Weakly Non-Linear Gaussian Fluctuations and the Edgeworth Expansion

Abstract

We calculate the cosmological evolution of the 1-point probability distribution function (PDF), using an analytic approximation that combines gravitational perturbation theory with the Edgeworth expansion of the PDF. Our method applies directly to a smoothed mass density field or to the divergence of a smoothed peculiar velocity field, provided that rms fluctuations are small compared to unity on the smoothing scale, and that the primordial fluctuations that seed the growth of structure are Gaussian. We use this `Edgeworth approximation' to compute the evolution of $<δ|δ|>$ and $<|delta|>$; these measures are similar to the skewness and kurtosis of the density field, but they are less sensitive to tails of the probability distribution, so they may be more accurately estimated from surveys of limited volume. We compare our analytic calculations to cosmological N-body simulations in order to assess their range of validity. When $σ\ll 1$, the numerical simulations and perturbation theory agree precisely, demonstrating that the N-body method can yield accurate results in the regime of weakly non-linear clustering. We show analytically that `biased' galaxy formation preserves the relation $<δ^3> \propto <δ^2>^2$ predicted by second-order perturbation theory, provided that the galaxy density is a local function of the underlying mass density.