1. THE LARGE-SCALE STRUCTURE OF THE PRESENT-DAY UNIVERSE

Transcription

1 Section 1 1. THE LARGE-SCALE STRUCTURE OF THE PRESENT-DAY UNIVERSE We have discussed during the Cosmology course of the Astronomy Degree and in Sect. 0 the properties of an expanding Universe under the assumptions formalized in the Cosmological Principle: the Universe can be described as a completely homogeneous and isotropic entity. We have discussed there in some detail the observational evidences supporting the principle. We have seen that such simplified assumptions entail enormous simplifications in the set-up of the mathematical tools to treat the geometry of the expanding space-time, in the definition of suitable distance measurements (the redshift), etc. Based on these assumptions, we have first derived the (Robertson-Walker) metric tensor, we have then derived the dynamical equations ruling the cosmic evolution of the Universe as a whole and, finally, developed (Sect.11) a brief historical overview of the main phases of the expansion from the Big Bang to the present time and mentioned a few of the physical processes taking place at the various epochs. Of course, would such be the exact conditions at any point/epoch of the space-time, we would have no way to get any complexity in the Universe, such as galaxies, stars, planets, life, and fundamental observers eventually. It is just because, for reasons to be understood, some deviations from perfect homogeneity in the cosmic fluid have developed at some stages that all these structures could have originated. The main purpose of this Master Degree course is to discuss both observational evidences and theoretical analysis tools concerning the development of the cosmological large-scale structure, starting from early highly homogeneous and undifferentiated conditions traced by the Cosmic Microwave Background to the highly differentiated and structured present-day Universe. We will also make an extensive attempt to understand physical mechanisms originating this evolution, which refer to the very early phases of the cosmic expansion. The observations and theories showing and explaining this entire story make one of the triumphs of modern science. Large Scale Structure of the Universe 1. 1

2 Chapter THE CORRELATION FUNCTIONS SPATIAL AND ANGULAR CORRELATION FUNCTIONS At first sight, one may be surprised that we have accepted the validity of the Cosmological Principle and at the same time, we are here to discuss the structures in the current-day highly inhomogeneous Universe and their evolution. We have just qualitatively mentioned the explanation of this apparent paradox in the previous course: the inhomogeneity in the matter distribution appears to be very strong on small cosmological scales, but degrade quickly on larger scales, such that by averaging the cosmic field on scales larger than a few hundred Mpc we obtain an essentially homogeneous Universe. This is again qualitatively illustrated by the plots of the 3-dimensional distributions of galaxies in Figure 1 and, which refer to two different samplings of the galaxy distribution with different depths. The first figure shows the projected distribution of galaxies from the CfA redshift survey contained in a strip of sky between 8.5 and 44.5 degrees declination and within a distance of Km/s (from Hubble law this means a distance of 400*z=400*15000/ Mpc 10 Mpc). On scales of about several tens of Mpc we see an highly structured distribution of galaxies, with filaments, voids, walls of increased density of galaxies, and the well-known fingers of God, showing regions containing structures with strong orbital motions of galaxies (the closest to the center is the Virgo cluster of galaxies). Figure shows similarly projected distributions of galaxies from the Las Campanas redshift survey, the main difference here compared to the previous being that the limiting distance is now Km/s (800 Mpc): if we look at this on small scales we see a similar pattern of clusters and voids in the galaxy distribution, but on the largest scales the distribution appears now as highly homogeneous. Perhaps the strongest evidence of homogeneity in the distribution of cosmic sources on the largest scales is that of radiosources at an average redshift z 1 in Figure 3, showing indeed a completely random distribution of points. In our previous discussion, galaxies and radio sources are assumed to be good tracers of the cosmic structure. Of course, this is a strong assumption, because these objects make only a minority fraction of the mass-energy content of the Universe (by far dominated energetically by dark matter and dark energy on the largest scales). However, this is quite a common one, and we will see later how to parametrize the uncertainties related with this assumption. Large Scale Structure of the Universe 1.

3 What we have seen so far are just general qualitative considerations on the large scale structure. What we need here and for all our subsequent discussions are quantitative measurements of it. This topic makes an important subject of modern cosmology, the statistical cosmology. Large Scale Structure of the Universe 1. 3

4 The simplest mathematical tool to quantify the cosmological large scale structure are the correlation functions. The use of these functions was first introduced by Totsuji and Kihara in 1969 with reference to stellar surveys, but its systematic exploitation was performed by Peebles and collaborators. Generally speaking, the correlation functions provide a description of the clustering properties of a distribution of points in multi-dimensional spaces: for our purposes, we will assume that such points correspond to the positions of galaxies or cosmic sources in the sky. As we will see, Large Scale Structure of the Universe 1. 4

5 the correlation functions have a very similar meaning to the covariance functions, the only difference being that the former refer to discrete sources, the latter to continuous distributions of "fluids". Figure 3. Distribution of the brightest radiogalaxies in the northern emisphere at wavelengths of λ=6 cm [from Gregory & Condon 1991]. North Celestial Pole at the center. Following Coles and Lucchin (00), let us first define the two-point spatial correlation function, ξ ( r1), in terms of the joint probability δ P to find one galaxy in a differential volume dv 1 and a second galaxy at a distance r 1 in a differential volume dv : d P = n [1 + ξ( r )] dv dv [1.1] V 1 1 where nv is the mean number density of sources. The definition and calculation of this quantity requires the information on the 3-dimensional distribution of objects, that is the angular position in the sky and the distance. Because of the general isotropy of the Universe stated by the cosmological principle, we have made in [1.1] the simplifying assumption that r 1 is just a scalar and not a vector, which entails that the clustering pattern is the same on average in any directions. Of course, we may envisage more general situations in which this is not the case and that argument has to be treated as a vector. The meaning of [1.1] is that if we consider a random distribution of points, then ξ ( r ) = 0, in case of a positive clustering (sources tending to stay together, like 1 Large Scale Structure of the Universe 1. 5

6 ξ ( r ) > 0, on the contrary this becomes a negative quantity friends in a train), than 1 for negative clustering (sources tend to stay apart one from the other). The function tells us the excess or defect of probability to have closeby objects. The mean number of galaxies within a distance r of a given object is : 4π r 3 N() r = nvr + 4 πnv ξ( r 0 1) r1 dr1 3 [1.] where the second addendum gives us the excess (or defect) in the number of sources compared to a random distribution. Again because of the cosmological principle, if we average [1.] over large enough volumes we get the condition that the second addendum should converge to zero, or, saying this otherwise, n V is defined as the source density averaged over large enough volumes to make the second addendum to converge to zero. So if the -point correlation function assumes positive values on a given range of scales, it should get negative values on other scales to bring the integral to zero on average. The definition of the -point spatial correlation requires the knowledge of the distance for all catalogued objects. If instead one has only the -dimensional catalogue positions projected on the sky, one can define the two-point angular correlation function w( ϑ ) in relation with the combined probability to find a galaxy dω (a source) in a elementary cell of area dω and a second one in a cell 1 separated by an angle ϑ : 1 d P = n [1 + w( ϑ )] dω dω [1.3] Ω 1 1 where n Ω is the areal density of sources projected in the sky per unit solid angle. In a very much analogous manner, we can define correlation functions of higher order. With reference to the definition in [1.1], we define the n-point correlation function as the joint probability to find n galaxies in n elementary volumes dv n, separated by r n distances one from the other. Of course, this probability includes contributions from correlation functions of lower-order than n. The reduced correlation function ς quantifies that part of the probability not dependent on the lower-order terms: for the 3-dimensional case, we have for example: d P = n [1 + ξ( r ) + ξ( r ) + ξ( r ) + V( r, r, r )] dv dv dv [1.4] V The reduced function ς ( r ij ) [or ξ ( ) (3) r ij ] then provides the excess probability to get triplets of sources for a given set of separations r ij with respect to the case of a simple in which there is distribution with two-point correlation given by the function ξ ( ) r ij Large Scale Structure of the Universe 1. 6

7 no favor or disfavor for arrangements with specific triangular forms. The three point angular correlation function is given, in a completely analogous way, by d P = n [1 + w( ϑ ) + w( ϑ ) + w( ϑ ) + z( ϑ, ϑ, ϑ )] dω dω dω Ω If the source positional distributions are Gaussian-distributed 1, then they are completely characterized by their two-point correlation function, or equivalently, by their power spectrum (see Chap. ). If they are non-gaussian, there is additional statistical information in the higher-order correlation functions, which is not captured by the two-point correlation function. In particular, the 3-point correlation function (or its Fourier counterpart, the bi-spectrum, see below) is important because it is the lowest-order statistic that can distinguish between Gaussian and non-gaussian perturbations. In principle, we can demonstrate that the knowledge of all n-point correlation functions completely defines the clustering properties of a distribution of points. However, in practice the calculation of correlation functions of order higher than is cumbersome (though simple in principle), so that the -point function is typically used for the quantification of the large scale structure of the Universe. In all the above, the clustering in the Universe is considered to be statistically homogenous, in that, on average, the source clustering does only depend on the source-to-source relative distances and not on the absolute sky position where to estimate the w( ϑ ) or ξ ( r 1 1) functions. If in addition it is Gaussian, this also implies that the variables ϑ and r 1 1 of the two functions are not vectors, but just the moduli of vectors, because otherwise the basic assumption about general homogeneity and isotropy of the Universe would be violated. Also in this sense, the so-called ergodicity assumption about the random field, that is the ensemble average (averaging over many realizations of the distribution, e.g. over many Universes) can be replaced by a spatial average. Only this assumption allows us to really measure the correlation functions (and the cosmological large-scale structure itself), as discussed in the next Sect. 1 We refer to the term of Gaussianity to indicate a specific condition of symmetry in the mass distribution, that is that, on average, for any positive fluctuation of a given size and amplitude (an excess over the average density), there should be an equivalent negative one with the same properties and negative sign. See Chapt. 3.3 below for further discussion of this topic. Large Scale Structure of the Universe 1. 7

