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

358 Statistics of Galaxy Clustering

Figure 16.5 Numerical simulation of galaxy clustering (left) together with a version generated randomly reshu ing the phases between Fourier modes of the original picture (right). The reshu ing operation preserves the reality of the original image.

Gaussian random fields possess Fourier modes whose real and imaginary parts are independently distributed. In other words, they have phase angles φk that are independently distributed and uniformly random on the interval [0, 2π]. When fluctuations are small, i.e. during the linear regime, the Fourier modes evolve independently and their phases remain random. In the later stages of evolution, however, wave modes begin to couple together. In this regime the phases become non-random and the density field becomes highly non-Gaussian (Coles and Chiang 2000). Phase coupling is therefore a key consequence of nonlinear gravitational processes if the initial conditions are Gaussian. Such phenomena consequently display a potentially powerful signature to exploit in statistical tests of this class of models.

A graphic demonstration of the importance of phases in patterns generally is given in Figure 16.5. The power spectrum P(k) is formally defined by an expression of the form

to take account of the fact that the density field is real we have that δk = δ−k. Since the amplitude of each Fourier mode is unchanged in the phase-reshu ing operation shown in Figure 16.5, the two pictures have exactly the same power spectrum, P(k). In fact, they have more than that: they have exactly the same amplitudes for all k. They also have totally di erent morphology. The shortcomings of P(k) as a descriptor of pattern can be partly ameliorated by defining higher-order quantities such as the bispectrum (Peebles 1980; Matarrese et al. 1997; Scoccimarro et al. 1999). The bispectrum is simply a three-point correlation function in redshift space. By analogy with (16.9.5) we have

The bispectrum is zero unless the three vectors ki form a triangle. The function B(k1, k2, k3) is particularly useful in redshift space, a fact we shall revisit in more detail in Chapter 18.

This idea can be generalised to arbitrary order correlations in Fourier space –

˜ 2 the polyspectra. Alternatively, one can study correlations of quantities like δ(k)

(Stirling and Peacock 1996). This is a special case of a four-point correlation function in Fourier space.

16.10 Percolation Analysis

Useful though the correlation functions and related quantities undoubtedly are, their interpretation is problematic, except perhaps in the framework of a model such as the hierarchical model. In particular, it is di cult to give a geometrical interpretation to the correlation functions. For this reason, it is useful to develop a di erent kind of statistical description of galaxy clustering which is more directly related to geometry. We would be interested particularly in a descriptor which revealed whether the distribution has a significant tendency to cluster in sheets, filaments or isolated clumps.

One possible such description is furnished by percolation analysis, which we now describe (Shandarin 1983; Dekel and West 1985). Imagine we have a cubic sample of the Universe of side L, containing N 1 points (galaxies, clusters, etc.).

the mean interparticle distance. If the spheres around two points overlap with each other, then we connect the two points: they become ‘friends’. If one of the spheres connects with another point, then those two points become ‘friends’ also. Applying the principle ‘the friend of my friend is also my friend’, all three points now become connected. At a given value of b, therefore, the distribution will consist of some isolated points and some connected ‘clusters’ (sets of ‘friends of friends’). For very small b all points will be isolated (nobody has any friends), while, for large b, all points will be connected (everybody is friends with everybody else). As b increases the number of clusters therefore decreases from N to 1, while the typical number of points per cluster increases from 1 to N. For a particular value, say bc (at least) one cluster forms which can connect two opposite faces of the cube. At this point the system is said to have percolated, and bc is the percolation parameter. (Sometimes in the literature the quantity Bc = 4πbc3/3 is called the percolation parameter.) The value of bc depends on the geometry of the spatial distribution of the points, on N and on L. Let us illustrate this with some simple examples.

