Gravitational Evolution of the Large-Scale Probability Density Distribution: The Edgeworth & Gamma Expansions

TL;DR

This paper compares Edgeworth and Gamma expansions for modeling the cosmic density PDF evolution, finding the Gamma expansion more accurate and well-behaved in the distribution tails, especially for positive densities.

Contribution

It introduces a Gamma-based expansion as a robust alternative to the Edgeworth expansion for modeling non-Gaussian cosmic PDFs, especially in the tails.

Findings

Gamma expansion converges better in distribution tails

Gamma expansion maintains positive densities

Edgeworth expansion fails in the tails

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 non-linear growth of fluctuations around the peak of the cosmic PDF, it fails to reproduce the expected behaviour in the tails of the distribution. Besides, this expansion is ill-defined as 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. The proposed expansion is better suited for describing a real PDF as 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 dynamical range making use of cosmological N-body simulations and assess 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.