Experimental Cosmic Statistics II : Distribution

Colombi et al. 1999 (paper I) investigated the counts-in-cells statistics and their respective errors in the τCDM Virgo Hubble Volume simulation. This extremely large N -body experiment also allows a numerical investigation of the cosmic distribution function, Υ(Ã) itself for the first time. For a statistic A, Υ(Ã) is the probability density of measuring the value à in a finite galaxy catalog. Υ was evaluated for the distribution of counts-in-cells, PN , the factorial moments, Fk, and the cumulants, ξ and SN ’s, using the same subsamples as paper I. While paper I concentrated on the first two moments of Υ, i.e. the mean, the cosmic error and the cross-correlations, here the function Υ is studied in its full generality, including a preliminary analysis of joint distributions Υ(Ã, B̃). The most significant, and reassuring result for the analyses of future galaxy data is that the cosmic distribution function is nearly Gaussian provided its variance is small. A good practical criterion for the relative cosmic error is that ∆A/A < ∼ 0.2. This means that for accurate measurements, the theory of the cosmic errors, presented by Szapudi & Colombi (1996) and Szapudi, Colombi & Bernardeau (1999), and confirmed empirically by paper I, is sufficient for a full statistical description and thus for a maximum likelihood rating of models. As the cosmic error increases, the cosmic distribution function Υ becomes increasingly skewed and is well described by a generalization of the lognormal distribution. The cosmic skewness is introduced as an additional free parameter. The deviation from Gaussianity of Υ(F̃k) and Υ(S̃N ) increases with order k, N , and similarly for Υ(P̃N ) when N is far from the maximum of PN , or when the scale approaches the size of the catalog. For our particular experiment, Υ(F̃k) and Υ(ξ̃) are well approximated with the standard lognormal distribution, as evidenced by both the distribution itself, and the comparison of the measured skewness with that of the lognormal distribution.