8 1.1. OBSERVATIONS OF THE GALAXY ANGULAR CORRELATION FUNCTIONS Studies of the correlation functions of magnitude-limited samples of galaxies selected from Schmidt telescope's plates over large sky areas (e.g. from the Lick Observatory plates or the Palomar Observatory Sky Survey, POSS) already during the '70, '80 and '90's have offered the first systematic means of quantification of the cosmological large scale structure. The results of these analyses have offered precise measurements and proofs of the validity of the Cosmological Principle, whose basic assumptions date back by years to Einstein, Milne and others. The usual method to estimate the angular correlation function is to take a flux- (or magnitude-) limited sample of N galaxies in the area, for each galaxy in the sample to calculate the angular distances to all other objects in the sample, and to repeat the measurements for every sample object to finally get N N angular distance measurements, whose distribution is binned in distance intervals. Such distribution is compared to that of a random distribution of points on the celestial sphere obtained with random-number generators. The estimate is then performed by calculating ndd( ϑ) w( ϑ) 1 n ( ϑ) [1.5] or, more robustly against effects at the border of the survey sky region, ndd( ϑ) w( ϑ) 1 n ( ϑ) ndd( ϑ) nrr( ϑ) or w( ϑ) 1 [1.5b] n ( ϑ) The various n values above correspond to the number of pairs in the angular distance bin ϑ both for the real (D) and the random (R) samples. Most naturally, the number of observed and simulated objects may be the same, but this needs not to be. The RR DR DR Between 1947 and 1957, W.W. Shane and C.A. Wirtanen photographed with the Lick Observatory 0-inch Double Astrograph wide-angle photographic camera the northern 3/4 of the sky, taking first-epoch plates for 1,390 fields from the North Celestial Pole down to 33 degrees south declination. (all of the sky visible from Mount Hamilton) in the framework of a proper motion survey (Shane expected that 50 years would have to pass before the stars would have moved enough to warrant taking second-epoch photographs). Meanwhile, the Lick plates could be used for a pioneering project in observational cosmology: mapping the distribution of galaxies on the sky. Between 1948 and 1959, Shane and Wirtanen identified and counted nearly one million faint, distant galaxies on the Lick plates. Sky maps and statistical analysis, published in 1967, revealed the remarkable discovery that galaxies clump and cluster into great networks and filaments, with comparatively empty space in between. As the first large, wide-angle galaxy survey, the Shane-Wirtanen galaxy counts became the fundamental database for a whole generation of cosmological research. Similar investigations have been conducted by F. Zwicky based on the POSS plates. Large Scale Structure of the Universe 1. 8

9 various ways to estimate the correlation function give the same results for large enough samples. Note finally that the same algorithms are used to estimate the spatial correlation function ξ ( r ij ) when the observer to galaxy distances would be known. Similarly, from [1.4] we can estimate the 3-point correlation function operatively as n ς( r, r, r ) = ξ( r ) ξ( r ) ξ( r ) 1 DDD nrrr Figure 4 shows some of the results when these techniques are applied to large galaxy samples selected from Schmidt plates using the UK Automated Plate Machine (Maddox et al. 1990). The analysis of these projected catalogues has shown that, over a large interval of angular values ϑ, the -point angular correlation function has a power-law shape w( ϑ). Aϑ δ ( ϑ ϑ ϑ ; δ ) [1.6] min max Large Scale Structure of the Universe 1. 9

10 At ϑ ϑ the shape of the function steepens, and for even larger values it needs to max go negative. The amplitude A depends on the characteristic distance of the galaxy sample. In the left panel of the figure data-points with the various symbols refer to different magnitude limits, corresponding to different average distances of the sample (the higher data-points correspond to fainter magnitude limits). We see that the shapes of the w( ϑ ) functions do not change (statistically homogenous clustering). On the right hand panel all w( ϑ ) have been re-scaled to the average distance of the Lick galaxy sample, assuming homogeneous clustering, with the scaling law: w d d w d d 0 ( ϑ, ) = L ϑ, 0 d d0 [1.7] This scaling relation is easily proven by considering, as in the graphical scheme below, that the same spatial scale l will appear under different angles when put at different distances, according to: θ = θd d. 0 Indeed in an Euclidean universe that we assume here, hence limited to low redshifts, we have: d d d0 d w( ϑ, d) dϑ = d0 wl( ϑ, d0) dϑ w( ϑ, d) = wl ϑ, d0 d0 d 0 d that proves eq. [1.7]. This includes another scaling factor coming from the fact that, if the source volume density keeps constant on average with distance, as assumed here, the excess (or defect) on number of pairs compared to a random distribution 3 will be proportional to the survey volume d. With the scaling law from [1.7] and the luminosity distance (A.4), it is easy matter to re-scale the observed w( ϑ ) functions corresponding to different survey magnitude limits, as it is done in Fig. 4b. Relation [1.7] can be alternatively retrieved from the Limber equation below [1.8] as well. Large Scale Structure of the Universe 1. 10

11 Note that the observed scaling relations of the angular correlation function with the survey depth, for galaxy surveys up to z , offer a further proof of the general homogeneity of the Universe on large scales and of the reproducibility of the clustering pattern of galaxies in different regions of the Universe THE LIMBER RELATION AND THE LOCAL GALAXY CORRELATION FUNCTIONS To investigate the properties of the universal large scale structure, we cannot be satisfied with the above results on the angular correlation only: we need of course to know the intrinsic spatial correlation functions, because the angular one are just relative to the observer. Knowing the former requires the information on all source distances, which is hard to obtain in many cases, while the angular function requires information on the source s sky positions only. However, there is a nice useful relation between the spatial and angular functions, allowing us to simply estimate the former from the latter based on simple assumptions. We defer to the Appendix 1.A.1 for a detailed derivation of the relation, and report here the main assumptions. Let us define as Φ ( L) the luminosity function for the galaxy population under consideration; this quantity has been defined in the basic course to provide us information about the number of objects per unit volume and unit luminosity interval. Such luminosity functions will have typically Schechter-function shapes, that is power-laws with exponential cutoff at high luminosities, at L * values of about L ʘ. The main assumption made is that the clustering properties of the population are independent on the source luminosity (the Limber hypothesis): this is a rather strong assumption, that may become invalid for some particular classes of galaxies (for example, cd galaxies, the most luminous known, are found close to the centre of rich galaxy clusters and are consequently strongly clustered). In general, however, the Limber assumption is a fairly good one. Another assumption made is that the angle 1 ϑ in the sky over which the correlation function is calculated should be <<1 rad, and, finally, that the physical distance corresponding to the angle 1 ϑ between any galaxy pairs in the sky should be much smaller than that from the observer to the average source position. If all this is OK, then we can exploit the following simple relation between the spatial and angular correlation functions: Large Scale Structure of the Universe 1. 11

12 ( + ) ϑ 5 * 1 y ( x) x dx dyx d xϑ1 1 y 0 1 w( ϑ ) [1.8] y ( x) x dx 0 where ψ is a galaxy visibility function defined in [1A.5], y is defined in [1A.8] and * * the x parameter is the reduced distance x d d, where d is the average distance from observer to the source population. The ψ ( x) function essentially provides us with the number of sources falling within the selection function and gives the probability for a source to be selected. From [1.8] it is immediate to derive the relation [1.7]. More important, eq. [1.8] allows us to guess the behavior of the spatial correlation function from data on the angular one. The power-law shape of the latter suggests that a reasonable guess might be a power-law form ξ() r = Br γ. [1.9] Indeed, as detailed in Appendix A eq. [A.11], if we introduce this into [1.8], we get the expected power-law as in [1.6] w( ϑ) Aϑ d with an index d γ 1 and a constant factor A that we can obtain by eq. [A.11]: [ γ ] Γ(1/ ) Γ ( 1/ ) * γ 5 γ A= B d ψ ( x) x dx ψ( x) x dx Γ( γ / ) [1.10] 0 0 Then the observations on the angular functions require a spatial correlation function with power-law index of γ. 1.8 at least within a range of spatial scales. The form [1.9] can be more suitably phrased as r ξ () r = r 0g g [1.11] where r 0g is the galaxy correlation length, for which the observations show r0 g 5h Mpc in the range of spatial scales 0.1h Mpc < r < 10h Mpc 3. At larger scales the correlation function converges faster to zero and to negative values. The correlation length r 0g gives us the spatial scale over which the correlation 3 The parameter h is a convention to scale all measured quantities in cosmology by the exact value of the Hubble constant, h H 100 Km / sec/ Mpc 0. Large Scale Structure of the Universe 1. 1

13 function values ξ ( r0 g ) = 1, where there is a factor excess probability to find nearby galaxies to any sample galaxy compared to a random distribution. This roughly makes a transition from the linear to the non-linear regime in the clustering process. The validity of this solution can be directly proven with direct estimates of the spatial correlation functions based on redshift surveys, as in Sect. 1.5 below. The length r 0g is a measure of the correlation strength. We report in the Figure below correlation functions for local galaxies of different colors and luminosities. Fig. 5. The projected correlation function of galaxies in the SDSS. The left-hand panel shows the results for three separate luminosity bins. The corresponding ranges in Mr 5logh (here Mr is the SDSS r-band K- corrected to a redshift z = 0.1) are indicated in brackets. Note that brighter galaxies are more strongly clustered. The right-hand panel shows the results for red and blue galaxies with 0 Mr 5logh> 1; clearly red galaxies are more strongly clustered than blue galaxies. This reflects the fact that red galaxies are more frequently found in clusters of galaxies and the latter show high correlation lengths (Chap. 1.4). [Wang et al. (007)]. In the case that the source population includes objects at large cosmological redshifts (as discussed in Chap. 1.6 below), the Limber relation [1.10] has to be slightly modified and generalized. The function ψ ( x), providing us with the probability for a source to be selected, should be changed with the redshift distribution dn dz of the source population for the given flux-limited sample for which we calculate the correlation properties. This redshift distribution will depend on the survey depth and the redshift-dependent luminosity functions. Then the relation of the spatial to the angular -point correlation functions gets Large Scale Structure of the Universe 1. 13

