The Effects of Smoothing on the Statistical Properties of the Large-Scale Cosmic Fileds

It has been shown that the large--scale correlation functions of the density field (and velocity divergence field) follow a specific hierarchy in the quasilinear regime and for Gaussian initial conditions (Bernardeau 1992). The exact relationships between the cumulants of the probability distribution functions (the so-called $S_p$ 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 extremely accurate.