Measuring our Universe from Galaxy Redshift Surveys

The cosmological principle

Our current Universe exhibits a wealth of nonlinear structures, but the zero-th order description of our Universe is based on the assumption that the Universe is homogeneous and isotropic smoothed over sufficiently large scales. This statement is usually referred to as the cosmological principle. In fact, the cosmological principle was first adopted when observational cosmology was in its infancy; it was then little more than a conjecture, embodying’ Occam’s razor’ for the simplest possible model.

Rudnicki [73] summarized various forms of cosmological principles in modern-day language, which were stated over different periods in human history based on philosophical and aesthetic considerations rather than on fundamental physical laws:

• The ancient Indian cosmological principle:

  The Universe is infinite in space and time and is infinitely heterogeneous.

• The ancient Greek cosmological principle:

  Our Earth is the natural center of the Universe.

• The Copernican cosmological principle:

  The Universe as observed from any planet looks much the same.

• The (generalized) cosmological principle:

  The Universe is (roughly) homogeneous and isotropic.

• The perfect cosmological principle:

  The Universe is (roughly) homogeneous in space and time, and is isotropic in space.

• The anthropic principle:

  A human being, as he/she is, can exist only in the Universe as it is.

We note that the ancient Indian principle may be viewed as a sort of ‘fractal model’. The perfect cosmological principle led to the steady state model, which although more symmetric than the (generalized) cosmological principle, was rejected on observational grounds. The anthropic principle is becoming popular again, e.g., in ‘explaining’ the non-zero value of the cosmological constant.

Like with any other idea about the physical world, we cannot prove a model, but only falsify it. Proving the homogeneity of the Universe is particularly difficult as we observe the Universe from one point in space, and we can only deduce isotropy indirectly. The practical methodology we adopt is to assume homogeneity and to assess the level of fluctuations relative to the mean, and hence to test for consistency with the underlying hypothesis. If the assumption of homogeneity turns out to be wrong, then there are numerous possibilities for inhomogeneous models, and each of them must be tested against the observations.

For that purpose, one needs observational data with good quality and quantity extending up to high redshifts. Let us mention some of those:

• CMB fluctuations

  Ehlers, Garen, and Sachs [18] showed that by combining the CMB isotropy with the Copernican principle one can deduce homogeneity. More formally their theorem (based on the Liouville theorem) states that “If the fundamental observers in a dust space-time see an isotropic radiation field, then the space-time is locally given by the Friedman-Robertson-Walker (FRW) metric”. The COBE (COsmic Background Explorer) measurements of temperature fluctuations (ΔT/T = 10⁻⁵ on scales of 10°) give via the Sachs-Wolfe effect and the Poisson equation r.m.s. density fluctuations of δρ/ρ ∼ 10⁻⁴ on 1000 h⁻¹ Mpc (see, e.g., [99]), which implies that the deviations from a smooth Universe are tiny.

• Galaxy redshift surveys

  The distribution of galaxies in local redshift surveys is highly clumpy, with the Supergalactic Plane seen in full glory. However, deeper surveys like 2dF and SDSS (see Section 6) show that the fluctuations decline as the length-scales increase. Peebles [69] has shown that the angular correlation functions for the Lick and APM (Automatic Plate Measuring) surveys scale with magnitude as expected in a Universe which approaches homogeneity on large scales. While redshift surveys can provide interesting estimates of the fluctuations on intermediate scales (see, e.g., [72]), the problems of biasing, evolution, and K-correction would limit the ability of those redshift surveys to ‘prove’ the cosmological principle. Despite these worries the measurement of the power spectrum of galaxies derived on the assumption of an underlying FRW metric shows good agreement with the A-CDM (cold dark matter) model.

• Radio sources

  Radio sources in surveys have a typical median redshift of , and hence are useful probes of clustering at high redshift. Unfortunately, it is difficult to obtain distance information from these surveys: The radio luminosity function is very broad, and it is difficult to measure optical redshifts of distant radio sources. Earlier studies claimed that the distribution of radio sources supports the cosmological principle. However, the wide range in intrinsic luminosities of radio sources would dilute any clustering when projected on the sky. Recent analyses of new deep radio surveys suggest that radio sources are actually clustered at least as strongly as local optical galaxies. Nevertheless, on very large scales the distribution of radio sources seems nearly isotropic.

• X-ray background

  The X-ray background (XRB) is likely to be due to sources at high redshift. The XRB sources are probably located at redshift z < 5, making them convenient tracers of the mass distribution on scales intermediate between those in the CMB as probed by COBE, and those probed by optical and IRAS redshift surveys. The interpretation of the results depends somewhat on the nature of the X-ray sources and their evolution. By comparing the predicted multipoles to those observed by HEAO1, Scharf et al. [75] estimate the amplitude of fluctuations for an assumed shape of the density fluctuations. The observed fluctuations in the XRB are roughly as expected from interpolating between the local galaxy surveys and the COBE and other CMB experiments. The r.m.s. fluctuations δρ/ρ on a scale of ∼ 600 h⁻¹ Mpc are less than 0.2%.

Since the (generalized) cosmological principle is now well supported by the above observations, we shall assume below that it holds over scales l > 100h⁻¹ Mpc.

The rest of the current section is devoted to a brief review of the homogeneous and isotropic cosmological model. Further details may be easily found in standard cosmology textbooks [96, 62, 69, 64, 10, 63].

The cosmological principle is mathematically paraphrased as that the metric of the Universe (in its zero-th order approximation) is given by

1

where x is the comoving coordinate, and where we use units in which the light velocity c = 1. The above Robertson-Walker metric is specified by a constant K, the spatial curvature, and a function of time a(t), the scale factor.

The homogeneous and isotropic assumption also implies that T_μν, the energy-momentum tensor of the matter field, should take the form of the ideal fluid:

2

where u_μ is the 4-velocity of the matter, ρ is the mean energy density, and p is the mean pressure.

From the Einstein equation to the Friedmann equation

The next task is to write down the Einstein equation,

3

using Equations (1) and (2). In this case one is left with the following two independent equations:

4

5

for the three independent functions a(t), ρ(t), and p(t).

Differentiation of Equation (4) with respect to t yields

6

Then eliminating ä with Equation (5), one obtains

7

This can be easily interpreted as the first law of thermodynamics, dQ = dU − _pdV = d(ρa³) − pd(a³) = 0, in the present context. Equations (4) and (7) are often used as the two independent basic equations for a(t), instead of Equations (4) and (5).

In either case, however, one needs another independent equation to solve for a(t). This is usually given by an equation of state of the form p = p(ρ). In cosmology, the following simple relation is assumed:

8

while the value of w may in principle change with redshift, it is often assumed that w is independent of time just for simplicity. Then substituting this equation of state into Equation (7) immediately yields

9

The non-relativistic matter (or dust), ultra-relativistic matter (or radiation), and the cosmological constant correspond to w = 0, 1/3, and −1, respectively.

If the Universe consists of different fluid species with w_i (i = 1,…, N), Equation (9) still holds independently as long as the species do not interact with each other. If one denotes the present energy density of the i-th component by ρi,0, then the total energy density of the Universe at the epoch corresponding to the scale factor of a(t) is given by

10

where the present value of the scale factor, a₀, is set to be unity without loss of generality. Thus, Equation (4) becomes

11

Note that those components with w_i = −1 may be equivalent to the conventional cosmological constant Λ at this level, although they may exhibit spatial variation unlike Λ.

Evaluating Equation (11) at the present epoch, one finds

12

where H₀ is the Hubble constant at the present epoch. The above equation is usually rewritten as follows:

13

where the density parameter for the i-th component is defined as

14

and similarly the dimensionless cosmological constant is

15

Incidentally Equation (13) clearly illustrates the Mach principle in the sense that the space curvature is simply determined by the amount of matter components in the Universe. In particular, the flat Universe (K = 0) implies that the sum of the density parameters is unity:

16

Finally the cosmic expansion is described by

17

As will be shown below, the present Universe is supposed to be dominated by non-relativistic matter (baryons and collisionless dark matter) and the cosmological constant. So in the present review, we approximate Equation (17) as

18

unless otherwise stated.

Expansion law and age of the Universe

Equation (18) has the following analytic solutions in several simple but practically important cases.

• Einstein-de Sitter model :

  19

• Open model with vanishing cosmological constant (Ω_m < 1, Ω_Λ = 0):

  20

  21

  22

• Spatially-flat model with cosmological constant (Ω_m < 1, Ω_Λ = 1 − Ω_m:

  23

  24

In the above, t₀ denotes the present age of the Universe,

25

and we adopt the initial condition that a = 0 at t = 0. The expression clearly indicates that t₀ increases as Ω_m decreases and/or Ω_Λ increases. Figure 1 plots the scale factor as a function of H₀(t − t₀), and Table 1 summarizes the age of the Universe.