14 π max w( ϑ) = dz( dn dz) H ( z) dπx( d ) ( ) 0 A ϑ π dz dn dz c [1.1a] where d A is the angular size distance (Sect. 0 eq. [0.7]) and π a variable along the line-of-sight in the reference system where the correlation function is measured. Here H( z ) is the Hubble parameter, that in the matter dominated era of interest here can be expressed (see eq. [0.1] Sect. 0) as 3 H( z) H0 m(1 z) K(1 z) Λ Ω + +Ω + +Ω. Then for a power-law correlation function as in [1.11] and if r 0 is the comoving correlation length of the population, the proportionality constant, instead of [1.10], becomes: [( γ 1) / ] π Γ 1 γ γ A= r0 dz( dn dz) H ( z) d A dz( dn dz) cγ( γ / ) 0 0 [1.1b] CORRELATION FUNCTIONS FOR OTHER COSMIC POPULATIONS. THE HIERARCHICAL CLUSTERING PARADIGM Analyses of the correlation properties for many source populations, in addition to galaxies, have been attempted following the same techniques as discussed above. Interesting results are found in relation with that particular class of cosmological objects that are the galaxy clusters. With this we refer to the positions of the centres of gravity of these large concentrations of galaxies. The -point correlation function of clusters in the Abell catalogue (including clusters with at least 65 members counted within a radius of 1.5 h -1 Mpc from the barycenter) shows the very same scaling relations as [1.11]: r ξc () r = r0 c with essentially the same slope as for galaxies, γ. 1.8, but with a much larger correlation length of r0 c 5h Mpc in the range 5h Mpc < r < 100h Mpc. Although these measurements are somewhat uncertain, there seems to be a relation of the correlation length with the cluster richness (number of galaxies and compactness), with the poor clusters and groups having lower values, and the rich clusters showing 1 the largest correlation lengths, r0 rc 50h Mpc. Low-redshift quasars have also γ Large Scale Structure of the Universe 1. 14

15 been measured clustering properties, and show values intermediate between those of 1 galaxies and clusters ( r0 QSO 10h Mpc ). Although a proper (and complex) dynamical study would be required to address this issue, it may be interesting to anticipate a discussion about this evidence that the correlation length and correlation strength increase with the environmental density. As originally discussed by Kaiser (1984), the likely origin of this effect should be found within the general interpretative scheme for the formation of the cosmological large scale structure, which is the Hierarchical Clustering paradigm. This idea assumes that luminous cosmic sources - galaxies, galaxy condensations, quasars - form inside dark-matter halos in a hierarchical fashion. This means that the sequence going from the smaller to the largest structures, that is from galaxies, to groups, poor clusters and rich clusters (the latter are the largest most massive virialized objects in the Universe) corresponds to one in which objects belong to increasingly higher and rarer peaks in the density field. On one side, large aggregations form by the collapse and coalescence of smaller entities (clusters from the aggregation of galaxies), so that they form later as a whole entity. On the other side, in this hierarchical paradigm larger structures have started to form earlier in cosmic time than lower mass objects. Rich clusters correspond to the highest density peaks that were the first to form and collapse from random fluctuations in the density field on time-scales corresponding to the (free-fall) collapse time as a function of density 1 t ff Gρ (G gravity constant, ρ density of gravitating matter). Higher density peaks start to form earlier within the cosmic fluid, so that they are structures that started to interact gravitationally with each other and with surrounding structures since longer time. This implies an higher correlation strength with respect to structures that started to differentiate from the homogeneous field at much later epochs CORRELATION FUNCTIONS IN REDSHIFT SPACE When large redshift surveys are available, as it started to be the case relatively recently, the correlation function ξ () r can be directly measured in the real 3D space. The correlation function in real space is expected to be isotropic, and is a function of the 3D separation only. When the object redshifts are used to derive the distances, the correlation function is distorted along the line-of-sight direction. This redshift-space distortion arises from two facts, operating respectively on small and large scales. On small scales, random peculiar velocities within galaxy groups or clusters cause radial elongations of observed structures, known as fingers of God like apparent in Fig. 1. Large Scale Structure of the Universe 1. 15

16 Another effect occurs at large scales due to the coherent motions of galaxies falling towards the potential well of large assembled structures. It usually leads not to a stretching but to a flattening of structures along the line-of-sight. Thus, peculiar motions prevent galaxies from following pure Hubble flow, affect measured redshifts and, hence, the estimated line-of-sight distances, while the galaxy's angular positions on the sky remain unaffected. Therefore, when the correlation function is computed from a redshift survey, it is usually measured as a function of the projected (transverse) separation, r p, and the line-of-sight (radial) separation, π. This function, ξ( r p, p ), is independent of direction in real space, but is strongly distorted in redshift space (Fig. 8), where the projected separations are the true measures of projected distance, while the radial separations are altered by peculiar velocities. In order to estimate ξ( r p, p ), eq. [1.5] and its analogous for the -point spatial function can still be used. Large Scale Structure of the Universe 1. 16

17 In practice, given the angular separation between two objects,, and their redshifts, z 1 and z, the comoving separations r p and π can be computed as follows. First, one computes the radial comoving distances, d c1 and d c, corresponding to the object redshifts of the galaxy couple, through the Mattig formula (Sect.0 eq. [8]) or similar. Then we have 4 while the 3-D (comoving) separation is [1.13b] [1.13] (we use a symbol different from r to make sure we understand that this is not the proper 3D separation, but just that inferred from redshifts). Fig. 8. The twodimensional correlation function w(rp; π) for the dfgrs, plotted as a function of transverse (rp) and radial (π) pair separation. The function was estimated by counting pairs in boxes and then smoothing with a Gaussian. Over-plotted lines correspond to the function calculated for a given theoretical model. This diagram clearly demonstrates the two effects of redshift distortions: fingers of God elongations at small scales and the coherent (Kaiser) flattening at large radii. The figure is adopted from Peacock et al.. From eq. (A.7) good approximation to s can also be found as. 4 Davis, M., Peebles, P. J. E., Large Scale Structure of galaxies. 1983, ApJ, 67, 465 Large Scale Structure of the Universe 1. 17

18 Results on the -point correlation function based on redshift surveys are reported in Fig. 6 above: we see results completely in agreement with those in Chap Figure 8 reports contours of the -point correlation functions of galaxies in redshift space [hotter colors correspond to higher values of ξ(, p )]. The redshift space distortions are well apparent here: they are graphically explained in Fig. 9. On the largest scales we are in the linear collapse regime, on the smaller the effects of the non-linear phase characterized by large orbital motions of galaxies shows up. r p Fig. 9. Illustration of the effects of redshift space distortions. On the largest scales the collapse of the large scale structures dominates, producing a radial flattening of the distances (and of the contours in Fig.8). On the shortest scales we have already virialized structures with large orbital motions and the "fingers of God" effects. An extreme situation happens at turn-around, when structures are just stopping the expansion and starting the collapse. To see how the pattern of galaxy clustering is distorted in redshift space, consider a simple spherical perturbation with initial over-density profile r() r r, that is the mean over-density within r (see also Mo et al Galaxy Formation & Evolution). The evolution of each mass shell follows the spherical collapse model predicted by the hierarchical clustering scenario. For a large radius within which the over-density contrast is small, the expansion of the mass shell is decelerated but its peculiar velocity is still too small to compensate for the Hubble expansion. In redshift space the mass shell will then appear squeezed along the line-of-sight when observed from a distance much larger than its size. This effect is illustrated at the top of Fig. 9. A Large Scale Structure of the Universe 1. 18

19 mass shell with linear over-density ρ / ρ 1 is just turning around at the time it is observed, so its peculiar infall velocity is exactly equal to the Hubble expansion velocity across its radius. In redshift space this shell appears completely collapsed to an observer at large distance, as shown in the middle panel of Fig. 9. Finally, a mass shell which has already turned around has a peculiar infall velocity which exceeds the Hubble expansion across its radius. If this infall velocity is less than twice the Hubble expansion velocity, the shell appears flattened along the line-ofsight, but with the nearer side having larger redshift distance than the farther side. At smaller radii the peculiar infall velocities of collapsing shells and the orbital motions of galaxies after virialization are much larger than the relevant Hubble velocities and are randomized by scattering effects during virialization. The structure then appears to be elongated along the line-of-sight in redshift space (a finger-of-god pointing at the observer). This is depicted in the bottom of Fig. 9. Effects of redshift distortions are different on different scales. As we understand, studies of the correlations functions in redshift space provide us also with important information on the galaxy peculiar motions and, eventually, on the dark-matter distribution. In conclusion, observations of the two-point angular correlation functions of galaxies display a double power-law behavior, as illustrated in Fig. 4. From the Limber inversion formula, we infer from this also a double power-law dependence of the spatial correlation function at low redshifts as: ξ () r r r small r large r [1.14] Large Scale Structure of the Universe 1. 19

20 1.1.6 TIME AND REDSHIFT DEPENDENCE OF THE CORRELATION FUNCTIONS The large scale structure evolves with time under the effects of the evolving gravitational field, as we will discuss in detail later (Sect. 8). This is reflected in the evolution of the correlation functions for source populations "tracing" the large scale structure. This is of course a very important subject of observational cosmology, for many reasons. The investigation of the correlation properties of high-redshift source populations informs us about the cosmic environment where this object's category is found, and this is a fundamental information to understand the astrophysical properties of the objects and their cosmological context. Secondly and more importantly, if we can identify classes of objects tracing the cosmological large scale structure at high redshift, then we can assess how in detail this structure evolves in time and how the present-day Universe was taking shape in the course of time. The estimate of the correlation function for high-redshift populations follows similar approaches as above for the local Universe. In particular, relations [1.13] are used for samples with redshift information. Fig. 10. Comoving correlation lengths of optical quasars, from Ross et al. (009). r 0QSO is close to 10h -1 Mpc and tends to increase at z>1.5. However, particularly for sources at high redshifts, for which the redshift information is hard to obtain spectroscopically because of their faintness, the clustering Large Scale Structure of the Universe 1. 0

