Gravitational Evolution of the Large‐Scale Probability Density Distribution: The Edgeworth and Gamma Expansions

Abstract

The gravitational evolution of the cosmic one-point probability distribution function (PDF) has been estimated using an analytic approximation that combines gravitational perturbation theory with the Edgeworth expansion around a Gaussian PDF. Despite the remarkable success of the Edgeworth expansion in modeling the weakly nonlinear growth of fluctuations around the peak of the cosmic PDF, it fails to reproduce the expected behavior in the tails of the distribution. This expansion is also ill defined, since it predicts negative densities and negative probabilities for the cosmic fields. This is a natural consequence of using an expansion around the Gaussian distribution, which is not rigorously well defined when describing a positive variate such as the density field. Here we present an alternative to the Edgeworth series based on an expansion around the gamma PDF. The gamma expansion is designed to converge when the PDF exhibits exponential tails, which are predicted by perturbation theory in the weakly nonlinear regime, and are found in numerical simulations from Gaussian initial conditions. The proposed expansion is better suited to describing a real PDF, since it always yields positive densities and the PDF is effectively positive-definite. We compare the performance of the Edgeworth and the gamma expansions for a wide dynamic range, making use of cosmological N-body simulations and assessing their range of validity. In general, the gamma expansion provides an interesting and simple alternative to the Edgeworth series, and it should be useful for modeling non-Gaussian PDFs in other contexts, such as in the cosmic microwave background.