Properties of the cosmological density distribution function

Abstract

The properties of the probability distribution function (PDF) of the cosmological continuous density field are studied. We focus our analysis on the quasi-linear regime where various calculations, based on dynamically motivated methods, have been presented: either by using the Zel'dovich approximation (ZA) or by using the perturbation theory to evaluate the behavior of the moments of the distribution function. We show how these two approaches are related to each other and that they can be used in a complementary way. For that respect the one-dimensional dynamics, where the ZA is exact solution, has first been used as a testing ground. In particular we show that, when the density PDF obtained with the ZA is regularized, its various moments exhibit the behavior expected by the perturbation theory applied to the ZA. We show that ZA approach can be used for arbitrary initial conditions (not only Gaussian) and that the nonlinear evolution of the moments can be obtained. The perturbation theory can be used for the exact dynamics. We take into account the final filtering of the density field both for ZA and perturbation theory. Applying these techniques we got the generating function of the moments for the one-dimensional dynamics, the three- dimensional ZA, with and without smoothing effects. We also suggest methods to build PDFs. One is based on the Laplace inverse transform of the moment generating function. The other, the Edgeworth expansion, is obtained when the previous generating function is truncated at a given order and allows to evaluate the PDF out of limited number of moments. It provides insight on the relationship between the moments and the shape of the density PDF. In particular it provides an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its maximum. Eventually, results obtained from a numerical simulation with CDM initial conditions have been used to validate the accuracy of the considered approximations. We explain the successful log- normal fit of the PDF from that simulation at moderate σ as mere fortune, but not as a universal form of density PDF in general.