Einstein’s static model and Lemaître’s model

So far we have shown that solutions of the Einstein equation are dynamical in general, i.e., the scale factor a is time-dependent. As a digression, let us examine why Einstein once introduced the Λ-term to obtain a static cosmological solution. This is mainly important for historical reasons, but is also interesting to observe how the operationally identical parameter (the Λ-term, the cosmological constant, the vacuum energy, the dark energy) shows up in completely different contexts in the course of the development of cosmological physics.

Consider first the case of Λ = 0 in Equations (4) and (5). Clearly the necessary and sufficient condition that the equations admit the solution of a = const. is given by

26

Namely, any static model requires that the Universe is dominated by matter with either negative pressure or negative density. This is physically unacceptable as long as one considers normal matter in the standard model of particle physics. If Λ ≠ 0 on the other hand, the condition for the static solution is

27

yielding

28

Thus both ρ and p can be positive if

29

In particular, if p = 0,

30

This represents the closed Universe (with positive spatial curvature), and corresponds to Einstein’s static model.

The above static model is a special case of Lemaître’s Universe model with Λ > 0 and K > 0. For simplicity, let us assume that the Universe is dominated by non-relativistic matter with negligible pressure, and consider the behavior of Lemaître’s model. First we define the values of the density and the scale factor corresponding to Einstein’s static model:

31

In order to study the stability of the model around the static model, consider a model in which the density at a = a_E is a factor of α(> 1) larger than ρ_E. Then

32

and Equations (4) and (5) reduce to

33

34

For the period of a ≪ a_E, Equation (33) indicates that a ∝ t^2/3 and the Universe is decelerating (ä < 0). When a reaches α^1/3a_E, ħ² takes the minimum value and the Universe becomes accelerating (ä > 0). Finally the Universe approaches the exponential expansion or de Sitter model: . If a becomes closer to unity, the minimum value reaches zero and the expansion of the Universe is effectively frozen. This phase is called the coasting period, and the case with α =1 corresponds to Einstein’s static model in which the coasting period continues forever. A similar consideration for α < 1 indicates that the Universe starts collapsing (ħ² =) before ä = 0. Thus the behavior of Lemaître’s model is crucially different if α is larger or smaller than unity. This suggests that Einstein’s static model (α = 1) is unstable.

Vacuum energy as an effective cosmological constant

So far we discussed the cosmological constant introduced in the l.h.s. of the Einstein equation. Formally one can move the Λ-term to the r.h.s. by assigning

35

This effective matter field, however, should satisfy an equation of state of p = −Ρ. Actually the following example presents a specific example for an effective cosmological constant. Consider a real scalar field whose Lagrangian density is given by

36

Its energy-momentum tensor is

37

and if the field is spatially homogeneous, its energy density and pressure are

38

Clearly if the evolution of the field is negligible, i.e., , p_ϕ ≈ − ρ_ϕ and the field acts as a cosmological constant. Of course this model is one of the simplest examples, and one may play with much more complicated models if needed.

If the Λ-term is introduced in the l.h.s., it should be constant to satisfy the energy-momentum conservation T_μν;^ν. Once it is regarded as a sort of matter field in the r.h.s., however, it does not have to be constant. In fact, the above example shows that the equation of state for the field has w = −1 only in special cases. This is why recent literature refers to the field as dark energy instead of the cosmological constant.

Gravitational instability

We have presented the zero-th order description of the Universe neglecting the inhomogeneity or spatial variation of matter inside. Now we are in a position to consider the evolution of matter in the Universe. For simplicity we focus on the non-relativistic regime where the Newtonian approximation is valid. Then the basic equations for the self-gravitating fluid are given by the continuity equation, Euler’s equation, and the Poisson equation:

39

40

41

We would like to rewrite those equations in the comoving frame. For this purpose, we introduce the position x in the comoving coordinate, the peculiar velocity v, density fluctuations δ(t, x), and the gravitational potential which are defined as

42

43

44

45

respectively. Then Equations (39) to (41) reduce to

46

47

48

where the dot and ∇ in the above equations are the time derivative for a given x and the spatial derivative with respect to x, i.e., defined in the comoving coordinate (while those in Equations (39, 40, 41) are defined in the proper coordinate).

A standard picture of the cosmic structure formation assumes that the initially tiny amplitude of density fluctuation grow according to Equations (46, 47, 48). Also the Universe smoothed over large scales approaches a homogeneous model. Thus at early epochs and/or on large scales, the nonlinear effect is small and one can linearize those equations with respect to δ and v:

49

50

51

where is the sound velocity squared.

As usual, we transform the above equations in k space using

52

then the equation for δ_k reduces to

53

if the signature of the third term is positive, δ_k has an unstable, or, monotonically increasing solution. This condition is equivalent to the Jeans criterion:

54

namely, the wavelength of the fluctuation is larger than the Jeans length λ_J which characterizes the scale that the sound wave can propagate within the dynamical time of the fluctuation Below the scale, the pressure wave can suppress the gravitational instability, and the fluctuation amplitude oscillates.

Linear growth rate of the density fluctuation

Most likely our Universe is dominated by collisionless dark matter, and thus λ_J is negligibly small. Thus, at most scales of cosmological interest, Equation (54) is well approximated as

55

For a given set of cosmological parameters, one can solve the above equation by substituting the expansion law for a(t) as described in Section 2.3. Since Equation (55) is the second-order differential equation with respect to t, there are two independent solutions; a decaying mode and a growing mode which monotonically decreases and increases as t, respectively. The former mode becomes negligibly small as the Universe expands, and thus one is usually interested in the growing mode alone.

More specifically those solutions are explicitly obtained as follows. First note that the l.h.s. of Equation (18) is the Hubble parameter at t, H(t) = ȧ/a:

56

57

The first and second differentiation of Equation (56) with respect to t yields

58

and

59

respectively. Thus the differential equation for H reduces to

60

This coincides with the linear perturbation equation for δ_k, Equation (55). Since H(t) is a decreasing function of t, this implies that H(t) is the decaying solution for Equation (55). Then the corresponding growing solution D(t) can be obtained according to the standard procedure: Subtracting Equation (55) from Equation (60) yields

61

and therefore the formal expression for the growing solution in linear theory is

62

It is often more useful to rewrite D(t) in terms of the redshift z as follows:

63

where the proportional factor is chosen so as to reproduce D(z) → 1/(1 + z) for z → ∞. Linear growth rates for the models described in Section 2.3 are summarized below:

• Einstein-de Sitter model (Ω_m = 1, Ω_Λ = 0):

  64

• Open model with vanishing cosmological constant (Ω_m < 1, Ω_Λ = 0):

  65

• Spatially-flat model with cosmological constant (Ω_m < 1, Ω_Λ = 1 − Ω_m):

  66

For most purposes, the following fitting formulae [67] provide sufficiently accurate approximations:

67

68

where

69

70

Note that Ω_m and Ω_Λ refer to the present values of the density parameter and the dimensionless cosmological constant, respectively, which will be frequently used in the rest of the review.

Figure 2 shows the comparison of the numerically computed growth rate (thick lines) against the above fitting formulae (thin lines), which are practically indistinguishable.

Gaussian random field

Consider the density contrast defined at the comoving position x_i. The density field is regarded as a stochastic variable, and thus forms a random field. The conventional assumption is that the primordial density field (in its linear regime) is Gaussian, i.e., its m-point joint probability distribution obeys the multi-variate Gaussian,

71

for an arbitrary positive integer m. Here M_ij ≡ 〈δ_iδ_j〉 is the covariance matrix, and M⁻¹ is its inverse. Since M_ij = ξ(x_i, x_j), Equation (71) implies that the statistical nature of the Gaussian density field is completely specified by the two-point correlation function ξ and its linear combination (including its derivative and integral). For an extensive discussion of the cosmological Gaussian density field, see [4].

The Gaussian nature of the primordial density field is preserved in its linear evolution stage, but this is not the case in the nonlinear stage. This is clear even from the definition of the Gaussian distribution: Equation (71) formally assumes that the density contrast distributes symmetrically in the range of −∞ < δ_i < ∞, but in the real density field δ_i cannot be less than −1. This assumption does not make any practical difference as long as the fluctuations are (infinitesimally) small, but it is invalid in the nonlinear regime where the typical amplitude of the fluctuations exceeds unity.

In describing linear theory of cosmological density fluctuations, the Fourier transform of the spatial density contrast is the most basic variable:

72

Since δ_k is a complex variable, it is decomposed by a set of two real variables, the amplitude D_k and the phase ϕ_k:

73

then linear perturbation equation reads

74

75

Equation (75) yields , and ϕ(t) rapidly converges to a constant value. Thus D_k evolves following the growing solution in linear theory.