21 information is often limited to the catalogued angular positions in sky of the source sample. Fig. 11. Comoving correlation lengths versus redshift for various cosmological source populations, as explained in the figure insert. We see quasars, galaxy clusters, Lyman Break Galaxies, radio AGNs, Extremely Red Objects. The fixed mass lines show the predicted clustering strength of dark matter halos of a given mass at any particular redshift. As we see, these strengths decrease with z, or the masses increase at decreasing redshift and fixed length. Shaded areas instead show how the comoving correlation lengths for environments identified at given redshifts would evolve with time according to different models of evolution of the -point correlation function, according to eq. [1.15] for the stable ε = γ 3 and linear ε = γ 1 evolution mode. [From Farrah et al. 006] Once a source population is identified at high redshifts by any mean, the typical procedure is then to calculate the angular correlation function with the usual methods of eqs. [1.5]. The information on the clustering in physical (proper) units is then achievable from [1.1a and b], that requires the knowledge of the selection function, in practice the distribution of redshifts for that particular source selection. The latter can be estimated for example using a sub-sample of the sources all with spectroscopic redshifts, or the photometric redshifts 5 for a representative subsample, or even modellistic predictions of how should look like. 5 Photometric redshifts are redshift estimates based on multi-band photometric imaging data representing the source spectrum in a very low-resolution mode and compared with synthetic galaxy spectra. Large Scale Structure of the Universe 1. 1

22 The first cosmic population known at very high redshift was optical quasar samples, that locally show moderately high correlation lengths (as we have seen in Chap. 1.4). The comoving correlation lengths r 0 for quasars selected in various samples by different means are reported in Fig. 10. While roughly stable at z<1.5, the lengths tend to increase at higher redshifts. More recently, many other classes of sources have been identified, whose correlation lengths are reported (and compared with those of QSOs) in Figure 11. To understand the cosmological environment hosting the source population, the values of the correlation lengths can finally be compared with the predicted values of models of the large scale structure, in two ways, both of them illustrated in Fig. 11. The first is to compare the observed r 0 values with the predicted clustering strength of dark matter halos of a given mass at any particular redshift. As we see, these strengths decrease in comoving units with z, or the masses increase at decreasing redshift at fixed correlation length. Or to calculate how these observed lengths at high redshifts would evolve with time according to different models of the evolution of the -point correlation function. A typical representation of the evolution of the spatial correlation function in comoving units is given as a factorized form: γ r c ξ( rc, z) = (1 + z) r0 γ (3 + ε) [1.15] where r 0 measures the strength of the clustering at z=0, γ 1.8 measures the scaledependence of clustering and ε parametrizes the evolution with redshift. Various special values of ε have particular interpretations (e.g. Waddington et al. 007). (i) ε = 0 is the stable clustering model, where the correlation function is fixed in proper coordinates and clustering grows stronger as the background mass distribution expands with the universe. (ii) ε = γ 3 is the comoving case, where clustering remains constant in comoving coordinates and simply expands with the universe. In this case halos of dark-matter simply expand with the universe. (iii) ε = γ 1 is the linear growth model, which corresponds to the application of linear perturbation theory to a scale-free power spectrum in an Einstein de Sitter γ 3 γ+ 1 universe, such that ξ ( r, z) (1 + z) (1 + z) (see Sect. 8 of the Course). c We note that all these models are qualitative indicators of possible evolution scenarios rather than realistic clustering models. However, model (iii) is the most physically motivated. This important issue will be further considered in Sect. 8. Large Scale Structure of the Universe 1.

23 Chapter THE POWER-SPECTRUM ANALYSIS OF THE LARGE SCALE STRUCTURE All we have seen so far concerns a statistical sampling of the cosmic fluid in terms of populations of cosmic point sources (typically galaxies), assumed to be tracers of the cosmological large scale structure. An alternative way to consider it is, instead of a limited set of test particles, a continuous fluid. Then the fundamental quantity to be defined is the fluctuation density field or the density-contrast field: ρ( x) ρ ( x) [1.0] where ρ( x ) is the density field at a given spatial point specified by x and ρ is an average value over a given volume V. Of course, in general the fluctuation and density fields will depend on both spatial and temporal position. Let us assume that the volume V makes a representative sampling of the Universe, which means that it is so large to include all relevant scales of interest for the large scale structure analysis. Instead of referring to the real field ( x ), it is common practice to make use of its Fourier representation. We have seen, when discussing of the galaxy spatial correlations, that the correlation strengths are a strong function of the spatial scale under consideration, in the sense that it strongly decreases at increasing scale. This was accurately quantified by the spatial correlation function ξ () r decreasing fast with the scale r. Also cutting the samples at different limiting fluxes, we have found w(θ) functions scaling with the average source distance (Fig..3a) in such a way to demonstrate that these structures repeat themselves homogeneously as a function of the survey depth, at least up to distances corresponding to substantial regression in cosmic time. In any case, this kind of periodic behaviour of ξ () r and a function ξ ( r) 0 at r >> r 0, corresponding to ( x) 0 for large x, show that good descriptions of the large scale structure can be obtained with Fourier representations of the ( x ) function, given that it converges to zero at large values of the argument (the requisite for a well-behaved Fourier development). In the light of the above we can represent the fluctuation field as: δ δ δ δ δ δ * δ ( x) = δ ( k )exp( ik x) = δ ( k )exp( ik x) [1.1] where the wavenumber vector k is given by k k ρ k k Large Scale Structure of the Universe 1. 3

24 π π π kx = nx ; ky = ny ; kz = nz ; [1.] L L L with n being integer numbers and L the maximum scale over which we can expect that the real function ( x ) is not null on average. The Fourier coefficients are, in general, complex numbers, given by the usual relation d 1 d d d d d k ( k ) = ( x)exp( ik x) dx V [1.3] V 3 = L. Remember that k 0 for k 0 V δ = = because the integral should vanish over the volume V. Since the field ( x * ) is real, we have that δk = δ k, and this is useful because it allows us to use only positive values for n and to consider the real part only of the Fourier coefficients. If, instead of the volume V, we had chosen a different one V' to sample the large scale structure, we would have obtained a different set of coefficients δ k. For different sampling volumes V, these coefficients may slightly vary in amplitude and phase. If the spatial average of the perturbation field ( x ) is null, its variance σ is not. It is immediate to find that δ δ 1 δ δ δ σ = ( x) = δk ( k) = δk ( k), with δk ( k) δk ( k) [1.4] V k k where the average is taken in principle over many sampling volumes. The quantity δ δ k ( k ) is the contribution to the variance by waves of wavenumber k. Equation [1.4] is a trivial application of the Parseval theorem on the Fourier transform of the real field δ ( x δ ): it states that the total field "power" is conserved from the real function to the transformed one: σ = = 1 d d 1 1 ( xdx ) = d d d ( kdk ) = d d ( k) 3 3 k k V π V V k The second integral is the average of the squares of the transformed field δ k. [1.5] Let us now assume that the large scale structure, the density field ( x ), and the clustering pattern are homogeneous and isotropic on the largest scales. This is a generalization of the Cosmological Principle to the situation of a clustered universe on small scales tending to homogeneity on the largest, but with no preferred directionality in the clustering pattern. Then what matters will not be the wavenumber vector k, but rather its modulus k = k, so that the variance simplifies to σ 1 1 = dk = ( ) V k π 0 P k k dk, [1.5b] Large Scale Structure of the Universe 1. 4

25 where we have defined Pk ( ) δk In complete analogy with what we did in many other occasions, we define this as the power-spectrum of the large scale structure. Note that the variance σ gives us the total average amplitude of the fluctuation field on a given spatial scale, but does not tell us which waves contribute to it inside the sampling volume V: the power spectrum tells us what scales contribute most to the field variance. The Parseval theorem [1.5] and [1.5b] makes also clear that the physical units of the powerspectrum are [volume]. Another way to represent such spectrum is through the quantity such that the variance may be re-written σ ( k) : 1 3 ( k) Pk ( ) k [1.6] π 1 = = π 0 0 Pkkdk ( ) ( kd ) lnk [1.7] The representation [1.6] is advantageous because it is dimensionless and it measures the contribution to the fluctuation power per logarithmic decade of k. As we will see, a power-law form of the power spectrum has a particular meaning n P( k) = Ak [1.8] with spectral index n. In general, of course, such representation will be valid over a restricted range of scales and the index n may vary as a function of the scale. 1.. THE COVARIANCE FUNCTION AND THE POWER-SPECTRUM Let us make now a comparison between what we have seen in terms of the correlation functions and the power-spectrum analysis. We may re-phrase the joint probability to find a second galaxy at a distance r with respect to another one in x for a continuous field ρ( x ) as d d d r( x) r ( x+ r) d P = dv 1dV m within the volumes dv1dv, with m the average galaxy mass. This can also be written with reference to a point-source sampling of the density field, as Large Scale Structure of the Universe 1. 5

