PROPERTIES OF THE COSMOLOGICAL DENSITY DISTRIBUTION FUNCTION

The properties of the probability distribution function (PDF) of the c osmological continuous density field are studied. We focus our analysi s on the quasi-linear regime where various calculations, based on dyna mically motivated methods, have been presented: either by using the Ze l'dovich approximation (ZA) or by using the perturbation theory to eva luate the behavior of the moments of the distribution function. We sho w 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 tes ting 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 Gauss ian) and that the nonlinear evolution of the moments can be obtained. The pertubation theory can be used for the exact dynamics. We take int o account the final filtering of the density field both for ZA and per turbation theory. Applying these technics we got the generating functi on of the moments for the one-dimensional dynamics, the three-dimensio nal ZA, with and without smoothing effects. We also suggest methods to build PDFs. One is based on the Laplace inverse transform of the mome nt generating function. The other, the Edgeworth expansion, is obtaine d when the previous generating function is truncated at a given order and allows to evaluate the PDF out of limited number of moments. It pr ovides insight on the relationship between the moments and the shape o f the density PDF. In particular it provides an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its max imum. Eventually, results obtained from a numerical simulation with CD M initial conditions have been used to validate the accuracy of the co nsidered approximations. We explain the successful log-normal fit of t he PDF from that simulation at moderate a as mere fortune, but not as a universal form of density PDF in general.