Perturbation Theory Confronts Observations~: Implications for the ``Initial'' Conditions and Omega

Abstract

This paper covers the material of our two talks. We describe a series of projects based upon perturbative expansions to follow the gravitational evolution of the one point probability distribution functions (PDFs) for the density contrast field and for the divergence of the corresponding velocity field. The Edgeworth expansion greatly simplifies the problem. Indeed, the PDF can be described through several low order moments, with the first non-trivial contribution coming from the skewness, or third moment. Gaussian initial conditions imply that the skewness of the smoothed density field is proportional to the square of the variance. The constant of proportionality is then a function of the logarithmic slope of the variance versus scale, and is otherwise independent of scale. This constant is also practically independent of the density parameter $Ω$, and has very nearly the same value when distances are estimated by redshifts. To show the latter property, we also briefly discuss our Lagrangian approach to perturbation theory. Finally, the observed skewness does depend, of course, on the relation between the distribution of light and mass; we describe the case of local biasing. For the divergence of the velocity field, the skewness of its PDF is also proportional to the square of its variance, but the ratio now strongly depends on the value of $Ω$. This offers a new method to determine the density parameter which does not involve comparisons