The most popular statistic of clustering in the Universe is the power spectrum of the density fluctuations,

76

which measures the amplitude of the mode of the wavenumber k. This is the Fourier transform of the two-point correlation function,

77

if the density field is globally homogeneous and isotropic (i.e., no preferred position or direction), Equation (77) reduces to

78

Since the above expression is obtained after the ensemble average, x does not denote an amplitude of the position vector, but a comoving wavelength 2π/k corresponding to the wavenumber k = ∣k∣. It should be noted that neither the power spectrum nor the two-point correlation function contains information for the phase ϕ_k. Thus in principle two clustering patterns may be completely different even if they have the identical two-point correlation functions. This implies the practical importance to describe the statistics of phases ϕ_k in addition to the amplitude D_k of clustering.

In the Gaussian field, however, one can directly show that Equation (71) reduces to the probability distribution function of ϕ_k and D_k that are explicitly written as

79

mutually independently of k. The phase distribution is uniform, and thus does not carry information. The above probability distribution function is also derived when the real and imaginary parts of the Fourier components δ_k are uncorrelated and Gaussian distributed (with the dispersion P(k)/2) independently of k. As is expected, the distribution function (79) is completely fixed if P(k) is specified. This rephrases the previous statement that the Gaussian field is completely specified by the two-point correlation function in real space.

Incidentally the one-point phase distribution turns out to be essentially uniform even in a strongly non-Gaussian field [81, 21]. Thus it is unlikely to extract useful information directly out of it mainly due to the cyclic property of the phase. Very recently, however, Matsubara [51] and Hikage et al. [31] succeeded in detecting a signature of phase correlations in Fourier modes of mass density fields induced by nonlinear gravitational clustering using the distribution function of the phase sum of the Fourier modes for triangle wavevectors. Several different statistics which carry the phase information have been also proposed in cosmology, including the void probability function [97], the genus statistics [26], and the Minkowski functionals [57, 76].

Log-normal distribution

A probability distribution function (PDF) of the cosmological density fluctuations is the most fundamental statistic characterizing the large-scale structure of the Universe. As long as the density fluctuations are in the linear regime, their PDF remains Gaussian. Once they reach the nonlinear stage, however, their PDF significantly deviates from the initial Gaussian shape due to the strong non-linear mode-coupling and the non-locality of the gravitational dynamics. The functional form for the resulting PDFs in nonlinear regimes are not known exactly, and a variety of phenomenological models have been proposed [34, 74, 9, 25].

Kayo et al. [40] showed that the one-point log-normal PDF

80

describes very accurately the cosmological density distribution even in the nonlinear regime (the r.m.s. variance σ_nl ≲ 4 and the over-density δ ≲ 100. The above function is characterized by a single parameter σ₁ which is related to the variance of δ. Since we use δ to represent the density fluctuation field smoothed over R, its variance is computed from its power spectrum P_nl explicitly as

81

Here we use subscripts “lin” and “nl” to distinguish the variables corresponding to the primordial (linear) and the evolved (nonlinear) density fields, respectively. Then σ₁ depends on the smoothing scale R alone and is given by

82

Given a set of cosmological parameters, one can compute σ_nl(R) and thus σ₁(R) very accurately using a fitting formula for P_nl(k) (see, e.g., [67]). In this sense, the above log-normal PDF is completely specified without any free parameter.

Figure 3 plots the one-point PDFs computed from cosmological N-body simulations in SCDM, LCDM, and OCDM (for Standard, Lambda, and Open CDM) models, respectively [36, 40]. The simulations employ N = 256³ dark matter particles in a periodic comoving cube (100 h⁻¹ Mpc)³. The density fields are smoothed over Gaussian (left panels) and Top-hat (right panels) windows with different smoothing lengths: R = 2h⁻¹ Mpc, 6 h⁻¹ Mpc, and 18 h⁻¹ Mpc. Solid lines show the log-normal PDFs adopting the value of σ_nl directly evaluated from simulations (shown in each panel). The agreement between the log-normal model and the simulation results is quite impressive. A small deviation is noticeable only for δ ≲ −0.5.

From an empirical point of view, Hubble [34] first noted that the galaxy distribution in angular cells on the celestial sphere may be approximated by a log-normal distribution, rather than a Gaussian. Theoretically the above log-normal function may be obtained from the one-to-one mapping between the linear random-Gaussian and the nonlinear density fields [9]. We define a linear density field g smoothed over R obeying the Gaussian PDF,

83

where the variance is computed from its linear power spectrum:

84

if one introduces a new field δ from g as

85

the PDF for δ is simply given by , which reduces to Equation (80).

At this point, the transformation (85) is nothing but a mathematical procedure to relate the Gaussian and the log-normal functions. Thus there is no physical reason to believe that the new field δ should be regarded as a nonlinear density field evolved from g even in an approximate sense. In fact it is physically unacceptable since the relation, if taken at face value, implies that the nonlinear density field is completely determined by its linear counterpart locally. We know, on the other hand, that the nonlinear gravitational evolution of cosmological density fluctuations proceeds in a quite nonlocal manner, and is sensitive to the surrounding mass distribution. Nevertheless the fact that the log-normal PDF provides a good fit to the simulation data, empirically implies that the transformation (85) somehow captures an important aspect of the nonlinear evolution in the real Universe.

Higher-order correlation functions

One of the most direct methods to evaluate the deviation from Gaussianity is to compute the higher-order correlation functions. Suppose that x_i now labels the position of the i-th object (galaxy). Then the two-point correlation function ξ₁₂ ≡ ξ(x₁, x₂) is defined also in terms of the joint probability of the pair of objects located in the volume elements of δV₁ and δV2,

86

where is the mean number density of the objects. This definition is generalized to three- and four-point correlation functions, ζ₁₂₃ ≡ ζ(x₁, x₂, x₃) and η₁₂₃₄ ≡ η(x₁, x₂, x₃, x₄), in a straightforward manner:

87

88

Apparently ξ₁₂, ζ₁₂₃, and η₁₂₃₄ are symmetric with respect to the change of the indices. Define the following quantities with the same symmetry properties:

89

90

91

92

then it is not unreasonable to suspect that the following relations hold:

93

94

95

where Q, R_a, R_b, and R_c are constants. In fact, the analysis of the two-dimensional galaxy catalogues [68] revealed

96

The generlization of those relations for N-point correlation functions is suspected to hold generally,

97

and is called the hierarchical clustering ansatz. Cosmological N-body simulations approximately support the validity of the above ansatz, but also detect the finite deviation from it [82].

Genus statistics

A complementary approach to characterize the clustering of the Universe beyond the two-point correlation functions is the genus statistics [26]. This is a mathematical measure of the topology of the isodensity surface. For definiteness, consider the density contrast field δ(x) at the position x in the survey volume V_all. This may be evaluated, for instance, by taking the ratio of the number of galaxies N(x, V_f) in the volume V_f centered at x to its average value :

98

where σ(V_f) is its r.m.s. value. Consider the isodensity surface parameterized by a value of ν ≡ δ(x, V_f)/σ(V_f). Genus is one of the topological numbers characterizing the surface defined as

99

where k is the Gaussian curvature of the isolated surface. The Gauss-Bonnet theorem implies that the value of g is indeed an integer and equal to the number of holes minus 1. This is qualitatively understood as follows: Expand an arbitrary two-dimensional surface around a point as

100

then the Gaussian curvature of the surface is defined by κ = κ₁κ₂. A surface topologically equivalent to a sphere (a torus) has κ = 1 (κ = 0), and thus Equation (99) yields g = −1 (g = 0) which coincides with the number of holes minus 1.

In reality, there are many disconnected isodensity surfaces for a given ν, and thus it is more convenient to define the genus density in the survey volume V_all using the additivity of the genus:

101

where the A_i(i = 1 ∼ I) denote the disconnected isodensity surfaces with the same value of ν = δ(x, V_f)/σ(V_f). Interestingly the Gaussian density field has an analytic expression for Equation (101):

102

where

103

is the moment of k² weighted over the power spectrum of fluctuations P(k) and the smoothing function (see, e.g., [4]). It should be noted that in the Gaussian density field the information of the power spectrum shows up only in the proportional constant of Equation (102), and its functional form is deterimined uniquely by the threshold value ν. This ν-dependence reflects the phase information which is ignored in the two-point correlation function and power spectrum. In this sense, genus statistics is a complementary measure of the clustering pattern of Universe.

