Gaussian random field

3.1 Gaussian random field

Consider the density contrast $δi ≡ δ(xi) = ρ(x)∕¯ρ − 1$ defined at the comoving position $xi$ . 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,

for an arbitrary positive integer $m$ . Here $Mij ≡ ⟨δiδj⟩$ is the covariance matrix, and $−1 M$ is its inverse. Since $Mij = ξ(xi,xj)$ , 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 $− ∞ < δi < ∞$ , but in the real density field $δi$ 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 $δ(x)$ $≡ ρ (x )∕⟨¯ρ⟩ − 1$ is the most basic variable:

Since $δk$ is a complex variable, it is decomposed by a set of two real variables, the amplitude $Dk$ and the phase $ϕk$ :

Then linear perturbation equation reads

˙a 2 D¨k + 2 -D˙k − (4πG ¯ρ + ˙ϕ )Dk = 0, (74 ) a ( ) ¨ ˙a- D˙k- ˙ ϕk + 2 a + D ϕk = 0. (75 ) k

Equation (75

) yields $ϕ˙(t) ∝ a− 2(t)D −k2(t)$ , and $ϕ(t)$ rapidly converges to a constant value. Thus $Dk$ 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,

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

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

) reduces to

Since the above expression is obtained after the ensemble average, $x$ does not denote an amplitude of the position vector, but a comoving wavelength $2 π∕k$ 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 $ϕk$ . 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 $ϕk$ in addition to the amplitude $Dk$ of clustering.

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

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 $δk$ 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 ].