3.1 Gaussian random
field
Consider the density contrast
defined at the comoving position
. 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
-point joint probability
distribution obeys the multi-variate Gaussian,
for an arbitrary positive integer
. Here
is the covariance matrix, and
is its inverse. Since
,
Equation (71) implies that the
statistical nature of the Gaussian density field is completely
specified by the two-point correlation function
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
, but in the real density field
cannot be less than
. 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:
Since
is a complex variable, it is decomposed by a set of
two real variables, the amplitude
and the phase
:
Then linear perturbation equation reads
Equation (75) yields
, and
rapidly converges
to a constant value. Thus
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
. 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,
does not denote an amplitude of the position vector,
but a comoving wavelength
corresponding to the
wavenumber
. It should be noted that neither the
power spectrum nor the two-point correlation function contains
information for the phase
. 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
in addition to the amplitude
of clustering.
In the Gaussian field, however, one can directly
show that Equation (71) reduces to the
probability distribution function of
and
that are explicitly written as
mutually independently of
. 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
are uncorrelated and
Gaussian distributed (with the dispersion
) independently of
. As is expected, the
distribution function (79) is completely fixed
if
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].