Annu. Rev. Astron. Astrophys. 1998. 36:
599-654
Copyright © 1998 by . All rights reserved |

**3.2. Statistics of Density Fields**

Turning next to statistics for continuous density fields, we note first
that *N*-point correlation functions and power spectra are naturally
defined and very useful in this case much as they are for point sets. A
family of new statistics, reviewed by
Melott (1990),
is based on the
two-dimensional surfaces of constant density (isodensity surfaces).
The best known of these is the genus per unit volume
*g*() as a function
of the standardized density contour level
=
/
(Gott et al 1986).
The total genus *G* of a surface is a topological invariant measuring
the number of "holes" (as in a doughnut)
minus the number of isolated regions. One of its attractions is the
fact that an exact prediction exists for the shape of
*G*() for
Gaussian random fields
(Doroshkevich 1970,
Bardeen et al 1986,
Hamilton et al 1986),
enabling a test of Gaussianity on large scales where the
matter distribution is expected to still approximate the linearly
growing initial conditions. When the smoothing scale is varied, the
amplitude of the genus curve varies in a spectrum-dependent way,
providing another useful clustering statistic. Computer programs for
computing the genus are given by
Melott (1990); an
alternative and simpler algorithm is provided by
Coles et al (1996).

Recent work has shown that the genus statistic is relatively insensitive to redshift-space distortions (Matsubara 1996), is slightly dependent on biasing (the relation of galaxies to mass, discussed below) (Park & Gott 1991), and is more sensitive to non-Gaussianity arising from nonlinear evolution of Gaussian initial conditions (e.g. Sahni et al 1997, Seto et al 1997) or from non-Gaussian initial conditions (Beaky et al 1992, Weinberg & Cole 1992, Matsubara & Yokoyama 1996, Avelino 1997). The area of isodensity contours provides an independent and useful statistic (Ryden 1988, Ryden et al 1989).

Minkowski functionals
(Mecke et al 1994)
have recently been introduced
in cosmology as a very powerful descriptor of the topology of isodensity
surfaces. In three dimensions, there are four Minkowski functionals
(*v*_{0}, *v*_{1},
*v*_{2}, *v*_{3});
two of them are the genus (actually, its relative
the Euler characteristic) and surface area statistics discussed above, and
the other two are the covered volume (related to the void probability
function) and integral mean curvature. Analytical results for Gaussian
random fields have been provided by
Schmalzing & Buchert
(1997), who also have
made available a computer program for computing these statistics from a
point process. The insight and unification provided by these recent results
suggests a promising future for Minkowski functionals as a statistic for
both cosmological simulations and redshift surveys.

Perhaps the simplest (though incomplete) test of Gaussianity is simply
to examine the one-point distribution function *f*
()
*d*. When the
density is defined by smoothing as a function of scale, one has
the continuous analog of the counts in cell distribution
*P _{N}*(

Low-order moments of *f*
(), particularly the
skewness (the third moment), have also been studied extensively (e.g.
Coles & Frenk 1991,
Coles et al 1993,
Juszkiewicz et al 1993,
Luo & Vishniac
1995;
see Lokas et al 1995
for the irreducible fourth moment, the kurtosis). With these,
Juszkiewicz et al
(1995)
obtained good approximations to the mildly nonlinear
*f* () by using
a series expansion.
Protogeros & Scherrer
(1997) considered a class
of local Lagrangian approximations in which the nonlinear density of
a mass element is a function only of its initial density contrast and
time. By choosing this function so that low-order moments agree well
with perturbation theory, they obtained a model for
*f*() that
agrees well with simulations in the mildly nonlinear regime
(Protogeros et al 1997).

Visual inspection of galaxy redshift surveys and projected catalogs gives a clear impression of filamentary and sheet-like structure. Consequently, several statistics have been developed that aim to quantify such structure. Recent work has been summarized by Davé et al (1997b), who explored moment-based shape statistics devised by several groups, as well as their own new filament statistics. These statistics degrade under sparse sampling but are promising discriminators of models, particularly with the large redshift surveys expected to become available within a few years.