26 d d d r( x) r( x+ r) d d P = nv dv 1dV = nv[1 + x ( r )] dv1dv r such that d x ( r ) = d d d r( x) r( x+ r) r 1. Let us now consider the quantity ( x) ( x+ r) = r( x) r r( x+ r) r r r [1.9] that is the (auto-)covariance function of the fluctuation field. By developing the products and the integrals we get, from [1.9] r( x) r( x+ r) r r r r r( x+ r) r ( x) ( x+ r) = + = r r r r [1.30] r( x) r( x+ r) = 1 = x ( r ) r which is a fundamental relation between the covariance function ( x ) ( x + r ) and the -point correlation function: these are essentially the same thing. We can now change this into a relation with the power-spectrum. If we now consider the Fourier representation of the fluctuation field [1.1], we can write 1 d d * ξ( r ) = ( x) ( x r ) dx k exp( ik x) k' exp ik ' ( x r ) dx V + = δ δ + = V V k k' d d d = d ( k ) exp( ik r ) k k because k = k, or equivalently 1 d d x ( r ) = P( k)exp( ik r ) dk = ( x) ( x + r ) 3 ( p ) V k [1.31] and the reverse relation also holds d 1 d d d d dk ( k ) = P( k) = x( r )exp( ik r ) dr V : [1.3] V k the power spectrum is just the Fourier transform of the covariance function and of the -point spatial correlation function, a theorem known as the Wiener-Khintchine theorem. It should be noticed that the expression [1.31] holds for a statistically homogeneous random field. Large Scale Structure of the Universe 1. 6

27 Although the power spectrum is just the Fourier transform of the two-point correlation function, it is more advantageous to work with the power spectrum than with the -point correlation function when studying galaxy clustering on large scales. The reason for this is that on large scales, where the density field is still in the linear regime δ ( x) << 1, different Fourier modes evolve independently one from the other (this will be demonstrated in Sect. 3), while the amplitude of the two-point correlation function is affected by many different modes. This will become evident when discussing the theory of the time evolution of perturbations in the cosmic fluid in Sect. 3. Consequently, power-spectrum amplitudes on large scales are less affected by small-scale structure than correlation function estimates, making observational results easier to interpret. Now, let us try to simplify [1.31]. First consider that the fluctuation field is a real ik r function, such that the complex argument e = cos( k r ) = cos( kr cos θ ) may be reduced to its real part, where the angle θ is that between the two vectors. Again under the assumptions of a statistically homogenous and isotropic structure on the largest scales, we can perform the integration over directions in [1.3] as Ω 1 1 π 90 ξ( r) cos( krcos θ) dω= dφ cos( krcos θ)sinθdθ = 0 90 π sin kr = cos( kr cos θ) d( kr cos θ) 4 π. kr = kr Once the (external) integral on directions is solved, from [1.31] this gives 1 sin kr r P k k dk π kr 0 ξ () = () [1.33] From this it is immediate to find the convergence of the -point function at small separations: for r 0, sin kr 1, and kr 1 ξ(0) P( k) k dk = σ π, [1.34] 0 which is the average total variance of the large scale structure. Considering now that, for kr << 1, sinkr kr 1 and for kr >> 1 sinkr kr 0, the factor sin kr kr in [1.33] operates as a window function, such that at the scale r only waves with wavenumbers k r 1 contribute to the fluctuation amplitude, while waves with larger wavenumbers, corresponding to small spatial scales, average out on that scale. Large Scale Structure of the Universe 1. 7

28 If we now take the form [1.8] to represent the power-spectrum, the integral [1.33] can be resolved in a simplified way as r n 3 n r (3 n) ξ () r P() k k dk k dk k r π 0, [1.35] 0 n which is a simple relation between the power-law representation P( k) = Ak and the correlation function. Particularly important the behaviour on the large spatial scales as revealed e.g. in Fig. 4 (3 + n) 4 and eq. [1.14], with ξ () r r r, hence requiring a value n 1. Relation [1.35] and [1.3] are also important because they allow us to immediately obtain information on the power-spectrum from -point correlation function measurements, and vice-versa. Note that, in general, the knowledge of the power spectrum of the large scale structure is not sufficient to unambiguously describe the properties of any statistical field in the same way as the correlation function ξ(r) only provides an incomplete characterization and knowledge of the higher order moments might be required. However, many situations with some degree of symmetry are uniquely characterized by the simple function P(k). These are the so-called Gaussian random fields, already mentioned in Sect. 1.1, which are characterized by a random distribution of the phases in the Fourier decomposition. For an ideally infinite number of stochastic realizations of the universe, for each value of the scale k the real value and the 3 imaginary value of the complex field ( k) = Pk ( ) k π form Gaussian distributions around a mean value = 0 with variance σ. Such Gaussian random fields play an important role in cosmology because it is expected that at very early epochs, the density field obeyed Gaussian statistics MEASUREMENTS OF THE COSMOLOGICAL POWER-SPECTRUM A qualitative summary of observational results on the density field fluctuations in the local Universe is reported in Figure 15. These results come from a variety of observations and techniques to sample the large scale structure. In particular, what is mentioned about the "SDSS galaxy clustering" refers to measurements of the -point spatial correlation function and power-spectrum from galaxy surveys. Concerning the other methods mentioned here, we will see them later in the course. Large Scale Structure of the Universe 1. 8

29 In the qualitative overview of the properties of the cosmological power-spectrum is illustrated in Fig. 15 we see confirmed here what we anticipated from the study of the spatial correlation function: the fluctuations in the density field are maximal on small scales and decrease steadily when considering larger and larger scales. Fig. 15. Representation of the power spectrum of density fluctuations in the Universe, as determined by different methods. In ordinate the quantity is reported, against spatial scale in abscissa. This is a qualitative plot just to illustrate how different samples and methods concur in allowing us the determination of this fundamental cosmological observable. Going from large to small scales, the results presented here are obtained from CMB temperature fluctuations, from the abundance of galaxy clusters, from the large-scale distribution of galaxies, from cosmic shear, and from the statistical properties of the Lyα forest. Some of these methods will be the subject of discussions in following Sections of this course. One can see that the power spectrum of a ΛCDM model is able to describe all these data over many orders of magnitude in scale. [From M. Tegmark] At this stage, however, we need quantitative measurements of the power-spectrum in the local Universe, because they are needed to develop a physically motivated and quantitative theory of the origin and development of the structures in the Universe. Large Scale Structure of the Universe 1. 9

30 Given a galaxy sample, it is straightforward to measure the power spectrum. In practice, the power-spectrum is rather straightforwardly calculated by first computing the density field from a flux-limited redshift survey. The second step is to smooth the discreteness of the distribution with a smoothing process (typically with a Gaussian kernel). The final, to calculate the fast Fourier transform of the density field, and bin it in wavenumber bins. Fig. 16. The power spectrum of density fluctuations in the Universe, as determined by different methods. The quantity is in the same units as defined in eq. [1.7]. In the right panel we see the averaged data. The k 1 and k scales are indicated with arrows. We report in Figures 16, 17 and 18 quantitative results on the cosmological powerspectrum at redshift zero, in some cases comparing them to predictions of physical models of the formation of the large scale structure. These models are based on the assumption that the dominant gravitational contribution in the Universe is (nonbaryonic) dark matter, called Cold Dark Matter, that is a particle field originated during the early evolutionary phases of the Universe, particles that were nonrelativistic at the moment of their de-coupling from the cosmic fluid (the de-coupling epoch is the epoch at which these particles and their anti-particles annihilated and the temperature of the photons was too low for them to generate new pairs of particles and anti-particles, see Sects. 7a and 8). Fluctuations in this dark matter field are mostly responsible for the gravitational potential determining the cosmological structure. As we see from Fig. 17 and 18, good fits to the low-wavenumber spectra (large spatial scales) can be obtained by assuming a cosmological density parameter of, and for the Hubble constant the value measured by HST of Large Scale Structure of the Universe 1. 30

31 1 1 H 70 Km s Mpc. This is a first indication in favour of a universal matter density 0 definitely lower than the critical density. This will need to be further elaborated and justified later in the Course. Fig. 17. The average power-spectrum of structures in the local Universe, compared with predictions of CDM models for two different normalizations σ 8 (see eq. [1.45] below). The k 1 and k scales in the fitting function [1.40] are indicated with arrows. Another important parameter appearing in Fig. 17 is the variance σ 8 of the density fluctuation field calculated by averaging on the spatial scale of 8 h -1 Mpc: 1 σ8 σ(8 h Mpc) (see below). The summary data reported in Fig. 17 show that σ. 8 1 For the moment, however, we do not want to enter into a detailed description of physical models of the large scale structure, which will be discussed in later Sects. of the Course, and are interested in just phenomenological descriptions of the data. As we see, the data appear to be consistent with a two-power-law behaviour with a 1 change of slope at about a wavenumber value k. 0.07h Mpc, corresponding to a spatial scale of about 0 Mpc. Altogether, the data are reasonably well fitted by the simple functional form Large Scale Structure of the Universe 1. 31

32 with k h Mpc and 1.8 ( kk) 1 ( k). [1.40]. 1 + ( kk) k h Mpc. Fig. 18. The average power-spectrum of the galaxy distribution derived from the dfgrs (black points). The curves show powerspectra from CDM models (including or excluding baryons from the simulations), for different values of the matter density parameter Ω m. h=0.7 (H 0 =70 Km/ s/ Mpc). 3 If we remember that Pk ( ) k ( k), then on the small spatial scales (large wavenumbers k 1) we have that the numerator dominates, and we have ( k) k P( k) k ( k) k. From [1.35], this corresponds to (3 + n) 1.8 n ξ ( r) r r where P( k) k and n= 1.. So, this is entirely consistent with the results we discussed in Chap. 1.3 about the form of the - point spatial correlation function. On the large spatial scales, small wavenumbers k, we have ( kk) k.. [1.41] ( k) k and Pk ( ) k ξ ( r) r ( kk) k k 1 On such large scales the correlation function then converges fast with the scale. Large Scale Structure of the Universe 1. 3

33 Another representation of the power-spectrum P(k) is reported in the Fig. 19 below, based on a variety of different data. Note that, while the power spectrum is the Fourier transform of the two-point correlation function, similar transforms of the N-point functions for higher order correlations would also provide descriptors of galaxy clustering. The Fourier transform of the three-point correlation function is known as the bispectrum, useful for detecting non-gaussian fluctuation statistics. Fig. 19. Another representation of the cosmological power spectrum of the large scale structure as the P(k) itself. This function shows a very characteristic maximum on a scale of 300 h -1 Mpc. Large Scale Structure of the Universe 1. 33

