Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
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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(nu) as a function of the standardized density contour level nu = delta / sigma (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(nu) 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 (v0, v1, v2, v3); 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 (delta) ddelta. When the density is defined by smoothing as a function of scale, one has the continuous analog of the counts in cell distribution PN(V). It has long been known that on scales of a few megaparsecs or less, f(delta) is strongly skewed toward positive values and is fit well by a lognormal distribution such that log (1 + delta) has a normal (Gaussian) distribution. Coles & Jones (1991) discussed the properties of lognormal random fields. While lognormal primordial fluctuations can be envisaged (Weinberg & Cole 1992), simulations show that nonlinear evolution of Gaussian initial conditions produces a f (delta) that is intriguingly well fitted by the lognormal form (Kofman et al 1994, Ueda & Yokoyama 1996). Bernardeau & Kofman (1995) have shown from perturbation theory and the Zel'dovich approximation that this agreement is fortuitous.

Low-order moments of f (delta), 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 (delta) 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(delta) 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.

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