For a uniform distribution of points on a cubic lattice it is clear that bc = 1. For a uniform distribution of particles in parallel planes of thickness h L, separated from each other by a distance λ, percolation will be completed in each plane at a value of the percolation parameter

360 Statistics of Galaxy Clustering

For a regular distribution on bars of square cross-section with side h L, separated by a distance λ, percolation again occurs simultaneously along each bar at a value of bc given by

Compared with a uniform distribution within a cube of side L, percolation occurs more easily, i.e. at a smaller value of bc, for a distribution on parallel planes and even more easily for a distribution on parallel bars.

For a uniform distribution in small cubes of side h L, separated by a distance

in this case percolation is more di cult than in the uniform case, or in the case of planes or bars.

It has been shown that, if the points are distributed randomly, the values of bc from sample to sample are distributed according to a Gaussian distribution with a mean value and dispersion which decrease as N increases; in particular we have bc,N→∞ 0.87.

A percolation analysis of the Local Supercluster has given an estimate bc 0.67, less than that expected for a random distribution. This is some empirical confirmation of the existence of some kind of geometrical structure, though it is di cult to say whether it means filaments or sheets. Indeed, according to N-body experiments, it seems that the values of b are not particularly sensitive to di erent choices of power spectrum, even for extremes such as HDM and CDM. This does not, however, mean that percolation analysis is not useful. There are many other diagnostics of the transition into the percolated regime in addition to bc. For example, it has been suggested that a useful method might be to look at the increase in the number of members of the second largest cluster as a function of b; the largest cluster essentially determines bc, but there will be many smaller clusters whose behaviour might be more sensitive to details of the spectrum than bc. One might also look at the distribution function of the sizes of percolated regions. Despite its simple geometrical interpretation and apparent e ectiveness, percolation theory is relatively neglected in cosmological studies, although it is used extensively, for example, in condensed matter physics; see Stau er and Aharony (1992). An example of the e ective use of percolation methods is given in Sahni et al. (1997).

Incidentally, a variant of percolation analysis is used in N-body simulations and in the making of catalogues of galaxy groups to identify overdense regions. In this context, particles are connected together by a friends-of-friends algorithm in the same way as was discussed above, but for these studies a value of b in the range 0.2–0.4 is usually used to define clusters and b is called the linking parameter in such applications.

We should also mention that many other statistics have been suggested for detecting and quantifying sheets and filaments in the galaxy distribution using

Topology 361

techniques from many diverse branches of mathematics, including graph theory and combinatorics; see, for example, Sahni et al. (1998). Although these have yet to yield dramatically interesting results, their likely sensitivity to high-order correlations makes it probable that they will come into their own when the next generation of very large-scale redshift surveys are available for analysis.

16.11 Topology

Interesting though the geometry of the galaxy distribution may be, such studies do not tell us about the topology of clustering or, in other words, its connectivity. One is typically interested in the question of how the individual filaments, sheets and voids join up and intersect to form the global pattern. Is the pattern cellular, having isolated voids surrounded by high-density sheets, or is it more like a sponge in which underand over-dense regions interlock?

Looking at ‘slice’ surveys gives the strong visual impression that we are dealing with bubbles; pencil beams (deep galaxy redshift surveys with a narrow field of view, in which the volume sampled therefore resembles a very narrow cone or ‘pencil’) reinforce this impression by suggesting that a line of sight intersects at more-or-less regular intervals with walls of a cellular pattern. One must be careful of such impressions, however, because of elementary topology. Any closed curve in two dimensions must have an inside and an outside, so that a slice through a sponge-like distribution will appear to exhibit isolated voids just like a slice through a cellular pattern. It is important therefore that we quantify this kind of property using well-defined topological descriptors.