Even if the primordial density field obeys the Gaussian statistics, the subsequent nonlinear gravitational evolution generates the significant non-Gaussianity. To distinguish the initial non-Gaussianity from that acquired by the nonlinear gravity is of fundamental importance in inferring the initial condition of the Universe in a standard gravitational instability picture of structure formation. In a weakly nonlinear regime, Matsubara derived an analytic expression for the non-Gaussianity emerging from the primordial Gaussian field [49]:

104

where

105

are the Hermite polynomials: H₁ = ν, H₂ = ν² − 1, H₃ = ν³ − 3ν, H₄ = ν⁴ − 6ν² +3, H₅ = ν⁵ − 10ν³ + 15ν, … The three quantities

106

denote the third-order moments of δ. This expression plays a key role in understanding if the non-Gaussianity in galaxy distribution is ascribed to the primordial departure from the Gaussian statistics.

Minkowski functionals

In fact, genus is one of the complete sets of N + 1 quantities, known as the Minkowski functionals (MFs), which determine the morphological properties of a pattern in N-dimensional space. In the analysis of galaxy redshift survey data, one considers isodensity contours from the three-dimensional density contrast field δ by taking its excursion set F_ν, i.e., the set of all points where the density contrast δ exceeds the threshold level ν as was the case in the case of genus described in the above subsection.

All MFs can be expressed as integrals over the excursion set. While the first MF is simply given by the volume integration of a Heaviside step function Θ normalized to the total volume V_tot,

107

the other MFs V_k(k = 1, 2, 3) are calculated by the surface integration of the local . The general expression is

108

with the local Minkowski functionals for k = 1, 2, 3 given by

109

110

111

where R₁ and R₂ are the principal radii of curvature of the isodensity surface.

For a 3-D Gaussian random field, the average MFs per unit volume can be expressed analytically as follows:

112

113

114

115

where , and δ is the density contrast.

The above MFs can be indeed interpreted as well-known geometric quantities: the volume fraction V₀(ν), the total surface area V₁(ν), the integral mean curvature V₂(ν), and the integral Gaussian curvature, i.e., the Euler characteristic V₃(ν). In our current definitions (see Equations (101, 108), or Equations (102, 115)), one can easily show that V₃(ν) reduces simply to −G(ν). The MFs were first introduced to cosmological studies by Mecke et al. [57], and further details may be found in [57, 32]. Analytic expressions of MFs in weakly non-Gaussian fields are derived in [52].

Concepts and definitions of biasing

As discussed above, luminous objects, such as galaxies and quasars, are not direct tracers of the mass in the Universe. In fact, a difference of the spatial distribution between luminous objects and dark matter, or a bias, has been indicated from a variety of observations. Galaxy biasing clearly exists. The fact that galaxies of different types cluster differently (see, e.g., [16]) implies that not all of them are exact tracers of the underlying mass distribution (see also Section 6).

In order to confront theoretical model predictions for the mass distribution against observational data, one needs a relation of density fields of mass and luminous objects. The biasing of density peaks in a Gaussian random field is well formulated [37, 4], and it provides the first theoretical framework for the origin of galaxy density biasing. In this scheme, the galaxy-galaxy and mass-mass correlation functions are related in the linear regime via

116

where the biasing parameter b is a constant independent of scale r. However, a much more specific linear biasing model is often assumed in common applications, in which the local density fluctuation fields of galaxies and mass are assumed to be deterministically related via the relation

117

Note that Equation (116) follows from Equation (117), but the reverse is not true.

The above deterministic linear biasing is not based on a reasonable physical motivation. If b > 1, it must break down in deep voids because values of δ_g below −1 are forbidden by definition. Even in the simple case of no evolution in comoving galaxy number density, the linear biasing relation is not preserved during the course of fluctuation growth. Non-linear biasing, where b varies with δ_m, is inevitable.

Indeed, an analytical model for biasing of halos on the basis of the extended Press-Schechter approximation [59] predicts that the biasing is nonlinear and provides a useful approximation for its behavior as a function of scale, time, and mass threshold. N-body simulations provide a more accurate description of the nonlinearity of the halo biasing confirming the validity of the Mo and White model [35, 103].

Modeling biasing

Biasing is likely to be stochastic, not deterministic [15]. An obvious part of this stochasticity can be attributed to the discrete sampling of the density field by galaxies, i.e., the shot noise. In addition, a statistical, physical scatter in the efficiency of galaxy formation as a function of δ_m is inevitable in any realistic scenario. For example, the random variations in the density on smaller scales is likely to be reflected in the efficiency of galaxy formation. As another example, the local geometry of the background structure, via the deformation tensor, must play a role too. Such ‘hidden variables’ would show up as physical scatter in the density-density relation [87].

Consider the density contrasts of visible objects and mass, δ_obj(x,z∣R) and δ_m(x,z∣R), at a position x and a redshift z smoothed over a scale R [86]. In general, the former should depend on various other auxiliary variables defined at different locations x′ and redshifts z′ smoothed over different scales R′ in addition to the mass density contrast at the same position, δ_m(x, z∣R). While this relation can be schematically expressed as

118

it is impossible even to specify the list of the astrophysical variables , and thus hopeless to predict the functional form in a rigorous manner. Therefore if one simply focuses on the relation between δ_obj(x,z∣R) and δ_m(x,z∣R), the relation becomes inevitably stochastic and nonlinear due to the dependence on unspecified auxiliary variables .

For illustrative purposes, we define the biasing factor as the ratio of the density contrasts of luminous objects and mass:

119

Only in very idealized situations, the above nonlocal stochastic nonlinear factor in terms of δ_m may be approximated by

• a local stochastic nonlinear bias,

  120

• a local deterministic nonlinear bias,

  121

  and
• a local deterministic linear bias,

  122

From the above point of view, the local deterministic linear bias is obviously unrealistic, but is still a widely used conventional model for biasing. In fact, the time- and scale-dependence of the linear bias factor b_obj(z, R) was neglected in many previous studies of biased galaxy formation until very recently. Currently, however, various models beyond the deterministic linear biasing have been seriously considered with particular emphasis on the nonlinear and stochastic aspects of the biasing [71, 15, 87, 86].

Density peaks and dark matter halos as toy models for galaxy biasing

Let us illustrate the biasing from numerical simulations by considering two specific and popular models: primordial density peaks and dark matter halos [86]. We use the N-body simulation data of L = 100 h⁻¹ Mpc again for this purpose [36]. We select density peaks with the threshold of the peak height ν_th = 1.0, 2.0, and 3.0. As for the dark matter halos, these are identified using the standard friend-of-friend algorithm with a linking length of 0.2 in units of the mean particle separation. We select halos of mass larger than the threshold M_th = 2.0 × 10¹² h⁻¹ M_⊙, M_th = 5.0 × 10¹² h⁻¹ M_⊙, and M_th = 1.0 × 10¹³ h⁻¹ M_⊙.

Figures 5 and 6 depict the distribution of dark matter particles (upper panel), peaks (middle panel), and halos (lower panel) in the LCDM model at z = 0 and z = 2.2 within a circular slice (comoving radius of 150 h⁻¹ Mpc and thickness of 15 h⁻¹ Mpc). We locate a fiducial observer in the center of the circle. Then the comoving position vector r for a particle with a comoving peculiar velocity v at a redshift z is observed at the position s in redshift space:

123

where H(z) is the Hubble parameter at z. The right panels in Figures 5 and 6 plot the observed distribution in redshift space, where the redshift-space distortion is quite visible: The coherent velocity field enhances the structure perpendicular to the line-of-sight of the observer (squashing) while the virialized clump becomes elongated along the line-of-sight (finger-of-God).

We use two-point correlation functions to quantify stochasticity and nonlinearity in biasing of peaks and halos, and explore the signature of the redshift-space distortion. Since we are interested in the relation of the biased objects and the dark matter, we introduce three different correlation functions: the auto-correlation functions of dark matter and the objects, ξ_mm and ξ_oo, and their cross-correlation function ξ_om. In the present case, the subscript o refers to either h (halos) or ν (peaks). We also use the superscripts R and S to distinguish quantities defined in real and redshift spaces, respectively. We estimate those correlation functions using the standard pair-count method. The correlation function ξ^(S) is evaluated under the distant-observer approximation.

Those correlation functions are plotted in Figures 7 and 8 for peaks and halos, respectively. The correlation functions of biased objects generally have larger amplitudes than those of mass. In nonlinear regimes (ξ > 1) the finger-of-God effect suppresses the amplitude of ξ^(S) relative to ξ^(R), while ξ^(S) is larger than ξ^(R) in linear regimes (ξ < 1) due to the coherent velocity field.

