THE EFFECTS OF SMOOTHING ON THE STATISTICAL PROPERTIES OF LARGE-SCALE COSMIC FIELDS

It has been shown that the large-scale correlation functions of the density field (and velocity divergence field) follow a specific hierarchy in the quasi-linear regime and for Gaussian initial conditions (Bernardeau 1992). The exact relationships between the cumulants of the probability distribution functions (the so-called Sp parameters) are however sensitive to the smoothing window function applied to the fields. In this paper, I present a method to derive the whole series of the S-p parameters when the density field is smoothed with a top-hat window function. The results are valid for any power spectrum and any cosmological parameters. Similar calculations are presented for the velocity divergence field. The resulting shapes of the one-point probability distribution functions of the cosmic density and the velocity divergence fields are given as a function of the power spectrum and Omega. Simple analytical fits are proposed when the index of the power spectrum is -1. Comparisons with numerical simulations prove these analytical results to be remarkably accurate.