Cosmic Statistics of Statistics

The errors on statistics measured in finite galaxy catalogs are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi (1996) is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly nonlinear to weakly nonlinear scales. For nonlinear functions of unbiased estimators, such as the cumulants, the phenomenon of cosmic bias is identified and computed. Since it is subdued by the cosmic errors in the range of applicability of the theory, correction for it is inconsequential. In addition, the method of Colombi, Szapudi & Szalay (1998) concerning sampling effects is generalized, adapting the theory for inhomogeneous galaxy catalogs. While previous work focused on the variance only, the present article calculates the cross-correlations between moments and connected moments as well for a statistically complete description. The final analytic formulae representing the full theory are explicit but somewhat complicated. Therefore as a companion to this paper we supply a FORTRAN program capable of calculating the described quantities numerically. An important special case is the evaluation of the errors on the two-point correlation function, for which this should be more accurate than any method put forward previously. This tool will be immensely useful in the future both for assessing the precision of measurements from existing catalogs, as well as aiding the design of new galaxy surveys. To illustrate the applicability of the results and to explore the numerical aspects of the theory qualitatively and quantitatively, the errors and cross-correlations are predicted under a wide range of assumptions for the future Sloan Digital Sky Survey. The principal results concerning the cumulants ξ, Q3 and, Q4, is that the relative error is expected to be smaller than 3, 5, and 15 percent, respectively, in the scale range of 1h Mpc−10h Mpc; the cosmic bias will be negligible.