3.4 Genus statistics

A complementary approach to characterize the clustering of the Universe beyond the two-point correlation functions is the genus statistics [26Jump To The Next Citation Point]. 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 Vall. This may be evaluated, for instance, by taking the ratio of the number of galaxies N (x, Vf) in the volume Vf centered at x to its average value --- N (Vf ):
N-(x,-Vf)- 2 1∕2 δ(x,Vf) = N-(V ) − 1, σ(Vf) = ⟨|δ(x,Vf )| ⟩ , (98 ) f
where σ(Vf) is its r.m.s. value. Consider the isodensity surface parameterized by a value of ν ≡ δ(x, Vf)∕σ (Vf). Genus is one of the topological numbers characterizing the surface defined as
1 ∫ g ≡ − --- κdA, (99 ) 4π
where κ 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
1- 2 1- 2 1- 2 1- 2 z = 2ax + bxy + 2cy = 2 κ1x1 + 2κ2x 2. (100 )
Then the Gaussian curvature of the surface is defined by κ = κ1 κ2. A surface topologically equivalent to a sphere (a torus) has κ = 1 (κ = 0), and thus Equation (99View Equation) 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 Vall using the additivity of the genus:

I I ∫ -1-∑ --1---∑ G(ν ) ≡ V gi = − 4πV κdA, (101 ) all i=1 alli=1 Ai
where the Ai (i = 1 ∼ I) denote the disconnected isodensity surfaces with the same value of ν = δ(x, V )∕σ (V ) f f. Interestingly the Gaussian density field has an analytic expression for Equation (101View Equation):
1 ( ⟨k2⟩)3 ∕2 2 G (ν) = ---2 ---- e−ν ∕2(1 − ν2), (102 ) 4 π 3
where
∫ 2 k2P (k)W&tidle; 2(kR )d3k ⟨k ⟩ ≡ --∫------&tidle;-2-----3--- (103 ) P (k)W (kR )d k
is the moment of k2 weighted over the power spectrum of fluctuations P (k) and the smoothing function W&tidle; 2(kR ) (see, e.g., [4Jump To The Next Citation Point]). 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 (102View Equation), 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.
View Image

Figure 4: Isodensity surfaces of dark matter distribution from N-body simulation: LCDM in (100h −1 Mpc )3 at ν = − 1.0 (upper left panel), ν = 0.0 (upper right panel), ν = 1.0 (lower left panel), and ν = 1.7 (lower right panel). (Figure taken from [54Jump To The Next Citation Point].)

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]:

( )3 ∕2 [ ( ) ] --1- ⟨k2⟩ −ν2∕2 S- 3T- G(ν ) = − 4 π2 3 e H2 (ν) + σ 6H5 (ν) + 2 H3 (ν) + 3U H1 (ν) , (104 )
where
( )n H (ν ) ≡ (− 1)neν2∕2 -d- e−ν2∕2 (105 ) n dν
are the Hermite polynomials: H1 = ν, 2 H2 = ν − 1, 3 H3 = ν − 3ν, 4 2 H4 = ν − 6ν + 3, H5 = ν5 − 10ν3 + 15ν, …. The three quantities
1 3 1 2 2 3 2 S = -4⟨δ ⟩, T = − ---2--4-⟨δ ∇ δ⟩, U = − ----22--4⟨∇ δ ⋅ ∇ δ∇ δ⟩ (106 ) σ 2⟨k ⟩σ 4 ⟨k ⟩σ
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.
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