34 1..4 THE MASS FLUCTUATION SPECTRUM. THE PRIMORDIAL POWER-SPECTRUM On the large spatial scales probed observationally in Figs , we have seen in eq. [1.41] that the power-spectrum of the local large-scale structure appears to have a particularly simple form, as in eq. [1.41]: n Pk ( ) k with n 1 [1.4] On such large scales, the fluctuations are in the so-called linear regime, which means that they can be treated as small perturbations in the density field. A treatment of such small perturbations, that we will later discuss in Sect. 3, shows that the perturbation spectrum in the linear regime evolves by keeping this spectral form unchanged. So it is believed that the spectral form observed on such large scales (>50 Mpc) is exactly the one originally present at very early epochs, that later evolved under the effect of self-gravity. The form [1.4] is called the primordial power-spectrum. It is a case of very special interest that will be discussed below and in following Sects. As an aside, it is customary to consider the power-spectrum as a function of the mass involved, instead of a function of the spatial scale or the wavenumber. Since the mass within the fluctuation is proportional to r 3 3, M r r, such that for a given 1/3 constant average density, r M, and the correlation function can be written ξ( M) ξ() r r M (3 + n) (3 + n)/3 and because ξ( M) σ ( R) σ ( M), the relative mass fluctuation will be dρ ( M) ( M) ξ ( M) M ρ 1/ (3 + n)/6 [1.43] /3 For n=1, ( M) M. This is the mass fluctuation spectrum, also parametrized by the spectral index n. A condition for the validity of the cosmological principle is that the mass fluctuation spectrum does not diverge on large scales; instead it should converge. A condition for this to happen is that : n > 3. If so, the amplitude of the mass fluctuation spectrum decreases at increasing mass (and scale). This is clearly the case for the spectra shown in [1.40], and in particular, for the primordial power spectrum. Origin of the primordial power spectrum. A very first and obvious question arises about the spectral shape [1.4]. A natural model for the origin of the primordial power spectrum would be to consider it as the effect of random statistical fluctuations in the distribution of the number N of particles on the scale r : in such a case, the Large Scale Structure of the Universe 1. 34

35 statistical probability distribution would be the Poisson one, in which the fluctuation 1/ in the number of particles would be dn N = 1/ N, or in terms of the mass fluctuation, assuming M N : dm ( M ) M According to the definitions in [1.43], this would correspond to a spectral index of the primordial power-spectrum of n = 0. This is the well-known white-noise power-spectrum, which is one with equal power noise on all scales. The 3 1 corresponding correlation functions would be ξ() r r ; ξ( M) M. It is all the more evident that this is not the spectral shape that we observe on the large scales, and particularly it is not the one corresponding to the primordial spectrum in [1.4]. The primordial power-spectrum has not to do with purely random fluctuations in the distribution of mass particles in the Universe. This particular spectral shape requires a specific physical process in operation (as we will see, probably operating during the very early Inflation era, Sect. 7). 1 M 1/ The Harrison-Zeldovich power spectrum. The spectrum in [1.4] is a fundamental achievement in theoretical cosmology, which agrees very well with observations. This spectrum is the very origin of what we called the cosmological large scale structure and has the needed form to match the observed properties of the LSS. If the spectral index were much smaller than n=1, there would have been excessively large metric perturbations on very small scales in the early Universe, which would inevitably have resulted in their collapse to form black holes. Also the n=1 spectrum does not diverge on large spatial scales, as we have seen. As we will show in Sect. 8, this spectrum has the important property that the density contrast ( M ) had the same amplitude on all scales when the perturbations came through their particle horizons. More on theoretical aspects of this functional form will be discussed in Sects. 7b and 8, particularly a possible physical origin for it as quantum fluctuations brought from microscopic to cosmological scales by the early inflationary expansion. Once we have assumed for the Initial Power Spectrum that in [1.4], and remembering that it is useful to work with the dimensionless quantity n ( k) kpk ( ) k π which expresses the contribution to the variance by the power in a unit logarithmic interval of k, we can similarly define the corresponding quantity for the gravitational potential as Large Scale Structure of the Universe 1. 35

36 1 3 4 n 1 Φ( k) k P Φ( k) k ( k) k, π which is independent of k for n = 1. [This difference between the mass fluctuation spectrum and the fluctuation of the gravitational potential is immediately understood considering that the potential has a spatial scale at denominator]. Thus, for the special case of n = 1, which is called the Harrison Zel dovich spectrum or scaleinvariant spectrum, the gravitational potential is finite on both large and small scales. This is clearly desirable, because divergence of the gravitational potential on small or large scales would lead to perturbations on these scales that are too large. The Normalization of the Power Spectrum. So far we have only discussed the shape of the power spectrum. To completely specify P(k), we also need to fix its overall amplitude. We do not have yet discussed a refined theory for the origin of the cosmological perturbations, and the amplitude of P(k) is not predicted a priori but rather has to be fixed by observations. Even for inflation models, where we can make detailed predictions for the shape of the initial power spectrum, the current theory has virtually no predictive power regarding the amplitude (see Sect. 8). For a power spectrum with a given shape, the amplitude is fixed if we know the value of P(k) at a given value of k, or the value of any statistics that depends only on P(k). Not surprisingly, many observational results can be used to normalize P(k). Different observations may probe the power spectrum at different scales, providing additional constraints on the shape of P(k). In fact, trying to determine the shape and amplitude of the linear power spectrum is one of the most important tasks of observational cosmology. The classical method for normalizing both the observational and the theoretical power spectra makes use of the variance of the galaxy distribution sampled with randomly placed spheres of radii R. The variance of the density field on a spatial scale R is related to the power spectrum by 1 s ( R) = P( k) W ( ) 0 R k k dk π [1.44] 3 with WR ( k) = [sin( kr) kr cos( kr)] ( kr) being the Fourier transform of the spherical top-hat window function of radius R. The observed dependence of the total variance on the spatial scale or the wavenumber is reported in the Figure 0 below. Note that for R 0, the variance σ ( R) σ. σ ( R) in [1.44] tends to the total variance Large Scale Structure of the Universe 1. 36

37 The value of σ(r) derived from the distribution of galaxies is close to unity for R=8h 1 Mpc. In what follows, the parameter defining the normalization of the power spectrum of the LSS will then be assumed to be: σ =. [1.45] 1 8 σ( R 8h Mpc) Fig. 0. Dependence of the total variance σ on the spatial scale R and wavenumber k. Then observations that we have discussed in Chapt..3 tell us that σ 8 1. A more precise current evaluation of the parameter, according to the data reported in Fig. 0, is σ , [1.45b] consistent also with all the data summarized in Fig. 15 and with those that we will discuss in Chapt. 3.5 of Sect.6 (CMB). 8 σ and the spectral index n make new fundamental cosmological parameters in addition to those discussed in Sect. 0. Large Scale Structure of the Universe 1. 37

38 Chapter OTHER STATISTICS THE COUNTS IN CELLS Counting the distributions of galaxies, or other sources, among elementary volumes defined inside a big and representative sky area offers an alternative simple but useful way of measuring the correlations properties of galaxies on large scales which does not suffer from some of the problems of the correlation functions. We will call this the counts-in-cell statistics, Pn ( V ), a symbolism making clear that this will depend essentially on the elementary volume V to which the distribution is referred. Indeed, P ( ) n V was one of the earliest quantitative methods for analysing the galaxy distributions adopted by Hubble himself. The Pn ( V ) in-cell statistics can be easily quantified by means of its first moments, that are the variance σ and the skewness γ, that are the second and third moments of the distribution respectively. These moments are simply related with other statistics that we have defined in previous chapters. For example, the derivation of the second moment is a simple matter. Let us define n= nv V the average number density of galaxies in V and n V the average volume density. Let us divide the V cell into infinitesimal sub-volumes such that every sub-cell may contain a number of galaxies n k that is either 1 or 0. Then over the volume V we have while the mean squared value is The first addendum is equal to k V n = n = ndv = n V n = n n = n + nn k l k k l k l k k l n n n k = k =, because of the k n (1 ξ[ r ]) dv dv V 1 1 V n values 0 or 1. The second term is equal to +, where r 1 is the distance from the elementary volume k to l. So we can express the second moment of the countsin-cells distribution Pn ( V ) as: σ n n n nn n = = + 1 = 1 n n n n and after substitution of the previous relation Large Scale Structure of the Universe 1. 38

39 σ V V V V dn n V n = = ξ( r ) dv dv n n V n V σ = + ξ( r 1) dv1dv n V [1.46] The term 1 n is due to Poisson fluctuations, it is a discreteness effect. The second term is an integral of the -point correlation function and provides us with the (positive or negative) contribution to the variance on the scale corresponding to the volume V due to the excess (or defect) of probability measured by the correlation function. The "graininess" of the matter distribution increases with the degree of correlation. Eq. [1.46] is very useful to measure the two-point correlation function and power-spectrum on very large spatial scales where the direct determination is noisy. And ultimately, it is a useful probe of the cosmological power-spectrum on the large scales. Now, the skewness γ (symmetry) of the galaxy distribution is the moment of order 3 of Pn ( V ). With a similar approach as above, it can be expressed in terms again of the Poisson and of the -point correlation function contributions, as well as of the contribution of the 3-point spatial correlation function ξ (3)( r1, r13, r3) defined in [1.4]: 3 dn 1 3σ 1 ξ 3 (3) ( r1, r13, r3) dv1dvdv3 n n n V [1.46bis] γ = = + + where ξ(3) ( r1, r13, r3) = ς( r1, r13, r3) is the reduced 3-point correlation function. In general, the skewness of a statistical distribution measures how much symmetric it is with respect to the average. Indeed, the skewness γ of the galaxy distribution is a quantity measuring how "symmetric" is the structure of filled parts and voids. For example, a Gaussian distribution (as previously discussed) is a completely symmetric one, in which γ = 0. Eq. [1.46bis] makes it clear that this kind of asymmetry can be measured by the 3-point function. Of course, in the case of more complex matter distributions, as in the real case, the first moments like in [1.46] and [1.46bis] only provide a crude description of the statistical distribution. In principle, we would need moments of all order to describe it completely. The alternative approach is that of considering not the sequence of moments, but the full Pn ( V ) distribution itself. In this case it is hard to obtain analytical expressions for it to compare with the data, so that a realistic solution is to perform numerical simulations based on some physical models and compare them to the observational data. This is exactly done in Figure for three different sampling volumes. The two models considered are a random distribution model (Poisson distribution in sky), and a Cold Dark Matter model. The data come from a redshift Large Scale Structure of the Universe 1. 39

