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Large Scale Structure PRAMANA ­c Indian Academy of Sciences Vol. 53, No. 6 — journal of December 1999 physics pp. 977–987 Large scale structure SOMNATH BHARADWAJ Department of Physics and Meteorology and Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur 721 302, India Email: [email protected] Abstract. We briefly discuss some aspects of the problem of forming large scale structures in the Universe. The basic picture that initially small perturbations generated by inflation grow by the process of gravitational instability to give the observed structures is largely consistent with the observations. The growth of the perturbations depends crucially on the contents of the Universe, and we discuss a few variants of the cold dark matter model. Many of these models are consistent with present observations. Future observations hold the possibility of deciding amongst these models. Keywords. Cosmology; theory; large scale structure. PACS Nos 98.65.Dx 1. Introduction Our present understanding of the Universe is based on two important assumptions 1. the Universe is homogeneous and isotropic on large scales 2. gravitation governs the dynamics of the Universe on large scales. The time-evolution of such a Universe can be described by just one function of time – the ´Øµ scale factor a which tells us about the expansion or contraction of the Universe (these ´Øµ being the only two possibilities which preserve assumption (1). The evolution of a is governed by the laws of gravitation eq. (4), and the observation that all the distant galax- ies are moving away from us leads us to the conclusion that the Universe is expanding [1]. A large variety of observational evidence supports the picture of an expanding Uni- verse that is very uniform on large scales [2], most important being the cosmic microwave background radiation (CMBR) [3] which is a nearly isotropic black body radiation of tem- :7¾6à perature ¾ . This radiation is believed to be cosmological in origin, and the high degree of isotropy of the CMBR is an indication that the Universe is indeed very uniform ½ Ì ´Øµ » a ´Øµ on large scales. In addition, the temperature of the radiation falls as CÅBÊ ´Øµ as the Universe expands, indicating that the Universe was hotter in the past when a was smaller. This model (the hot big bang model) has been extremely successful in explaining many features of the observed Universe, of which the accurate predictions of the abun- dances of the light nuclei and the number of neutrino families are of particular importance. This model is currently accepted as the standard model of cosmolgy. 977 Somnath Bharadwaj The observed Universe is not uniform and exhibits structures on a variety of scales, 5¼ ÃÔ c ½ÅÔc starting from galaxies , clusters of galaxies , voids and superclusters ¾¼ ÅÔ c and larger. There is some evidence that the Universe tends to homogeneity ½¼¼ ÅÔ c on scales > [2,4]. The problem at hand is to understand the formation of these structures within the framework of the standard model of cosmology. 2. Brief introduction to observations Before going on to the theory of structure formation, we very briefly touch upon a few of the observational aspects of the problem. The most direct method of observing the large scale structures (LSS) in the Universe is through observations of galaxies. The first step in this process is to identify and determine the angular positions of all the galaxies in a certain part of the sky with fluxes within a specified range. The next task is to determine the distances to the galaxies. Direct determination of the distances to galaxies is very difficult and one uses the fact that the radiation from distant galaxies gets stretched to the red due Þ =¡= to the expansion of the Universe to determine the distances. When the redshift e is small it can be interpreted as a Doppler shift and it is related to Ö , the distance to the emitting galaxy, cÞ = Ú = À Ö; ¼ (1) Ú À = ½¼¼h kÑ=×=ÅÔ c h ¼:65 where is the recession velocity of the galaxy and ¼ ; is the present value of the Hubble parameter [5]. These observations of angular positions and recession velocities (³ distance) constitute a redshift survey. Figure 1 shows the dis- tribution of galaxies in the Las Campanas redshift survey (LCRS) [6], which is one of the largest surveys currently available. The next step is to identify and quantify the structures in the distribution of galaxies. The deviations from a uniform distribution (fluctuations) are usually quantified using Æ ´Üµ Æ g ; ¡ ´Üµ= g (2) Æ Æ Æ ´Üµ Ü where g is the number density of galaxies at the position in the survey and is the ¡ ´Üµ mean number density