Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

338 Statistics of Galaxy Clustering

choose the appropriate statistical measure of fluctuations to compare the theory with the observations. Moreover, the linear approximation is only expected to be accurate on large scales where, because of the assumption of statistical homogeneity implicit in the Cosmological Principle, the fluctuation level will be small and therefore di cult to measure above sampling noise (statistical uncertainty due to finite survey size). Secondly, one needs to be sure that the sample of galaxies one uses to ‘measure’ clustering in our observed Universe is large enough to be, in some sense, representative of the Universe as a whole. If one extracts a statistical measure of clustering from a finite sample, then the value of the statistic would be di erent if one took a sample of the same size at a di erent place in the Universe. This e ect is generally known as ‘cosmic variance’, although this is not a particularly good term for the phenomenon it purports to describe. Important though these problems are, they are overshadowed by the obstacle presented by the existence of a bias, as described in Section 14.8. This means that, however accurately one can predict mass fluctuations analytically and however robustly one can measure galaxy fluctuations observationally, one cannot compare the two without assuming some ad hoc relationship between galaxies and mass like the linear bias model.

As we shall see, bias complicates all galaxy-clustering studies. If the bias is of the linear form described by Equation (14.8.10), then there is a simple constant multiplier between the ‘mass’ statistic and the ‘galaxies’ statistic so that, for example, the shape of the galaxy–galaxy correlation function and the shape of the matter autocovariance function are the same, but the amplitudes are di erent. In this case, knowing the multiplier b essentially eliminates the problem. On the other hand, the linear bias model is only expected to be applicable on very large scales (and perhaps not even then). Indeed, it is possible to imagine an extreme kind of bias which has the e ect that there is very little correlation between the positions of galaxies and concentrations of mass. This is especially the case in scenarios where the bulk of the matter of the Universe is in the form of non-baryonic and therefore non-luminous material. Fortunately, however, there are ways to circumvent the bias problem to achieve a normalisation of the power spectrum or, at least, constrain it.

One 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 generated by the whole mass distribution, not just by the luminous part. As we discussed in Section 4.6, the existence of peculiar motions means that the Hubble law is not exactly correct and consequently that a galaxy’s redshift is not directly proportional to its distance from the observer. Galaxy redshift surveys generally supply only the redshifts, which are tacitly assumed to translate directly into distances via the Hubble law. Statistical measurements based on redshift surveys are therefore ‘distorted’ by deviations from the Hubble flow. The direct use of measured peculiar velocities and the indirect use of redshift-space distortions are both discussed in detail in Chapter 18; in the present chapter we shall generally assume that we can measure the statistical quantities in question in real space without worrying about redshift space.

The other way to normalise the spectrum only recently became possible with the COBE discovery of fluctuations in the CMB temperature in 1992. These are generally thought to be due to the influence of primordial fluctuations at t trec, long before galaxy formation commenced. Knowing the amplitude of these fluctuations allows one, in principle, to compute the amplitude of the power spectrum at the present time without worrying about bias at all. We discuss this, and other issues connected to the CMB, in Chapter 17.

In the present chapter we shall concentrate on the statistical study of the clustering properties of galaxies and galaxy clusters and the relationship between observed statistical properties and theory. We shall use some of the tools introduced in Chapter 14 but will also introduce many new ones including, for example, techniques based on ideas from topology, dynamical systems and condensed matter physics. Di erent statistical descriptors measure di erent aspects of the clustering pattern revealed by a survey. Some quantities, such as the two-point correlation function (Section 16.2), the cell-count variance (Section 16.6) and the galaxy power spectrum (Section 16.7) are directly related to, and can therefore constrain, the fluctuation power spectrum. Other approaches, such as percolation analysis (Section 16.9) and topology (Section 16.10), test the morphology of the large-scale galaxy distribution and may therefore be sensitive to the existence of sheets and filaments predicted in the nonlinear phase of perturbation evolution or to features, such as bubbles, which may be connected with some form of nonGaussian perturbation (Section 14.10). These methods therefore constrain a different set of ‘ingredients’ of structure-formation models. Other methods, such as higher-order correlations (Section 16.4), can shed light on whether self-similarity is important in the origin of the observed structure. We shall also take the opportunity in this chapter to show specific examples of how recent analyses of the 2dF Galaxy Redshift Survey and Sloan data using these statistical tools have yielded important constraints on models of structure formation. We shall, however, try to place an emphasis on methods rather than existing results, since we anticipate that new data will add much to our understanding of galaxy clustering in the next few years.