40 surveys of IR-selected galaxies (the QDOT/IRAS galaxy survey). The point to stress here is, again, that a random point distribution is highly inconsistent with the data, as it provides a too sharply peaked distribution around the average number density value, while the data show a much broader one (many elementary volumes containing no galaxies and many containing a large number of galaxies). As a final note, the Pn ( V ) distribution appears highly asymmetric when referred to small volumes (a), and tends to an increased symmetry at increasing V. This is an interesting effect that will be further discussed in Chap Fig.. Distribution functions of galaxy counts obtained from the QDOT survey of IRAS galaxies; the counts are shown for "Gaussian spheres" of different radii (increasing from a to c) and compared to both a random (Poisson) distribution and CDM simulation. Large Scale Structure of the Universe 1. 40

41 1.3. FRACTAL ANALYSES An interesting consideration, today of mostly historical interest, is to what extent matter and sources in the real Universe follow some self-similar distributions that are characteristic of our current day experience or in some fields of physics. A mathematical way to represent such self-similar distributions is offered by the fractal sets. A fractal set is a distribution having the same structure on a variety of scales considered. One common application of this concept is to consider the structure of the ideal lines separating the ground from the seas and oceans on Earth, the coasts: the structure of these lines keep similar to each other independently on the scale considered, from the centimetres to the thousands of Km! Fractal analysis is mostly used in fields such as the physics of condensed matter. A question has emerged in cosmology particularly during the 80's and 90's of how much such structures may be characteristic also of the matter distribution in the Universe. We have seen, through for example the auto-correlation functions, that a kind of self-similarity is characteristic of the distribution of galaxies in space, at least on a limited range of scales. Let us translate these properties in the language of fractal analysis. Let us consider the mass contained in a small sphere of radius r around a given galaxy, and denote this mass distribution as M(<r). In the range of scales such that ξ () r >> 1, we have 3 D M( < r) r ξ() r r ; D = 3 γ [1.47] where D is a constant as far as ξ () r is a power-law with constant index, D = 3 γ. 1.. The exponent D is called the fractal dimension of the galaxy distribution. The constancy of the exponent is interpreted by saying that the galaxy distribution has a fractal structure on those scales. Note that, by analogy, if the matter is distributed along one-dimensional structures (filaments), D M() r r, D = 1; if in two-dimensional sheets, D = ; when filling homogeneously a full 3D volume, D=3. Note also that, would the Universe be strictly a fractal with constant dimension D, the correlation length would increase with the radius of the sample, and the mean density would decrease with the depth of the sample considered. Would the data demonstrate that the cosmological structure would correspond to a constant fractal dimension D. 1. on all scales, this would be a tragedy for cosmology, as the cosmological principle should be abandoned, will all consequences of the case. As a matter of fact, the Universe is not obeying such a structure, and a more realistic description requires that the dimension D changes with the spatial scale considered, and in particular it increases with the scale, tending to homogeneity. 3 For r r then M ( < r ) r and D=3, because ξ ( r ) 0. 0 Large Scale Structure of the Universe 1. 41

42 1.3.3 TOPOLOGICAL PROPERTIES 6 All what we have seen so far provides us with interesting insights about the matter distribution in the universe and quantitative measures of it that will be needed later. Fig. 3. Examples of excursion sets for simulated distributions of galaxies based on the Cold Dark Matter model. Various panels correspond to different threshold values normalized to the average density. 6 This Chaper. is partly taken from Coles & Lucchin, Cosmology. Large Scale Structure of the Universe 1. 4

43 However, we still lack descriptions of the connectivity properties of matter distribution, that is how different portions of the Universe, the filled parts and the voids, are connected one to each other. How the individual structures, filaments, voids, sheets and clusters contribute to form the global structure. Just to clarify the kind of questions we are posing in this Chapter, we will try to investigate if the matter distribution is condensed on the surfaces of cells surrounding void regions (cellular pattern), or it is concentrated in roughly spherical over-densities surrounded by vacuum, or if it is more like a sponge, that is under-dense and over-dense regions are inter-twined with no predominance of one over the other. The mathematical theory describing such connectivity properties is topology. Topological analyses of the matter distribution give interesting insights about some general properties of the large scale structure, like for example its gaussianity, or deviations from it, as we introduced it in Chap... The topological quantity that has been used for topological investigations is the genus. To obtain this from an observational sample, one must first smooth the galaxy distribution with a filter (usually a Gaussian filter is used) to remove the discrete nature of the distribution and produce a continuous density field. The second step is to define a set of threshold levels ν of the average matter density in this continuous field, and then construct excursion sets (sets of iso-density surfaces where the field density exceeds those values), for various levels ν, both positive and negative (density values above or below the average). An excursion set will typically consists of a number of regions, some of which will be simply connected, e.g. a deformed sphere, and others which will be multiply connected, like a torus (doubly connected). Excursion sets for model distributions of Dark Matter are reported in Figure 3. Each set is referred to a given threshold ν, corresponding to the number of standard deviations that the local density deviates from the average density in a representative volume of the Universe. ν takes negative (sub-dense) and positive (super-dense) values in the figure. Finally, the genus is defined as an integral over the intrinsic local Gaussian curvature K of the excursion surface over the volume 7 : 7 The Gaussian curvature to any surface is calculated with reference to the vector normal to the surface at any point. A bundle of planes containing the orthogonal vector to any given point will intersect the surface along many D curves, all having their own curvature k. In general, these sections will have different curvature values, that may be positive or negative. K is defined as the product of the maximum and the minimum of these curvature values (the principal curvatures): K max[ k] min[ k]. K can then be positive or negative. For all points of the hyperboloid below K<0, for the cylinder K=0 (either k or k being null), for the sphere K>0. 1 Large Scale Structure of the Universe 1. 43

44 4 π[1 G( ν)] = K ds [1.48] where the integral is taken over the excursion set surface S ν corresponding to the threshold ν. The genus so defined is essentially a measure of the number of "handles" a surface has: a sphere has zero handle, hence has zero genus, a torus has 1 handle, so genus G=1, and so on. In general, simply connected regions (spheres) have genus G<1, multiply connected surfaces have genus G>1. Another related quantity is the genus per unit volume, the genus density: S ν 1 gs ( ν) ( G[ ν] 1) V = K ds 4πV. [1.49] Sν "Gaussian" density fields (note the minus sign here compared to eq. [1.48]). At this stage, it is appropriate to better define the concept of Gaussian density fields. We have encountered them at various stages previously and we will still find them in the following. By Gaussian fields we mean a distribution that, independently from the specific form of the power-spectrum representing the field, to any positive fluctuation with respect to the mean density over a given spatial scale, there should be a negative fluctuations on the same scale and of the same amplitude, on average. There should be perfect symmetry forward-backward with respect to the mean density. Technically, we have seen in Chap.. that these Gaussian random fields are characterized by a random distribution of the phases of the waves in the Fourier development of eq. [1.3]. It can be proven that Gaussian density fields all have a characteristic distribution of the genus density: gs ( ) (1 )exp( ) ν = A ν ν [1.50] where A is a normalization constant depending only on the spectral form of the fluctuation field, e.g. on its power-spectrum. A comparison of the form in [1.50] with observations provides us a mean for testing the gaussianity of the large scale structure as a function of spatial scale: from what we said above about the evolution of fluctuations, we expect that on large scales the original gaussianity is maintained if originally present, while on small scales the non-linear collapse would introduce deviations from it. Such deviations can manifest themselves in producing 'meatball structures', that is an excess of high-density simply connected regions surrounded by low-density regions, compared with the Gaussian curve. The opposite tendency, usually called 'Swisscheese', is to have an excess of low density simply connected regions in a high density background. The latter would exactly be what one might expect to see if the cosmic explosions scenario 8, with the generation of bubbles in the cosmic fluid, would be responsible for the large-scale structure. 8 The theory of the cosmic explosions was proposed by Cowie and Ostriker and forsaw a process of sweeping of the baryonic fluid by a coordinated set of explosions of supernovae, this sweeping producing a void and the accumulation of gas to form stars on the walls surrounding the void. On physical grounds, a supernova explosion compresses the gas Large Scale Structure of the Universe 1. 44

45 Fig. 4. The genus distribution for galaxies from the Giovannelli & Heynes galaxy redshift survey and from the Center for Astrophysics CfA redshift survey as a function of the threshold density parameter ν. The lines are the predictions for a Gaussian density field. The value of zero for the density parameter ν corresponds to the average density in the sampled volume, positive values to regions of excess density, negative values to lower than average density. From the bottom to the top panels the genus distributions refer to smoothing over increasingly larger scales. The Gaussian prediction perfectly fits data on the largest scales (top). A small deviation towards meatball topology is seen in middle panel, and a stronger deviation in the bottom, referring to small smoothing scales of 600 km/s. [From Gott et al. 1989, ApJ 340, 65]. due to shock waves and hence triggers further star-formation in the high-density region. It is then a typically non-linear process, which may explain how this might produce effects on very large scales of tens Mpc. Large Scale Structure of the Universe 1. 45

