Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
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3. ANALYSIS TECHNIQUES

Cosmic structure formation models are stochastic: The initial conditions are random (albeit with well-specified statistical properties), and their evolution is chaotic. No realistic model would claim to predict the exact structures we see in our Universe; at best, theorists can hope to construct models of the Universe that "look like" the real thing. But how is this vague test quantified? Many statistics have been devised for this purpose. In this section, a brief summary of methods is presented; a more complete review was given by Coles (1992).

Table 1 lists the major statistics applied in cosmological simulations of structure formation. They are distinguished by whether they are most naturally applied to continuous fields such as the density fluctuation field delta(vector{x}) or to a set of discrete points (e.g. galaxies or simulation particles). In practice these applications are interchangeable because a point set can be convolved with a window function to produce a continuous field, and a continuous field may be Poisson sampled (with spatially-varying Poisson density) to produce a point set (e.g. Bertschinger 1992). Other statistics apply exclusively to the internal properties of galaxies, clusters, or dark matter halos. Some statistical measures associated with specific physical measurements (e.g. the column density distribution of Lyalpha absorption lines or the distribution of gravitational lens separations) are discussed in Section 5.

Table 1. Statistical measures applied to galaxies and numerical simulations of structure formation

Category Statistic Name Reference

Particle positions xi(r) Two-point correlation function Peebles 1980
P(k) Power spectrum Bertschinger 1992
zeta(r1, r2, r3) Three-point correlation function Groth & Peebles 1977
B(k1, k2, k3) Bispectrum Peebles 1980
xiN, bar{xi}N N-point correlation functions and moments Peebles 1980
P0(V), PN(V) Void probability function, cell counts White 1979
- Percolation, minimal spanning tree statistics Coles 1992
- Multifractal statistics Martínez et al 1990
Density fields G(nu) Genus of isodensity surfaces Melott 1990
- Area of isodensity surfaces Ryden 1988
vi(nu) Minkowski functionals Mecke et al 1994
f (delta) One-point density distribution Kofman et al 1994
<deltaNc> One-point cumulants (skewness, kurtosis, etc) Peebles 1980
- Shape statistics Davé et al 1997b
Velocity fields f(v) One-point velocity distribution (and moments) Inagaki et al 1992
M Mach number Ostriker & Suto 1990
f(theta) Velocity divergence distribution (and moments) Bernardeau et al 1985
f(v12), sigma12 Pairwise radial velocity distribution and dispersion Davis & Peebles 1983
Redshift space xi(rp, pi), xi(s) Redshift space correlation functions Davis & Peebles 1983
Ps(k, µ) Redshift space power spectrum Cole et al 1995
Clusters or halos n(m) Mass distribution Press & Schechter 1974
n(Vc) Circular velocity distribution Gelb & Bertschinger 1994a
n(sigma) Velocity dispersion distribution Evrard 1989
n(T), n(L) Temperature and X-ray luminosity distributions Cen & Ostriker 1994a

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