Biasing of galaxies in cosmological hydrodynamic simulations

Popular models of the biasing based on the peak or the dark halos are successful in capturing some essential features of biasing. None of the existing models of bias, however, seems to be sophisticated enough for the coming precision cosmology era. The development of a more detailed theoretical model of bias is needed. A straightforward next step is to resort to numerical simulations which take account of galaxy formation even if phenomenological at this point. We show an example of such approaches from Yoshikawa et al. [103] who apply cosmological smoothed particle hydrodynamic (SPH) simulations in the LCDM model with particular attention to the comparison of the biasing of dark halos and simulated galaxies (see also [78]).

Galaxies in their simulations are identified as clumps of cold and dense gas particles which satisfy the Jeans condition and have the SPH density more than 100 times the mean baryon density at each redshift. Dark halos are identified with a standard friend-of-friend algorithm; the linking length is 0.164 times the mean separation of dark matter particles, for instance, at z = 0. In addition, they identify the surviving high-density substructures in dark halos, DM cores (see [103] for further details).

Figure 9 illustrates the distribution of dark matter particles, gas particles, dark halos, and galaxies at z = 0 where galaxies are more strongly clustered than dark halos. Figure 10 depicts a close-up snapshot of the most massive cluster at z = 0 with a mass M ≃ 8 × 10¹⁴M_⊙. The circles in the lower panels indicate the positions of galaxies identified in our simulation.

Figure 11 shows the joint distribution of δ_h and δ_g with the mass density field δ_m at redshift z = 0, 1, and 2 smoothed over R_s = 12 h⁻¹ Mpc. The conditional mean relation computed directly from the simulation is plotted in solid lines, while dashed lines indicate theoretical predictions of halo biasing by Taruya and Suto [87]. For a given smoothing scale, the simulated halos exhibit positive biasing for relatively small δ_m in agreement with the predictions. On the other hand, they tend to be underpopulated for large δ_m, or anti-biased. This is mainly due to the exclusion effect of dark halos due to their finite volume size which is not taken into account in the theoretical model. Since our simulated galaxies have smaller spatial extent than the halos, the exclusion effect is not so serious. This is clearly illustrated in the lower panels in Figure 11, and indeed they show much better agreement with the theoretical model.

We turn next to a more conventional biasing parameter defined through the two-point statistics:

124

where ξ_ii(r) and ξ_mm (r) are two-point correlation functions of objects i and of dark matter, respectively. While the above biasing parameter is ill-defined where either ξ_ii(r) or ξ_mm(r) becomes negative, it is not the case at clustering scales of interest (< 10 h⁻¹ Mpc).

Figure 12 shows two-point correlation functions of dark matter, galaxies, dark halos, and DM cores (upper and middle panels), and the profiles of biasing parameters b_ξ(r) for those objects (lower panels) at z = 0, 1, and 2. In the lower panels, we also plot the parameter b_var,i ≡ σ_i/σ_m, which are defined in terms of the one-point statistics (variance), for comparison on smoothing scales R_s = 4 h⁻¹ Mpc, R_s = 8 h⁻¹ Mpc, and R_s = 12 h⁻¹ Mpc at r = R_s for each kind of objects by different symbols. In the upper panels, we show the correlation functions of DM cores identified with two different maximum linking lengths, l_max = 0.05 and l_max = b_h/2. Correlation functions of DM cores identified with l_max = 0.05 are similar to those of galaxies. On the other hand, those identified with l_max = b_h/2 exhibit much weaker correlation, and are rather similar to those of dark halos. This is due to the fact that the present algorithm of group identification with larger l_max tends to pick up lower mass halos which are poorly resolved in our numerical resolution.

The correlation functions of galaxies are almost unchanged with redshift, and the correlation functions of dark halos only slightly evolve between z = 0 and 2. By contrast, the amplitude of the dark matter correlation functions evolve rapidly by a factor of ∼ 10 from z = 2 to z = 0. The biasing parameter b_ξ,g is larger at a higher redshift, for example, b_ξ,g ≃ 2–2.5 at z = 2. The biasing parameter b_ξ,h for dark halos is systematically lower than that of galaxies and DM cores again due to the volume exclusion effect. At z = 0, galaxies and DM cores are slightly anti-biased relative to dark matter at r ≃ 1 ^h−1 Mpc. In lower panels, we also plot the one-point biasing parameter b_var,i ≡ σ_i/σ_m at r = R_s for comparison. In general we find that b_ξ,i is very close to b_var,i at z ∼ 0, but systematically lower than b_var,i at higher redshifts.

For each galaxy identified at z = 0, we define its formation redshift z_f by the epoch when half of its cooled gas particles satisfy our criteria of galaxy formation. Roughly speaking, z_f corresponds to the median formation redshift of stars in the present-day galaxies. We divide all simulated galaxies at z = 0 into two populations (the young population with z_f < 1.7 and the old population with z_f > 1.7) so as to approximate the observed number ratio of 3/1 for late-type and early-type galaxies.

The difference of the clustering amplitude can be also quantified by their two-point correlation functions at z = 0 as plotted in Figure 13. The old population indeed clusters more strongly than the mass, and the young population is anti-biased. The relative bias between the two populations ranges 1.5 and 2 for 1 h⁻¹ Mpc < r < 20 h⁻¹ Mpc, where ξ_young and ξ_old are the two-point correlation functions of the young and old populations.

Halo occupation function approach for galaxy biasing

Since the clustering of dark matter halos is well understood now, one can describe the galaxy biasing if the halo model is combined with the relation between the halos and luminous objects. This is another approach to galaxy biasing, halo occupation function (HOF), which has become very popular recently. Indeed the basic idea behind HOF has a long history, but the model predictions have been significantly improved with the recent accurate models for the mass function, the biasing and the density profile of dark matter halos. We refer the readers to an extensive review on the HOF by Cooray and Sheth [13]. Here we briefly outline this approach.

We adopt a simple parametric form for the average number of a given galaxy population as a function of the hosting halo mass:

125

The above statistical and empirical relation is the essential ingredient in the current modeling characterized by the minimum mass M_min of halos which host the population of galaxies, a normalization parameter which can be interpreted as the critical mass M₁ above which halos typically host more than one galaxy (note that M₁ may exceed M_min since the above relation represents the statistical expected value of number of galaxies), and the power-law index α of the mass dependence of the efficiency of galaxy formation. We will put constraints on the three parameters from the observed number density and clustering amplitude for each galaxy population. In short, the number density of galaxies is most sensitive to M₁ which changes the average number of galaxies per halo. The clustering amplitude on large scales is determined by the hosting halos and thus very sensitive to the mass of those halos, M_min. The clustering on smaller scales, on the other hand, depends on those three parameters in a fairly complicated fashion; roughly speaking, M_min changes the amplitude, while a, and to a lesser extent M₁ as well, change the slope.

With the above relation, the number density of the corresponding galaxy population at redshift z is given by

126

where n_halo(M) denotes the halo mass function.

The galaxy two-point correlation function on small scales is dominated by contributions of galaxy pairs located in the same halo. For instance, Bullock et al. [8] adopted the mean number of galaxy pairs 〈N_g(N_g − 1)〉(M) within a halo of mass M of the form:

127

In the framework of the halo model, the galaxy power spectrum consists of two contributions, one from galaxy pairs located in the same halo (1-halo term) and the other from galaxy pairs located in two different halos (2-halo term):

128

The 1-halo term is written as

129

Seljak [77] chose p = 2 for 〈N_g(N_g − 1)〉 > 1 and p = 1 for 〈N_g(N_g − 1)〉 < 1. The 2-halo term on the assumption of the linear halo bias model [59] reduces to

130

where P_lin(k) is the linear dark matter power spectrum, b(M) is the halo bias factor, and y(k, M) is the Fourier transform of the halo dark matter profile normalized by its mass, [77].

The halo occupation formalism, although simple, provides a useful framework in deriving constraints on galaxy formation models from large data sets of the upcoming galaxy redshift surveys. For example, Zehavi et al. [105] used the halo occupation formalism to model departures from a power law in the SDSS galaxy correlation function. They demonstrated that this is due to the transition from a large-scale regime dominated by galaxy pairs in different halos to a small-scale regime dominated by those in the same halo. Magliocchetti and Porciani [47] applied the halo occupation formalism to the 2dFGRS clustering results per spectral type of Madgwick et al. [45]. This provides constraints on the distribution of late-type and early-type galaxies within the dark matter halos of different mass.

Cosmological light-cone effect on the two-point correlation functions

Observing a distant patch of the Universe is equivalent to observing the past. Due to the finite light velocity, a line-of-sight direction of a redshift survey is along the time, as well as spatial, coordinate axis. Therefore the entire sample does not consist of objects on a constant-time hypersurface, but rather on a light-cone, i.e., a null hypersurface defined by observers at z = 0. This implies that many properties of the objects change across the depth of the survey volume, including the mean density, the amplitude of spatial clustering of dark matter, the bias of luminous objects with respect to mass, and the intrinsic evolution of the absolute magnitude and spectral energy distribution. These aspects should be properly taken into account in order to extract cosmological information from observed samples of redshift surveys.