In an influential series of papers, Gott and collaborators have developed a method for doing just this (Gott et al. 1986; Hamilton et al. 1986; Gott et al. 1989, 1990; Melott 1990). Briefly, the method makes use of a topological invariant known as the genus, related to the Euler–Poincaré characteristic, of the isodensity surfaces of the distribution. To extract this from a sample, one must first smooth the galaxy distribution with a filter (usually a Gaussian is used; see Section 14.3) to remove the discrete nature of the distribution and produce a continuous density field. By defining a threshold level on the continuous field, one can construct excursion sets (sets where the field exceeds the threshold level) for various density levels. An excursion set will typically consist of a number of regions, some of which will be simply connected, e.g. a deformed sphere, and others which will be multiply connected, e.g. a deformed torus is doubly connected. If the density threshold is labelled by ν, the number of standard deviations of the density away from the mean, then one can construct a graph of the genus of the excursion sets at ν as a function of ν: we call this function G(ν). The genus can be formally expressed as an integral over the intrinsic curvature K of the excursion set surfaces, Sν , by means of the Gauss–Bonnet theorem.

The general form of this theorem applies to any two-dimensional manifold M with any (one-dimensional) boundary ∂M which is piecewise smooth. This latter condition implies that there are a finite number n vertices in the boundary at

362 Statistics of Galaxy Clustering

which points it is not di erentiable. The Gauss–Bonnet theorem states that

where the αi are the angle deficits at the vertices (the n interior angles at points where the boundary is not di erentiable), kg is the geodesic curvature of the boundary in between the vertices and k is the Gaussian curvature of the manifold itself. Clearly ds is an element of length taken along the boundary and dA is an area element within the manifold M. The right-hand side of Equation (16.11.1) is the Euler–Poincaré characteristic, χE, of the manifold.

This probably seems very abstract but the definition (16.11.1) allows us to construct useful quantities for both twoand three-dimensional examples. If we have an excursion set as described above in three dimensions, then its surface can be taken to define such a manifold. The boundary is just where the excursion set intersects the limits of the survey and it will be taken to be smooth. Ignoring this, we see that the Euler–Poincaré characteristic is just the integral of the Gaussian curvature over all the compact bits of the surface of the excursion set. Hence, in this case,

2πχE = K dS = 4π[1 − G(ν)]. (16.11.2)

Sν

Roughly speaking, the quantity G is the genus, which for a single surface is the number of ‘handles’ the surface possesses; a sphere has no handles and has zero genus, a torus has one and therefore has a genus of one. For technical reasons to do with the e ect of boundaries, it has become conventional not to use G but GS = G − 1. In terms of this definition, multiply connected surfaces have GS 0 and simply connected surfaces have GS < 0. One usually divides the total genus GS by the volume of the sample to produce gS, the genus per unit volume.

One of the great advantages of using the genus measure to study large-scale structure, aside from its robustness to errors in the sample, is that all Gaussian density fields have the same form of gS(ν):

where A is a spectrum-dependent normalisation constant. This means that, if one smooths the field enough to remove the e ect of nonlinear displacements of galaxy positions, the genus curve should look Gaussian for any model evolved from Gaussian initial conditions, regardless of the form of the initial power spectrum, which only enters through the normalisation factor A. This makes it a potentially powerful test of non-Gaussian initial fluctuations, or of models which invoke non-gravitational physics to form large-scale structure. The observations support the interpretation that the initial conditions were Gaussian, although the distribution looks non-Gaussian on smaller scales. The nomenclature for the nonGaussian distortion one sees is a ‘meatball shift’: nonlinear clustering tends to produce an excess of high-density simply connected regions, compared with the

Figure 16.6 Genus curve for galaxies in the IRAS PSCz survey. The noisy curve is the smoothed galaxy distribution, while the solid line is the best-fitting curve for a Gaussian field, Equation (16.11.3). Picture courtesy of the PSCz team.

Gaussian curve. The opposite tendency, usually called ‘Swiss cheese’, is to have an excess of low-density simply connected regions in a high-density background, which is what one might expect to see if cosmic explosions or bubbles formed the large-scale structure. What one would expect to see in the standard picture of gravitational instability from Gaussian initial conditions is a ‘meatball’ topology when the smoothing scale is small, changing to a sponge as the smoothing scale is increased. This is indeed what seems to be seen in the observations so there is no evidence of bubbles; an example is shown in Figure 16.6.

