Gravitational evolution of the large-scale probability density distribution: The edgeworth and gamma expansions

The gravitational evolution of the cosmic one-point probability distributio n function (PDF) has been estimated using an analytic approximation that co mbines gravitational perturbation theory with the Edgeworth expansion aroun d 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 o f the distribution. This expansion is also ill defined, since it predicts n egative densities and negative probabilities for the cosmic fields. This is a natural consequence of using an expansion around the Gaussian distributi on, which is not rigorously well defined when describing a positive variate such as the density held. Here we present an alternative to the Edgeworth series based on an expansion around the gamma PDF. The gamma expansion is d esigned to converge when the PDF exhibits exponential tails, which are pred icted by perturbation theory in the weakly nonlinear regime, and are found in numerical simulations from Gaussian initial conditions. The proposed exp ansion is better suited to describing a real PDF, since it always yields po sitive densities and the PDF is effectively positive-definite. We compare t he performance of the Edgeworth and the gamma expansions for a wide dynamic range, making use of cosmological N-body simulations and assessing their r ange of validity. In general, the gamma expansion provides an interesting a nd simple alternative to the Edgeworth series, and it should be useful for modeling non-Gaussian PDFs in other contexts, such as in the cosmic microwa ve background.