In order to predict quantitatively the two-point statistics of objects on the light-cone, one must take account of

1. nonlinear gravitational evolution,

2. linear redshift-space distortion,

3. nonlinear redshift-space distortion,

4. weighted averaging over the light-cone,

5. cosmological redshift-space distortion due to the geometry of the Universe, and

6. object-dependent clustering bias.

The Effect 5 comes from our ignorance of the correct cosmological parameters, and Effect 6 is rather sensitive to the objects which one has in mind. Thus the latter two effects will be discussed in the next Section 5.2.

Nonlinear gravitational evolution of mass density fluctuations is now well understood, at least for two-point statistics. In practice, we adopt an accurate fitting formula [67] for the nonlinear power spectrum in terms of its linear counterpart. If one assumes a scale-independent deterministic linear bias, furthermore, the power spectrum distorted by the peculiar velocity field is known to be well approximated by the following expression:

131

where k_⊥ and k_∥ are the comoving wavenumber perpendicular and parallel to the line-of-sight of an observer, and is the mass power spectrum in real space. The second factor on the r.h.s. comes from the linear redshift-space distortion [38], and the last factor is a phenomenological correction for the non-linear velocity effect [67]. In the above, we introduce

132

We assume that the pair-wise velocity distribution in real space is approximated by

133

with σ_P being the 1-dimensional pair-wise peculiar velocity dispersion. Then the finger-of-God effect is modeled by the damping function D_vel [k_∥σ_P(z)]:

134

where μ is the direction cosine in k-space, and the dimensionless wavenumber k is related to the peculiar velocity dispersion σ_P in the physical velocity units:

135

Since we are mainly interested in the scales around 1 h⁻¹ Mpc, we adopt the following fitting formula throughout the analysis below which better approximates the small-scale dispersions in physical units:

136

Integrating Equation (131) over μ, one obtains the direction-averaged power spectrum in redshift space:

137

138

139

140

Adopting those approximations, the direction-averaged correlation functions on the light-cone are finally computed as

141

where z_min and z_max denote the redshift range of the survey, and

142

Throughout the present analysis, we assume a standard Robertson-Walker metric of the form

143

where S_K(χ) is determined by the sign of the curvature K as

144

where the present scale factor a₀ is normalized as unity, and the spatial curvature K is given as

145

(see Equation (13)). The radial comoving distance χ(z) is computed by

146

The comoving angular diameter distance D_c(z) at redshift z is equivalent to S⁻¹(χ(z)), and, in the case of Ω_Λ = 0, is explicitly given by Mattig’s formula:

147

then dV_c/dz, the comoving volume element per unit solid angle, is explicitly given as

148

Evaluating two-point correlation functions from N-body simulation data

The theoretical modeling described above was tested against simulation results by Hamana, Colombi, and Suto [28]. Using cosmological N-body simulations in SCDM and Λ-CDM models, they generated light-cone samples as follows: First, they adopt a distance observer approximation and assume that the line-of-sight direction is parallel to the Z-axis regardless of its (X, Y) position. Second, they periodically duplicate the simulation box along the Z-direction so that at a redshift z, the position and velocity of those particles locating within an interval χ(z) ± Δχ(z) are dumped, where Δχ(z) is determined by the output time-interval of the original N-body simulation. Finally they extract five independent (non-overlapping) cone-shape samples with the angular radius of 1 degree (the field-of-view of π degree²). In this manner, they have generated mock data samples on the light-cone continuously extending up to z = 0.4 (relevant for galaxy samples) and z = 2.0 (relevant for QSO samples) from the small and large boxes, respectively.

The two-point correlation function is estimated by the conventional pair-count adopting the following estimator [43]:

149

The comoving separation x₁₂ of two objects located at z₁ and z₂ with an angular separation θ₁₂ is given by

150

where x₁ ≡ D_c(z₁) and x₂ ≡ D_c(z₂).

In redshift space, the observed redshift z_obs for each object differs from the “real” one z_real due to the velocity distortion effect:

151

where v_pec is the line of sight relative peculiar velocity between the object and the observer in physical units. Then the comoving separation s₁₂ of two objects in redshift space is computed as

152

where s₁ ≡ D_c(z_obs,1) and s₂ = D_c(z_obs,2).

In properly predicting the power spectra on the light-cone, the selection function should be specified. For galaxies, we adopt a B-band luminosity function of the APM galaxies fitted to the Schechter function [44]. For quasars, we adopt the B-band luminosity function from the 2dF QSO survey data [7]. To compute the B-band apparent magnitude from a quasar of absolute magnitude M_B at z (with the luminosity distance d_L(z)), we applied the K-correction,

153

for the quasar energy spectrum L_ν ∝ ν^−p (we use p = 0.5). In practice, we adopt the galaxy selection function ϕ_gal(< B_lim, z) with B_lim = 19 and z_min = 0.01 for the small box realizations, and the QSO selection function ϕ_qso(< B_lim, z) with B_lim = 21 and z_min = 0.2 for the large box realizations. We do not introduce the spatial biasing between selected particles and the underlying dark matter.

Figures 14 and 15 show the two-point correlation functions in SCDM and Λ-CDM, respectively, taking account of the selection functions. It is clear that the simulation results and the predictions are in good agreement.

Cosmological redshift-space distortion

Consider a spherical object at high redshift. If the wrong cosmology is assumed in interpreting the distance-redshift relation along the line of sight and in the transverse direction, the sphere will appear distorted. Alcock and Paczynski [2] pointed out that this curvature effect could be used to estimate the cosmological constant. Matsubara and Suto [54] and Ballinger, Peacock, and Heavens [3] developed a theoretical framework to describe the geometrical distortion effect (cosmological redshift distortion) in the two-point correlation function and the power spectrum of distant objects, respectively. Certain studies were less optimistic than others about the possibility of measuring this Alcock-Paczynski effect. For example, Ballinger, Peacock, and Heavens [3] argued that the geometrical distortion could be confused with the dynamical redshift distortions caused by peculiar velocities and characterized by the linear theory parameter . Matsubara and Szalay [55, 56] showed that the typical SDSS and 2dF samples of normal galaxies at low redshift (z ∼ 0.1) have sufficiently low signal-to-noise, but they are too shallow to detect the Alcock-Paczynski effect. On the other hand, the quasar SDSS and 2dFGRS surveys are at a useful redshift, but they are too sparse. A more promising sample is the SDSS Luminous Red Galaxies survey (out to redshift z ∼ 0.5) which turns out to be optimal in terms of both depth and density.

While this analysis is promising, it remains to be tested if non-linear clustering and complicated biasing (which is quite plausible for red galaxies) would not ‘contaminate’ the measurement of the equation of state. Even if the Alcock-Paczynski test turns out to be less accurate than other cosmological tests (e.g., CMB and SN Ia), the effect itself is an interesting and important ingredient in analyzing the clustering pattern of galaxies at high redshifts. We shall now present the formalism for this effect.

Due to a general-relativistic effect through the geometry of the Universe, the observable separations perpendicular and parallel to the line-of-sight direction, x_s⊥ = (c/H₀)zδθ and x_s∥ = (c/H₀)δz, are mapped differently to the corresponding comoving separations in real space x_⊥ and x_∥:

154

155

with d_A(z) being the angular diameter distance. The difference between c_⊥(z) and c_∥(z) generates an apparent anisotropy in the clustering statistics, which should be isotropic in the comoving space. Then the power spectrum in cosmological redshift space P^(CRD) is related to P^(S) defined in the comoving redshift space as

156

where the first factor comes from the Jacobian of the volume element , and k_s⊥ = c_⊥(z)k_⊥ and k_s∥ = c_∥(z)k_∥ are the wavenumber perpendicular and parallel to the line-of-sight direction.

Using Equation (131), Equation (156) reduces to

157

where

158

Figure 16 shows anisotropic power spectra P^(CRD)(k_s,μ_k; z = 2.2). As specific examples, we consider SCDM, LCDM, and OCDM models, which have (Ω_m, Ω_Λ, h, σ₈) = (1.0, 0.0, 0.5, 0.6), (0.3, 0.7, 0.7, 1.0), and (0.3, 0.0, 0.7, 1.0), respectively. Clearly the linear theory predictions (σ_P = 0; top panels) are quite different from the results of N-body simulations (bottom panels), indicating the importance of the nonlinear velocity effects (σ_P computed according to [58]; middle panels).