The smoothing required also 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 by virtue of the central limit theorem. The power of this method is therefore limited by the smoothing required, which, in turn, depends on the spacedensity of galaxies. An example is shown in Figure 16.6, which shows the genus curve for the PSCz survey of IRAS galaxies.

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 Center for Astrophysics (CfA) compilations). Here the excursion sets one deals with are just regions of the plane where the (surface) density exceeds some threshold. In this case we imagine the manifold referred to in the statement of the Gauss–Bonnet theorem to be not the surface of the excursion set but the surface upon which the set is defined (i.e. the sky). For reasonably small angles this can be taken to be a flat plane so that the Gaussian curvature of M is everywhere zero. (The generalisation to large

364 Statistics of Galaxy Clustering

angles is trivial; it just adds a constant-curvature term.) The Euler characteristic is then simply an integral of the line curvature around the boundaries of the excursion set:

2πχE = kg ds. (16.11.4)

In this case the Euler–Poincaré characteristic is simply the number of isolated regions in the excursion set minus the number of holes in such regions.

This is analogous to the genus, but has the interesting property that it is an odd function of ν for a two-dimensional Gaussian random field, unlike G(ν) which is even. In fact the mean value of χ per unit area on the sky takes the form

where B is a constant which depends only on the (two-dimensional) power spectrum of the random field. Notice that χ < 0 for ν < 0 and χ > 0 for ν > 0. A curve shifted to the left with respect to this would be a meatball topology, and to the right would be a Swiss cheese.

There are some subtleties with this. Firstly, as discussed above, two-dimensional topology does not really distinguish between ‘sponge’ and ‘Swiss cheese’ alternatives. Indeed, there is no two-dimensional equivalent of a sponge topology: a slice through a sponge is topologically equivalent to a slice through Swiss cheese. Nevertheless, it is possible to assess whether, for example, the mean density level (ν = 0) is dominated by underdense or overdense regions so that one can distinguish Swiss cheese and meatball alternatives to some extent. The most obviously useful application of this method is to look at projected catalogues, the main problem being that, if the catalogue is very deep, each line of sight contains a superposition of many three-dimensional structures. This projection acts to suppress departures from Gaussian statistics by virtue of the central limit theorem. Nevertheless, useful information is obtainable from projected data simply because of the size of the data sets available; as is the case with three-dimensional studies, the analysis reveals a clear meatball shift, which is what one expects in the gravitational instability picture. 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.

More recently, this approach has been generalised to include not just the Euler– Poincaré distribution but all possible topological invariants, i.e. all characteristic quantities that satisfy the requirements that they be additive, continuous, translation invariant and rotation invariant. For an excursion set defined in d dimensions there are d + 1 such quantities that can be regarded as independent. Any characteristic satisfying these invariance properties can be expressed in terms of linear combinations of these four independent quantities. These are usually called Minkowski functionals. Their use in the analysis of galaxy-clustering studies was advocated by Mecke et al. (1994) and has become widespread since then.

In three dimensions there are four Minkowski functionals. One of these is the integrated Gaussian curvature (equivalent to the genus we discussed above).

Comments 365

Another is the mean curvature, H, defined by

In this expression R1 and R2 are the principal radii of curvature at any point in the surface; the Gaussian curvature is 1/(R1R2) in terms of these variables. The other two Minkowski functionals are more straightforward. They are the surface area of the set and its volume. These four quantities give a ‘complete’ topological description of the excursion sets.

16.12 Comments

In this chapter we have attempted to give a reasonably complete, though by no means exhaustive, overview of the statistical analysis of galaxy clustering. In addition to those 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. More are sure to be developed in the future, and the next generation of galaxy redshift surveys will surely furnish more accurate estimators of those statistics we have had space to describe here. By way of a summary, it is useful to delineate some common strands revealed by the various statistical approaches described in this chapter.