16.2 Correlation Functions

We begin our study of statistical cosmology by describing the correlation functions which have, for many years, been the standard way of describing the clustering of galaxies and galaxy clusters in cosmology. The use of these functions was first suggested in the 1960s by Totsuji and Kihara (1969), but their most influential advocate has been Peebles, who, along with several colleagues in the 1970s, carried out a program to extract estimates of these functions from the Lick galaxy catalogue and other data sets; see Peebles (1980) and references therein for details.

The correlation functions furnish a description of the clustering properties of a set of points distributed in space. The space can be three dimensional, but useful results are also obtainable for two-dimensional distributions of positions on the

340 Statistics of Galaxy Clustering

celestial sphere; see Section 16.3. We shall assume in this section that our ‘points’ are galaxies but this need not be the case. Indeed, this technique has been applied not only to various di erent kinds of galaxies (optical, infrared, radio) but also to quasars and clusters of galaxies; these latter objects are particularly important, for reasons we shall describe in Section 16.5. We shall also see that the correlation functions are closely related to the functions we described in Section 13.9 as the covariance functions, the di erence between covariance and correlation functions being that the former describe properties of a continuous density field while the latter describe properties of a clustered set of points.

We have met the simplest correlation function already, in Section 13.9, but we give a more complete definition here. The joint probability δ2P2 of finding one galaxy in a small volume δV1 and another in the volume δV2, separated by a vector r12, if one chooses the two volumes randomly within a large (representative) volume of the Universe, is given by

where nV is the mean number-density of galaxies and the function ξ(r) is called the two-point galaxy–galaxy spatial correlation function. Because of statistical homogeneity and isotropy, ξ depends only on the modulus of the vector r12 (which we have written r12 in the equation) and not on its direction. If the galaxies are sprinkled completely randomly in space, then it is clear that ξ(r12) ≡ 0; this means that ξ represents the excess probability, compared with a uniform random distribution, of finding another galaxy at a distance r12 from a given galaxy. If ξ(r) > 0, then galaxies are clustered, and if ξ(r) < 0, they tend to avoid each other. For reasons we explained in Section 14.9, if the correlation function is positive at r12 0, it must change sign at large r12 so that its volume integral over all r12 does not diverge. Equation (16.2.1) implies, for example, that the mean number of galaxies within a distance r of a given galaxy is

the second term on the right-hand side of this equation represents the excess number compared with a uniform random distribution.

The two-point correlation function of a self-gravitating distribution of matter evolves rapidly in the nonlinear regime. This means that the shape of ξ(r) in the regime where ξ 1 or greater will be very di erent from that of the primordial correlation function, and the amplitude will be di erent from that expected from linear theory. For this reason one cannot expect to use observations of ξ(r) directly to normalise the spectrum. Notice, however, that the second term on the right-hand side of Equation (16.2.2) is an integral over ξ which is weighted to large r, and hence to regions of small ξ(r). This motivates the use of the quantity J3, defined by

with R up to several tens of Mpc, to obtain the normalisation; WTH is the top-hat window function introduced in Section 13.3. This kind of normalisation was used frequently before the discovery of CMB temperature fluctuations.

Let us stress again that ξ(r) measures the correlations between galaxies, not the correlations of the mass distribution. These might be equal if galaxies trace the mass, but if galaxy formation is biased they will di er. In the linear bias model – equation (14.8.10) – the galaxy–galaxy correlations will be a factor b2 higher than the mass correlations.

If one only has a two-dimensional (projected) catalogue, then one can define the two-point galaxy–galaxy angular correlation function, w(ϑ), by

which, in analogy with (16.2.1), is just the probability of finding two galaxies in small elements of solid angle δΩ1 and δΩ2, separated by an angle ϑ12 on the celestial sphere; nΩ is the mean number of galaxies per unit solid angle on the sky.

In an analogous manner one can define the correlation functions for N > 2 points; we mentioned this in Section 13.9. The definition proceeds from equation (13.8.15), which gives the probability of finding N galaxies in the N (disjoint) volumes δVi in terms of the total N-point correlation function ξ(N). This function, however, contains contributions from correlations of lower order than N and a more useful statistic is the reduced or connected correlation function, which is simply that part of ξ(N) which does not depend on correlations of lower order; we shall use ξ(N) for the connected part of ξ(N). One can illustrate the way to extract the reduced correlation function simply using the three-point function as an example. Using the cluster expansion in the form given by equation (13.8.13) and, as instructed in Section 13.9, interpreting the single partitions δi as having the value of unity for point distributions rather than the zero value one uses in the case for continuous fields, we find