Next we decompose the power spectrum into harmonics,

159

where L_l(μ_k) are the l-th order Legendre polynomials. Similarly, the two-point correlation function is decomposed as

160

using the direction cosine μ_x between the separation vector and the line-of-sight. The above multipole moments satisfy the following relations:

161

162

with j_l(kx) being spherical Bessel functions. Substituting P^(CRD)(k_s, μ_k; z) in Equation (159) yields , and then ξ^(CRD)(x_s; z) can be computed from Equation (161).

A comparison of the monopoles and quadrupoles from simulations and model predictions exhibits how the results are sensitive to the cosmological parameters, which in turn may put potentially useful constraints on (Ω_m, Ω_Λ). Figure 17 indicates the feasibility, which interestingly results in a constraint fairly orthogonal to that from the supernovae Ia Hubble diagram.

Two-point clustering statistics on a light-cone in cosmological red-shift space

In order to explore the relation between the two-point statistics on a constant-time hypersurface in real space and that on a light-cone hypersurface in cosmological redshift space, we simply consider the case of the deterministic, linear, and scale-independent bias:

163

In what follows, we explicitly use the subscript ‘mass’ to indicate the quantities related to the mass density field, while those without the subscript correspond to objects satisfying Equation (163).

Using Equation (157), the two-point correlation function in the cosmological redshift space, ξ^(CRD)(x_s⊥, x_s∥; z), is computed as

164

165

166

where ξ^(S)(x_⊥, x_∥; z) is the redshift-space correlation function defined through Equation (131).

Since and are defined in redshift space, the proper weight should be

167

where and denote the number densities of the objects in cosmological redshift space and comoving space, respectively, and ϕ(z) is the selection function determined by the observational target selection and the luminosity function of the objects. Then, the final expressions [84] reduce to

168

169

where z_min and z_max denote the redshift range of the survey, is the comoving volume element per unit solid angle.

Note that k_s and x_s, defined in and , are related to their comoving counterparts at z through Equations (158) and (154), while those in and are not specifically related to any comoving wavenumber and separation. Rather, they correspond to the quantities averaged over the range of z satisfying the observable conditions and k_s = 2π/x_s.

Let us show specific examples of the two-point clustering statistics on a light-cone in cosmological redshift space. We consider SCDM and LCDM models, and take into account the selection functions relevant to the upcoming SDSS spectroscopic samples of galaxies and quasars by adopting the B-band limiting magnitudes of 19 and 20, respectively.

Figure 18 compares the predictions for the angle-averaged (monopole) power spectra under various approximations. The upper and lower panels adopt the selection functions appropriate for galaxies in 0 < z < z_max = 0.2 and QSOs in 0 < z < z_max = 5, respectively. The left and right panels present the results in SCDM and LCDM models. For simplicity we adopt a scale-independent linear bias model [23]:

170

with b(k, z = 0) = 1 and 1.5 for galaxies and quasars, respectively.

The upper and lower panels correspond to magnitude-limited samples of galaxies (B < 19 in 0 < z < z_max = 0.2; no bias model) and QSOs (B < 20 in 0 < z < z_max = 5; Fry’s linear bias model), respectively. We present the results normalized by the real-space power spectrum in linear theory P^(R,lin)(k; z) [4], and and are computed using the nonlinear power spectrum [67].

Consider first the results for the galaxy sample (upper panels). On linear scales (k <; 0.1 h Mpc−1), plotted in dashed lines is enhanced relative to that in real space, mainly due to a linear redshift-space distortion (the Kaiser factor in Equation (131)). For nonlinear scales, the nonlinear gravitational evolution increases the power spectrum in real space, while the finger-of-God effect suppresses that in redshift space. Thus, the net result is sensitive to the shape and the amplitude of the fluctuation spectrum, itself; in the LCDM model that we adopted, the nonlinear gravitational growth in real space is stronger than the suppression due to the finger-of-God effect. Thus, becomes larger than its real-space counterpart in linear theory. In the SCDM model, however, this is opposite and becomes smaller.

The power spectra at z = 0.2 (dash-dotted lines) are smaller than those at z = 0 by the corresponding growth factor of the fluctuations, and one might expect that the amplitude of the power spectra on the light-cone (solid lines) would be in-between the two. While this is correct, if we use the comoving wavenumber, the actual observation on the light-cone in the cosmological redshift space should be expressed in terms of k_s (see Equation (158)). If we plot the power spectra at z = 0.2 taking into account the geometrical distortion, in the dotted lines becomes significantly larger than . Therefore, should take a value between those of . This explains the qualitative features shown in the upper panels of Figure 18. As a result, both the cosmological redshift-space distortion and the light-cone effect substantially change the predicted shape and amplitude of the power spectra, even for the galaxy sample [60]. The results for the QSO sample can be basically understood in a similar manner, except that the evolution of the bias makes a significant difference, since the sample extends to much higher redshifts.

Figure 19 shows the results for the angle-averaged (monopole) two-point correlation functions, exactly corresponding to those in Figure 18. The results in this figure can also be understood by an analogy of those presented in Figure 18 at k ∼ 2π/x. Unlike the power spectra, however, two-point correlation functions are not positive definite. The funny features in Figure 19 on scales larger than 30 h⁻¹ Mpc (100 h⁻¹ Mpc) in SCDM (LCDM) originate from the fact that ξ^(R,lin)(x, z = 0) becomes negative there.

In fact, since the resulting predictions are sensitive to the bias, which is unlikely to quantitatively be specified by theory, the present methodology will find two completely different applications. For relatively shallower catalogues, like galaxy samples, the evolution of bias is not supposed to be so strong. Thus, one may estimate the cosmological parameters from the observed degree of the redshift distortion, as has been conducted conventionally. Most importantly, we can correct for the systematics due to the light-cone and geometrical distortion effects, which affect the estimate of the parameters by ∼ 10%. Alternatively, for deeper catalogues like high-redshift quasar samples, one can extract information on the object-dependent bias only by correcting the observed data on the basis of our formulae.

In a sense, the former approach uses the light-cone and geometrical distortion effects as real cosmological signals, while the latter regards them as inevitable, but physically removable, noise. In both cases, the present methodology is important in properly interpreting the observations of the Universe at high redshifts.

The latest galaxy redshift surveys

Redshifts surveys in the 1980s and the 1990s (e.g., the CfA, IRAS, and Las campanas surveys) measured thousands to tens of thousands galaxy redshifts. Multifibre technology now allows us to measure redshifts of millions of galaxies. Below we summarize briefly the properties of the main new surveys 2dFGRS, SDSS, 6dF, VIRMOS, DEEP2, and we discuss key results from 2dFGRS and SDSS. Further analysis of these surveys is currently underway.

Cosmological parameters from 2dFGRS

Combining 2dFGRS and CMB

While the CMB probes the fluctuations in matter, the galaxy redshift surveys measure the perturbations in the light distribution of particular tracer (e.g., galaxies of certain type). Therefore, for a fixed set of cosmological parameters, a combination of the two can better constrain cosmological parameters, and it can also provide important information on the way galaxies are ‘biased’ relative to the mass fluctuations,

The CMB fluctuations are commonly represented by the spherical harmonics C_ℓ. The connection between the harmonic ℓ and k is roughly

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for a spatially-flat Universe. For Ω_m = 0.3, the 2dFGRS range 0.02 < k < 0.15 h Mpc⁻¹ corresponds approximately to 200 < ℓ < 1500, which is well covered by the recent CMB experiments.

Recent CMB measurements have been used in combination with the 2dF power spectrum. Efstathiou et al. [17] showed that 2dFGRS+CMB provide evidence for a positive cosmological constant Ω_Λ ∼ 0.7 (assuming w = −1), independently of the studies of supernovae Ia. As explained in [72], the shapes of the CMB and the 2dFGRS power spectra are insensitive to Dark Energy. The main important effect of the dark energy is to alter the angular diameter distance to the last scattering, and thus the position of the first acoustic peak. Indeed, the latest result from a combination of WMAP with 2dFGRS and other probes gives , σ₈ = 0.84 ± 0.04, Ω_tot = 1.02 ± 0.02, and w < −0.78 (95% CL, assuming w ≥ −1) [79].

The bi-spectrum and higher moments

It is well established that important information on the non-linear growth of structure is encoded at the high order moments, e.g., the skewness or its Fourier version, the bi-spectrum. Verde et al. [95] computed the bi-spectrum of 2dFGRS and used it to measure the bias parameter of the galaxies. They assumed a specific quadratic biasing model:

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By analysing 80 million triangle configurations in the wavenumber range 0.1 < k < 0.5 h Mpc⁻¹ they found b₁ = 1.04 ± 0.11 and b₂ = −0.054 ± 0.08, in support of no biasing on large scale. This is a non-trivial result, as the analysis covers non-linear scales. Baugh et al. [5] and Croton et al. [14] measured the moments of the galaxy count probability distribution function in 2dFGRG up to order p = 6 (order p = 2 is the variance, p = 3 is the skewness, etc.). They demonstrated the hierarchical scaling of the averaged p-point galaxy correlation functions. However, they found that the higher moments are strongly affected by the presence of two massive superclusters in the 2dFGRS volume. This poses the question of whether 2dFGRS is a’ fair sample’ for high order moments.

