Experimental cosmic statistics – II. Distribution

Colombi et al. (Paper I) investigated the counts‐in‐cells statistics and their respective errors in the τ CDM Virgo Hubble Volume simulation. This extremely large N ‐body experiment also allows a numerical investigation of the cosmic distribution function , ϒ, itself for the first time. For a statistic A , ϒ is the probability density of measuring the value in a finite galaxy catalogue. ϒ was evaluated for the distribution of counts‐in‐cells, P N , the factorial moments, F k , and the cumulants, and S N 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 ,  ). 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 Δ A A ≲0.2. This means that for accurate measurements, the theory of the cosmic errors, presented by Szapudi & Colombi and Szapudi, Colombi & Bernardeau, 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( k ) and ϒ( N ) increases with order k , N and similarly for ϒ( N ) when N is far from the maximum of P N , or when the scale approaches the size of the catalogue. For our particular experiment, ϒ( k ) and ϒ( ) 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.