46 If instead the hierarchical clustering scenario, based on the gravitational instability from an initial Gaussian fluctuation spectrum, is responsible, we would expect a meatball topology on the small scales where the non-linear collapse has operated, and a Gaussian symmetric topology on the large scales. This is indeed what the data suggest (see Fig. 4). So, the topological analysis is essential to disentangle these various hypotheses. The smoothing required however poses a problem, however, because present redshift surveys sample space only rather sparsely and one needs to smooth rather heavily to construct a continuous field. A smoothing on scales much larger than the scale at which correlations are significant will tend to produce a Gaussian distribution because of the central limit theorem. The power of this method is therefore limited by the smoothing required, which, in turn, depends on the space-density of the galaxy sample (if the latter is too low, the method looses its effectiveness to reveal departures from gaussianity). This is particularly a problem in the topological analysis of the QDOT redshift survey (Coles & Lucchin). Topological information can also be obtained from two-dimensional data sets, whether these are simply projected galaxy positions on the sky (such as the Lick map, or the APM survey) or 'slices' (such as the various CfA compilations). There are some subtleties, however. Firstly, two-dimensional information topology does not distinguish between 'sponge' and 'Swiss-cheese' alternatives. Nevertheless, it is possible to assess whether, for example, the mean density level (ν = 0) is dominated by under-dense or over-dense regions so that one can distinguish Swiss-cheese and meatball alternatives to some extent. Note also that the expression corresponding to [1.50] for the Gaussian-distributed samples in D is slightly different, the probability distribution of Genus in this case is rather: g ( ν) = Bν exp( ν ). The methods used for the study of two-dimensional galaxy clustering can also be used to analyse the pattern of fluctuations on the sky seen in the cosmic microwave background. S Large Scale Structure of the Universe 1. 46

47 Chapter DISCUSSION In this Section we have offered an overview of the most important statistical analyses of galaxy clustering and quantitative measurements of the cosmological large-scale structure. We have summarized the main statistical descriptors and their properties. To begin with, a variety of methods give relatively direct constraints on the power spectrum of the matter fluctuations; the two-point correlation function, the galaxy power spectrum and the variance of the counts-in-cells distribution are all related in a relatively simple way to this. In addition to those that we have described here, many other statistical descriptors have been employed in this field, particularly with respect to the problem of detecting filaments, sheets and voids in the large-scale distribution. Friends-of-friends and percolation analyses. Just to mention it, one useful statistics is offered by the percolation analysis, based on the friend-of-friend concept. Assume to consider a representative volume of the universe of side L containing N sampling points (e.g. galaxies) and consider to draw a sphere of radius R around each. This will define galaxies as friends if their sphere intercept one the other. By increasing R the friend-of-friends increase till they fill the whole volume. We define BC blthe percolation parameter such that with R = b the collection of all defined clusters of friends touch two opposite sides of the volume. The distribution of B C quantifies the structure of the matter distribution, if preferentially homogeneous, in - dimensional surfaces, or preferentially in bars and filaments, because the B C parameter decreases correspondingly. Another potentially useful statistics is the voids statistics, i.e. counting the size distribution of void regions in space, that we do not cover here. We finish this Section by mentioning here, in any case, some general problems about all these analyses. The cosmological bias. One of the problems is the ubiquitous effect of the so-called cosmological bias. This concerns how well the different tracers of the large scale structure provide a correct view of it, or else how much biased is the sampling they provide us. This means that, however robustly one can measure galaxy fluctuations observationally, one cannot interpret these data in terms of a physical model of the mass distribution without assuming some ad hoc relationship between galaxies and (mostly dark) mass. The simplest assumption we can make about this is that the bias effect is linear. If the bias is of the linear form, then there is a simple constant factor between the mass Large Scale Structure of the Universe 1. 47

48 statistic and the galaxies statistic, so that, for example, the shape of the galaxy galaxy correlation function and the shape of the matter auto-covariance function are the same, but the amplitudes are different. In this case, the uncertainty related with the bias problem is parametrized with the constant such that δρ b δρ ξ r = b ξ r s = b s k = b k. gal mass ( ) ( ), and gal mass, gal ( ) mass ( ) gal mass In the simplest conceivable case of a linear bias, the various statistics extracted from galaxy clustering, ξ(r), Δ (k) and σ, are all a factor b higher (or lower) than the corresponding quantities for the mass fluctuations. More in general, the relationship between galaxy and mass statistics may be quite more complex than this. Of course, different tracers of the large-scale structure will have different values of the bias parameter. For example, red optical galaxies have different bias (and typically b > 1) to the IRAS galaxies (typically b 1), as we have seen in Chap This problem can be addressed and hopefully resolved by comparing the properties of the large-scale structure as mapped by the different tracers. On this regard, a particular importance have sampling methods of the structure directly sensitive to the total content of gravitating matter, hence to dark-matter, that will be discussed in Sect.. One other way is to look not just at the positions of galaxies, but also at their peculiar motions. These motions are generated by gravity which, in turn, is produced by the whole mass distribution, not just by the luminous part. And this will be addressed in our subsequent Sect. 4. Redshift space distortions. The second problem, as we just mentioned, is that we have dealt largely with the distribution of galaxies in redshift space. The existence of peculiar motions makes the relationship between real space and redshift space rather complicated. This problem is, however, potentially useful in some cases, because the distortion of various statistics in redshift space relative to real space can give us information rather directly about the peculiar velocities (see e.g. Chap. 1.5 above) and hence, indirectly, about the distribution of mass fluctuations through the continuity equation. As we anticipated, we return to this matter in Sect. 4. Cosmic variance. The third, even more fundamental, problem goes under the name of cosmic variance: one needs to be sure that the sample of galaxies and the survey volume used for the statistical analyses to measure clustering in our observed Universe are large enough Large Scale Structure of the Universe 1. 48

49 to be, in some sense, representative of the Universe as a whole. If one extracts a statistical measure of clustering from a finite and too small sample, then the value of the statistic would be different if one took a sample of the same size at a different place in the Universe. So, even under the assumptions of the cosmological principle, and of the ergodic hypothesis (end Sect. 1.1), one might not obtain a good sampling of the large-scale structure if the sampled volumes are too small. This problem may become insurmountable in some specific situations where we would need extremely large volumes to measure the quantity under investigation. One such an example is when we want to determine the luminosity functions of very luminous and rare objects, like very massive galaxies, cluster of galaxies, or luminous quasars at low redshifts: the available volume of the Universe might be too small for reaching enough statistics. One other is the cosmic power-spectrum at very large scales: we would need many universes to achieve enough global volume to properly sample it. Another classical situation is about the CMB background fluctuations on very large angular scales, of several tens degrees: too few independent sky pixels are available over the whole 4π of sky to measure these fluctuations credibly, and large error-bars on data-points are then unavoidable. We will return to this in Sect. 6. In spite of all these problems, most of which can however be reduced or even solved by matching results from various independent methodologies, a rather clear picture is emerging about the general pattern of cosmological structures. To summarize: the reference framework remains the Cosmological Principle and the general homogeneity and isotropy it implies on very large scales; a high degree of scale-invariance revealed by the power-spectrum (and galaxy correlation functions); a very high degree of symmetry proven by the lack of any evidence for deviations from gaussianity on large scales shown by the topological investigations, together with the evidence for the filamentary and meatball structures turning on at progressively smaller and smaller scales, induced by the non-linear collapse. All this, together with the quantitative determinations of the cosmological powerspectrum, makes essential information for our subsequent analyses. Large Scale Structure of the Universe 1. 49

50 APPENDIX 1.A.1 DERIVATION OF THE LIMBER EQUATION This relation between the angular and spatial correlation function was first discussed by Rubin (1954) and Limber (1954). Following Coles and Lucchin (1995), let us take Φ ( L) as the galaxy luminosity function, typically a Schechter function with characteristic luminosity L *, such that Φ ( L) will converge rapidly at larger luminosities (note that a very similar derivation would be made in terms of the magnitudes). The combined probability to have a galaxy with luminosity L 1 in the differential element dl 1 and in a differential volume dv 1 is given by d P = Φ( L) dl dv [A.1] (this expression defines the luminosity function Φ ( L) ). The joint probability to have a second one within dl and dv at a distance r 1 may be written as d P = [ Φ( L ) Φ ( L ) + G( L, L, r )] dl dv dl dv [A.] where the function G expresses the probability of spatial correlation between the two. If we adopt now the Limber hypothesis, that the correlation properties are completely independent on the luminosity classes, then we can factorize this function as GL (, L, r ) = Φ( L) Φ ( L) ξ ( r ). [A.3] We need now to simulate to some extent the observational effects when looking at a projected sky distribution of objects. If we refer to a flux-limited sample, then we will detect all galaxies with flux density brighter than a threshold value S 0 within a given sky area. To be slightly more general, let us introduce a selection function for our galaxy sample dependent only on the galaxy flux: f(s/s 0 ) gives us the probability for a galaxy to enter the selection criterion. A good sample would correspond to an f function null for S<S 0 and f=1 for S>S 0. In practical situations, the f function will smoothly vary from 1 to 0 (this was referred to as the sky-coverage in Sect.3 of the Third Year course). Assuming that we analyse samples at low redshifts, such that we are confined to the local Euclidean sub-space, we can now define a typical (luminosity) distance of the source sample, given the limiting flux S 0 and the characteristic luminosity L * : * * d L /4πS0 = [A.4] and a similar relation in terms of the luminosity distance if working at higher redshifts. We can easily express the number of sources detected per unit solid angle as Large Scale Structure of the Universe 1. 50

51 where and [A.5] The function essentially provides us with the number of sources falling within the selection criterion and then gives us the probability for a source of any luminosity L to be selected. Now from [1.3], [A.] and [A.3] [A.6] (where there is a residual dependence on right-side integral due to the variable such that this one is the inner of a double integration). We need now to express the distance r 1. Following the graph below, we have [A.7] Large Scale Structure of the Universe 1. 51