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. Two problems arise here, however. One is the ubiquitous problem of bias we discussed in Chapter 15. In the simplest conceivable case of a linear bias, the various statistics extracted from galaxy clustering, ξ(r), ∆2(k) and σ2, are all a factor b2 higher than the corresponding quantities for the mass fluctuations. In a more complicated biasing model, the relationship between galaxy and mass statistics may be considerably more obscure than this. The second problem is that we have dealt almost exclusively 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, at least in principle, give information indirectly about the peculiar velocities and hence about the distribution of mass fluctuations through the continuity equation; we return to this matter in Chapter 18. Within the uncertainties introduced by these factors, a consensus has emerged from these studies that the power spectrum of galaxy clustering is consistent with the shape described by Equation (16.8.4), i.e. with a di erent shape to the standard CDM scenario, but approximately fitted by a low-density CDM transfer function.

Measures of the topology and geometry of galaxy clustering are less e ective at constraining the power spectrum, but relate to di erent ingredients of models of structure formation. Percolation analysis, and other pattern descriptors

366 Statistics of Galaxy Clustering

not mentioned here, give qualitative confirmation of the existence of Zel’dovich pancakes and filaments as expected in gravitational instability theories. The behaviour of higher-order moments lends further credence to the this picture. Large-scale topology has failed to show up any significant departures from Gaussian behaviour. It seems reasonable therefore to describe all this evidence as being consistent with the basic scenario of structure formation by gravitational instability which we have sought to describe in this book. We shall see that further support for this general picture is furnished by fluctuations in the CMB temperature (Chapter 17) and studies of galaxy-peculiar motions (Chapter 18).

Bibliographic Notes on Chapter 16

The classic reference work for statistical cosmology is Peebles (1980). A more modern survey of statistical methods for cosmology applications is given by Martínez and Saar (2002). Further useful sources are Saslaw (1985) and Peacock (1992). Fall (1979) is also full of interesting ideas.

Problems

1.Suppose the Universe consists of a spherically symmetric distribution of galaxies with density profile n = n0r−α. Using an appropriate definition of the two-point correlation function, ξ(r), show that

ξ(r) r3−2α.

2.Assume the galaxy distribution consists of a collection of spherical clusters con-

taining di erent numbers of galaxies n. Let the number of clusters per unit volume as a function of n be proportional to n−β and assume all clusters containing exactly n galaxies have a radius rn nα. Show that, for ξ(r) 1 and r small,

ξ(r) r−3+(3/α)−β/α.

3.Enumerate the twelve distinct snake graphs and the four distinct star graphs for N = 4, as shown in Figure 16.3.

4.Show that, for a hierarchical distribution, the skewness of the cell-count fluctuations, γ, is related to the variance, σ2, via γ = 3Qσ4.

5.Identify the three Minkowski functionals needed to characterise an excursion set in two dimensions.

17

The Cosmic

Microwave

Background

17.1Introduction

The detection of fluctuations in the sky temperature of the cosmic microwave background (CMB) in 1992 by the COBE team led by George Smoot was an important milestone in the development of cosmology (Bennett et al. 1992; Smoot et al. 1992; Wright et al. 1992). Aside from the discovery of the CMB itself, it was probably the most important event in this field since Hubble’s discovery of the expansion of the Universe in the 1920s. The importance of the COBE detection lies in the way these fluctuations are supposed to have been generated. As we shall explain in Section 17.4, the variations in temperature are thought to be associated with density perturbations existing at trec. If this is the correct interpretation, then we can actually look back directly at the power spectrum of density fluctuations at early times, before it was modified by nonlinear evolution and without having to worry about the possible bias of galaxy power spectra.

The search for anisotropies in the CMB has been going on for around 35 years. As the experiments got better and better, and the upper limits placed on the possible anisotropy got lower and lower, theorists concentrated upon constructing models which predicted the smallest possible temperature fluctuations. The baryon-only models were discarded primarily because they could not be modified to produce low enough CMB fluctuations. The introduction of dark matter allowed such a reduction and the culmination of this process was the introduction of bias,