δ3P3 = n3V [1 + ξ(r12) + ξ(r23) + ξ(r31) + ζ(r12, r23, r31)]δV1δV2δV3, (16.2.5)

where ζ ≡ ξ(3) is the reduced three-point function. The terms ξ(rij) represent the excess number of triplets one gets compared with a random distribution (described by the ‘1’) just by virtue of having more pairs than in a random distribution; the term ζ is the number of triplets above that expected for a distribution with a given two-point correlation function. From now on we shall drop the term ‘connected’ or ‘reduced’; whenever we use an N-point correlation function, it will be assumed to be the reduced one. The three-point angular correlation function z is defined in an analogous manner:

δ3P3 = n3Ω[1+w(ϑ12)+w(ϑ23)+w(ϑ31)+z(ϑ12, ϑ23, ϑ31)]δΩ1δΩ2δΩ3, (16.2.6)

which is the probability of finding galaxies in the three solid-angle elements δΩ1, δΩ2 and δΩ3, separated by angles ϑ12, ϑ23 and ϑ31 on the celestial sphere. For

342 Statistics of Galaxy Clustering

N = 4 the spatial correlation function η ≡ ξ(4) is defined by

δ4P4 = n4V [1 + ξ(r12) + ξ(r13) + ξ(r14) + ξ(r23) + ξ(r24) + ξ(r34)

+ξ(r12)ξ(r34) + ξ(r13)ξ(r24) + ξ(r14)ξ(r23)

+ζ(r12, r23, r31) + ζ(r12, r24, r41) + ζ(r13, r34, r41)

+ζ(r23, r34, r42) + η(r12, r13, r14, r23, r24, r34)]δV1δV2δV3δV4

(16.2.7)

in an obvious notation; one can also define the four-point angular function u in an appropriate manner. The usual notation for the five-point spatial function is τ ≡ ξ(5) and, for its angular version, t.

16.3 The Limber Equation

One of the most useful aspects of the correlation functions, particularly the twopoint correlation function, is that its spatial and angular versions have a relatively simple relationship between them. This allows one to extract an estimate of the spatial function from the angular version. In Section 4.5 we introduced the luminosity function Φ(L). Let us convert this into a function of magnitude M, as described in Section 1.8, via Ψ(M) = Φ(L)|dL/dM|. This allows us to write

which is the probability of finding a galaxy with absolute magnitude between M and M +δM in the volume δV. By analogy with Equation (16.2.1) we can also write the joint probability of finding two galaxies, one in δV1 with magnitude between M1 and M1 +δM1 and the other in δV2 with magnitude between M2 and M2 +δM2, separated by a distance r12, as

where the function G takes account of the correlations between the galaxies. We now suppose that the absolute magnitude of a galaxy is statistically independent of its position with respect to other galaxies, that is to say that Ψ(M) is independent of the strength of clustering. This hypothesis, called the Limber hypothesis, seems to be verified by observations but is actually quite a strong assumption: it means, for example, that there is no variation of the luminosity properties of galaxies with the density of their environment. We then write

Projected catalogues generally collect the positions of galaxies brighter than a certain apparent magnitude limit m0 within some well-defined region on the celestial sphere. To take account of systematic observational errors concerning the objects with apparent magnitude m m0, one introduces a selection function f(m−m0)

which is the probability that an observer includes a galaxy with apparent magnitude m in the catalogue. The function f should be equal to unity for m m0 (galaxies much brighter than m0), and practically zero for m m0. A good catalogue will also have a sharp cut-o at m m0, though this is not always realised in practice. The luminosity function of galaxies has a characteristic magnitude at M −19.5 + 5 log h and tends rapidly to zero for M < M . Let us assume that the typical distance from the observer of galaxies in the catalogue is D , the distance at which a galaxy with absolute magnitude M is seen with an apparent magnitude m0; from Equation (1.8.3) we have

The number of galaxies in a certain catalogue per unit solid angle, from Equations (16.3.1) and (16.3.4), is given by

The function ψ(x) represents the number of galaxies per unit volume, at a distance given by r = xD , belonging to the catalogue. This function is given to a good approximation by

From Equations (16.3.2) and (16.3.3) one can recover Equation (16.2.4):

δ2P2 = n2Ω[1 + w(ϑ12)]δΩ1δΩ2

Because the catalogue is assumed to be a ‘fair’ sample of the Universe, the typical length scale of correlations must be much less than D . For this reason the main

344 Statistics of Galaxy Clustering

contribution to the integral over ξ(r12) in (16.3.8) comes from points with x1 x2 1, separated by a small angle ϑ12. For this reason (16.3.9) becomes

and the Equations (16.3.8) and (16.3.11) furnish the relation

ϑ12 ∞ ψ2(x)x5 dx +∞ ξ[D xϑ12(1 + y2)1/2] dy

w(ϑ12) 0 ∞ −∞ , (16.3.12)

[ 0 ψ(x)x2 dx]2

called the Limber equation (obtained by Limber (1953, 1954) to analyse the correlations of stars in our Galaxy). This relationship has the interesting scaling property that

where w and w are the correlation functions corresponding to two catalogues with characteristic distances D and D , respectively.

