Physical processes producing too great deviations must be rejected. (See for example Ref. , or [120,121]). Observational constraints on the anisotropies set rel- evant 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 obser- vational 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.
Looking for the imprints of non-linear structures on CMB 471
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 mea- surements [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 
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...