of galaxies. The Fourier transform of g Z ik¡Ü ¿ Æ ´kµ= e ¡ ´Üµd Ü g g (3) ¾ È ´k µ=jÆ ´kµj g is used to define the power spectrum g which is very commonly used to characterize the statistical properies of the LSS [7,8]. Figure 2 shows the power spectrum for the LCRS. It also shows the power spectrum for the APM survey [10] which has ob- servations of only the angular positions of the galaxies and does not have any information about the distances. ¿ È ´k µ The quantity k gives a rough estimate of the fluctuations in the distribution of ½ galaxies at the length-scale k . Using galaxies as tracers to map the LSS has the problem that the relation between the distribution of galaxies and the distribution of matter (which is believed to be mostly 978 Pramana – J. Phys., Vol. 53, No. 6, December 1999 Large scale structure Figure 1. This figure shows the distribution of galaxies in the Las Campanas redshift survey. invisible i.e. dark matter) is not known. There are reasons to believe that the two distri- butions might be different and the simplest (not the most realistic)way this may be taken into account is through a linear bias parameter b which relates the fluctuations in the matter ¡´Üµ ¡´Üµ=b¡ ´Üµ distribution to the fluctuations in the distribution of galaxies i.e. g . The power spectrum for the matter distribution is then related to the galaxy power spectrum ¾ È ´k µ È ´k µ=b È ´kµ g g through the relation (e.g. [11]). Another very important effect arises because the inhomogeneities in the matter distribu- tion effects the motion of the galaxies. As a consequence, the velocity of any galaxy has Ú = Ú · Ú Ú È À two parts À ,where is the part which arises due to the expansion of the Ú Universe eq. (1) and the peculiar velocity È which is due to the inhomogeneities in the matter distribution. This affects the redshifts and hence the distance determination to the Pramana – J. Phys., Vol. 53, No. 6, December 1999 979 Somnath Bharadwaj 1000 CDM 100 10 0.001 0.01 0.1 1 Figure 2. The APM power spectrum (filled squares) and the best fit to the LCRS power =½:¾ spectrum (dashed-dotted thick curve). We assume b for both of these. We also = ¼:5 use h . The power spectra for several decaying neutrino models all obtained ¾ Ñ Ø Ñ Ø d d by varying and keeping a constant and the CDM power spectrum are also shown. galaxies. The final effect is that the galaxy power spectrum determined from redshift sur- veys turns out to be anisotropic in its k, whereas one expects the actual matter and galaxy power spectra to be isotropic. This effect has to be taken into account when analyzing redshift surveys, and it can be utilized to extract some very important cosmological infor- mation [8,12]. 3. Theory It is currently believed that the observed LSS was formed from some initially small fluc- ´¡´Ü;Ø µ ½µ tuations i which grew by the process of gravitational instability to produce the structures which are observed now. 980 Pramana – J. Phys., Vol. 53, No. 6, December 1999 Large scale structure 3.1 Initial fluctuations A number of possible mechanisms have been proposed to explain how the initial fluctua- tions were generated, and inflation has been most successful amongst them. Most models of inflation make three generic predictions [13] ¡´Ü;Ø µ 1. the initial density fluctuation i is a Gaussian random field i.e. the phase Ô i´k;Ø µ i ´k;Ø µ Æ ´k;Ø µ= È´k;Ø µe i i i in is random, with no correlation between the phases at different values of k. È ´k; Ø µ=Ak 2. The initial power spectrum is of the form i . 3. The Universe is spatially flat. 3.2 Time evolution of the fluctuations The dynamics of the Universe is governed by the gravitational field produced by the dif- fernet constituents of the Universe. For a uniform Universe this gives us the equation governing the evolution of the scale factor Ö Ô d 8G 4 !´µ: a´ µ= ´µa ´µ=À ¼ (4) d ¿ Ê Ø ¼ ¼ ´½=a´Ø µµ dØ ´Øµ= ´ µ where is the conformal time. and is the total energy density ¼ of the Universe. In general this will be the sum of the contributions from the different È ´ µ= ´µ constituents of the Universe i.e. a . We next introduce the critical density a ¾ ´ µ =¿À =8G c¼ c which has the value at present. It is often convenient to express ¼ a ¼ the present value of the density a of any particular component in terms of the critical ª = = ¼ a¼ c¼ density using the density parameter a . The normalization of the scale factor is chosen so that its present value is ½. The quantity 4 ! ´ µ=´µa ´µ= ¼ c used in (4) is introduced as a convenient way of parametrizing the time evolution of the density and it has been chosen so that it has the value ½ at present.
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