One can extend the Limber equation to higher-order correlations N > 2, still assuming the Limber hypothesis. It is thus possible to relate the angular and spatial N-point functions for N > 2. We shall spare the reader the details, but just mention some of the results in the next section.

16.4 Correlation Functions: Results

16.4.1 Two-point correlations

The analysis of two-dimensional catalogues of the projected positions of galaxies on the sky (chiefly the Lick map and, more recently, the APM and COSMOS surveys) has shown that, over a suitable interval of angles ϑ, the angular two-point correlation function w(ϑ) is well approximated by a power law

where the amplitude A depends on the characteristic distance D of the galaxies in the catalogue, and the angular interval over which the relationship (16.4.1) holds corresponds to a spatial separation 0.1h−1 Mpc r 10h−1 Mpc at this distance. One can use the scaling relation (16.3.13) to compare the correlation functions of catalogues with di erent values of D and so check the assumptions upon which the analysis is based. Beyond the power-law regime the angular correlation function breaks and rapidly falls to zero.

If one makes the assumption that, over a certain interval of scale, the two-point spatial correlation function is given by

Figure 16.1 The dots with error bars show determinations of w(ϑ) from the APM survey, while the solid lines show a family of CDM models labelled by the shape parameter Γ . Figure courtesy of Steve Maddox.

(Γ is the Euler gamma function). The assumption (16.4.2) therefore appears consistent with the angular correlation function (16.4.1) if

with r0g 5h−1 Mpc and γ 1.8 in the range 0.1h−1 Mpc r 10h−1 Mpc (e.g. Shanks et al. 1989); on larger scales the correlation function tends rapidly towards zero and is di cult to measure above statistical noise. The form of ξ(r) given in (16.4.5) is confirmed by direct, i.e. three-dimensional, determinations from galaxy surveys, as shown in Figure 16.2. The quantity r0g, where ξ = 1, is often called the correlation length of the galaxy distribution; it marks, roughly speaking, the transition between linear and nonlinear regimes.

The usual method for estimating ξ(r), or w(ϑ), employs a random Poisson point process generated with the same sample boundary and selection function

346 Statistics of Galaxy Clustering

as the real data; one can then estimate ξ straightforwardly according to

where nDD(r), nRR(r) and nDR(r) are the number of pairs with separation r in the actual data catalogue, in the random catalogue and with one member in the data and one in the random catalogue, respectively. In Equations (16.4.6) and (16.4.7) we have assumed, for simplicity, that the real and random catalogues have the same number of points (which they need not). The second of these estimators is more robust to boundary e ects (e.g. if a cluster lies near the edge of the survey region), but they both give the same result for large samples.

16.5 The Hierarchical Model

The problem with the higher-order correlation functions ξ(N) is that they are functions of all the distances separating the N points and are consequently much more di cult to interpret than ξ = ξ(2), which is a function of only one variable. It therefore helps to have a model for the higher-order correlations which one can use to interpret the results. The fact that the two-point correlation function has a power-law behaviour suggests that one might look for a hierarchical model, i.e. for a self-similar behaviour of the ξ(N) in which the Nth function is related to the (N − 1)th function and thence all the way down to the two-point function, according to some simple scaling rule. Notice that this assumption is conceptually distinct from the simplified treatment of hierarchical clustering we presented in Section 14.4, i.e. the hierarchical model for correlations does not automatically follow from that discussion. In fact, the hierarchical model here rests on the assumption of scale invariance, i.e. that the higher-order correlations possess no characteristic scale. The appropriate model for the three-point function is

where Q is a constant. This form does indeed appear to fit observations fairly well, with a value Q 1 over the range 50h−1 kpc < r < 5h−1 Mpc. The appropriate generalisation of Equation (16.5.1) to N > 3 is more complicated, and involves a bit of combinatorial analysis:

Figure 16.2 Estimates of ξ(r) from di erent redshift surveys, including the Las Campanas Redshift Survey shown in Figure 4.6. The variable s is shown instead of r to denote determination in redshift space, rather than real space; see Section 18.5. Figure courtesy of Tom Shanks.