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

8 pages, 6 figures http://www.ieec.fcr.es/cosmo-www/gamma5.ps

Cited by in corpus (24)

• Large-Scale Structure of the Universe and Cosmological Perturbation Theory
• Effects of Scale-Dependent Non-Gaussianity on Cosmological Structures
• Nearest Neighbor distributions: new statistical measures for cosmological clustering
• The 2dF Galaxy Redshift Survey: voids and hierarchical scaling models
• Back in the saddle: Large-deviation statistics of the cosmic log-density field
• Multi-parameter generalization of nonextensive statistical mechanics
• Game of cones: A nulling strategy for modelling lensing convergence in cones with large deviation theory
• Density distribution of the cosmological matter field
• Parameter information from nonlinear cosmological fields
• Linear and non-linear bias: predictions vs. measurements
• A question of separation: disentangling tracer bias and gravitational nonlinearity with counts-in-cells statistics
• Non-Gaussian gravitational clustering field statistics
• Higher Order Clustering in the Durham/UKST and Stromlo-APM Galaxy Redshift Surveys
• The VIMOS Public Extragalactic Redshift Survey (VIPERS): On the correct recovery of the count-in-cell probability distribution function
• Are the 2dFGRS superstructures a problem for hierarchical models?
• The position-dependent matter density probability distribution function
• Evolution of skewness and kurtosis of cosmic density fields
• The Real and Redshift Space Density Distribution Function for Large-Scale Structure in the Spherical Collapse Approximation
• Evolution of the probability distribution function of galaxies in redshift-space
• High-precision Monte-Carlo modelling of galaxy distribution
• One-point probability distribution function from spherical collapse: Early Dark Energy (EDE) vs. $\Lambda$CDM
• Exact Moment Scaling from Multiplicative Noise
• Cosmology: Techniques and Observations
• Structure of cosmic web in non-linear regime: the nearest neighbour and spherical contact distributions