Luminosity and spectral-type dependence of galaxy clustering

Although biasing was commonly neglected until the early 1980s, it has become evident obser-vationally that on scales ≲ 10 h⁻¹ Mpc different galaxy populations exhibit different clustering amplitudes, the so-called morphology-density relation [16]. As discussed in Section 4, galaxy biasing is naturally predicted from a variety of theoretical considerations as well as direct numerical simulations [37, 59, 15, 87, 86, 103]. Thus, in this Section we summarize the extent to which the galaxy clustering is dependent on the luminosity, spectral-type, and color of the galaxy sample from the 2dFGRS and SDSS.

SDSS: Three-point correlation functions and the nonlinear biasing of galaxies per luminosity and color

Let us move next to the three-point correlation functions (3PCF) of galaxies, which are the lowest-order unambiguous statistic to characterize non-Gaussianities due to nonlinear gravitational evolution of dark matter density fields, formation of luminous galaxies, and their subsequent evolution. The determination of the 3PCF of galaxies was pioneered by Peebles and Groth [70] and Groth and Peebles [27] using the Lick and Zwicky angular catalogs of galaxies. They found that the 3PCF ζ (r₁₂,r₂₃, r₃₁) obeys the hierarchical relation:

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with Q_r being a constant. The value of Q_r in real space deprojected from these angular catalogues is 1.29 ± 0.21 for r < 3 h⁻¹ Mpc. Subsequent analyses of redshift catalogs confirmed the hierarchical relation, at least approximately, but the value of Q_z (in redshift space) appears to be smaller, Q_z ∼ 0.5−1.

As we have seen in Section 6.3.2, galaxy clustering is sensitive to the intrinsic properties of the galaxy samples under consideration, including their morphological types, colors, and luminosities. Nevertheless the previous analyses were not able to examine those dependences of 3PCFs because of the limited number of galaxies. Indeed Kayo et al. [39] were the first to perform the detailed analysis of 3PCFs explicitly taking account of the morphology, color, and luminosity dependence. They constructed volume-limited samples from a subset of the SDSS galaxy redshift data, ‘Large-scale Structure Sample 12’. Specifically they divided each volume limited sample into color subsamples of red (blue) galaxies, which consist of 7949 (8329), 8930 (8155), and 3706 (3829) galaxies for −22 < M_r − 5 log h < −21, −21 < M_r − 5 log h < −20, and −20 < M_r − 5 log h < −19, respectively.

Figure 29 indicates the dimensionless amplitude of the 3PCFs of SDSS galaxies in redshift space,

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for the equilateral triplets of galaxies. The overall conclusion is that Q_z is almost scale-independent and ranges between 0.5 and 1.0, and that no systematic dependence is noticeable on luminosity and color. This implies that the 3PCF itself does depend on the galaxy properties since two-point correlation functions (2PCFs) exhibit clear dependence on luminosity and color. Previous simulations and theoretical models [82, 53, 50, 85] indicate that Q decreases with scale in both real and redshift spaces. This trend is not seen in the observational results.

Figure 29.

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In order to demonstrate the expected dependence in the current samples, they compute the biasing parameters estimated from the 2PCFs,

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where the index i runs over each sample of galaxies with different colors and luminosities. The predictions of the mass 2PCFs in redshift space, ξ_z,ΛCDM(s), in the Λ cold dark matter model are computed following [28]

As an illustrative example, consider a simple bias model in which the galaxy density field δ_g,i for the i-th population of galaxies is given by

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if both b_g,i(1) and b_g,i(1) are constant and the mass density field δ_mass ≪ 1, Equation (174) implies that

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Thus the linear bias model (b_g,i(2) = 0) simply implies that Q_g,i is inversely proportional to b_g,i(1), which is plotted in Figure 30. A comparison of Figures 29 and 30 indicates that the biasing in the 3PCFs seems to compensate the difference of Q_g purely due to that in the 2PCFs.

Figure 30.

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Such behavior is unlikely to be explained by any simple model inspired by the perturbative expansion like Equation (176). Rather it indeed points to a kind of regularity or universality of the clustering hierarchy behind galaxy formation and evolution processes. Thus the galaxy biasing seems much more complex than the simple deterministic and linear model. More precise measurements of 3PCFs and even higher-order statistics with future SDSS datasets would be indeed valuable to gain more specific insights into the empirical biasing model.

Topology of the Universe: Analysis of SDSS galaxies in terms of Minkowski functionals

All the observational results presented in the preceding Sections 6.1, 6.2, and 6.3 were restricted to the two-point statistics. As emphasized in Section 3, the clustering pattern of galaxies has much richer content than the two-point statistics can probe. Historically the primary goal of the topological analysis of galaxy catalogues was to test Gaussianity of the primordial density fluctuations. Although the major role for that goal has been superseded by the CMB map analysis [41], the proper characterization of the morphology of large-scale structure beyond the two-point statistics is of fundamental importance in cosmology. In order to illustrate a possibility to explore the topology of the Universe by utilizing the new large surveys, we summarize the results of the Minkowski Functionals (MF) analysis of SDSS galaxy data [32].

In an apparent-magnitude limited catalogue of galaxies, the average number density of galaxies decreases with distance because only increasingly bright galaxies are included in the sample at larger distance. With the large redshift surveys it is possible to avoid this systematic change in both density and galaxy luminosity by constructing volume-limited samples of galaxies, with cuts on both absolute-magnitude and redshift. This is in particular useful for analyses such as MF and was carried out in the analysis shown here.

Figure 31 shows the MFs as a function of ν_f defined from the volume fraction [26]:

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This is intended to map the threshold so that the volume fraction on the high-density side of the isodensity surface is identical to the volume in regions with density contrast δ = υ_f σ, for a Gaussian random field with r.m.s. density fluctuations σ. If the evolved density field may approximately have a good one-to-one correspondence with the initial random-Gaussian field, then this transformation removes the effect of evolution of the PDF of the density field. Under this assumption, the MFs as a function of volume fraction would be sensitive only to the topology of the isodensity contours rather than evolution with time of the density threshold assigned to a contour. While the limitations of the approximation of monotonicity in the relation between initial and evolved density fields are well recognized [40], we plot the result in this way for simplicity.

The good match between the observed MFs and the mock predictions based on the LCDM model with the initial random-Gaussianity, as illustrated in Figure 31, might be interpreted to imply that the primordial Gaussianity is confirmed. A more conservative interpretation is that, given the size of the estimated uncertainties, these data do not provide evidence for initial non-Gaussianity, i.e., the data are consistent with primordial Gaussianity. Unfortunately, due to the statistical limitation of the current SDSS data, it is not easy to put a more quantitative statement concerning the initial Gaussianity. Moreover, in order to go further and place more quantitative constraints on primordial Gaussianity with upcoming data, one needs a more precise and reliable theoretical model for the MFs, which properly describes the nonlinear gravitational effect possibly as well as galaxy biasing beyond the simple mapping on the basis of the volume fraction. In fact, galaxy biasing is a major source of uncertainty for relating the observed MFs to those obtained from the mock samples for dark matter distributions. If LCDM is the correct cosmological model, the good match of the MFs for mock samples from the LCDM simulations to the observed SDSS MFs may indicate that nonlinearity in the galaxy biasing is relatively small, at least small enough that it does not significantly affect the MFs (the MFs as a function of ν_σ remain unchanged for the linear biasing).

Other statistical measures

In this section, we have presented the results on the basis of particular statistical measures including the two-point correlation functions, power spectrum, redshift distortion, and Minkowski functionals. Of course there are other useful approaches in analysing redshift surveys: the void probability function, counts-in-cells, Voronoi cells, percolation, and minimal spanning trees. Another area not covered here is optimal reconstruction of density field (e.g, using the Wiener filtering). The reader is referred to a good summary of those and other methodology in the book by Martinez and Saar [48].

Admittedly the results that we presented here are rather observational and phenomenological, and far from being well-understood theoretically. It is quite likely that when other on-going and future surveys are being analysed in great detail, the nature of galaxy clustering will be revealed in a much more quantitative manner. They are supposed to act as a bridge between cosmological framework and galaxy formation operating in the Universe. While the proper understanding of physics of galaxy formation is still far away, the future redshift survey data will present interesting challenges for constructing models of galaxy formation.