NONLINEAR APPROXIMATIONS TO GRAVITATIONAL-INSTABILITY - A COMPARISON IN THE QUASI-LINEAR REGIME

We compare different nonlinear approximations to gravitational cluster ing in the weakly nonlinear regime, using as a comparative statistic t he evolution of non-Gaussianity which can be characterized by a set of numbers S-p describing connected moments of the density field at the lowest order in [delta(2)]:[delta(n)](c) similar or equal to S-n[delta (2)](n-1). Generalizing earlier work by Bernardeau (1992) we develop a n Ansatz to evaluate all S-p in a given approximation by means of a ge nerating function which can be shown to satisfy the equations of motio n of a homogeneous spherical density enhancement in that approximation . On the basis of the values of S-p we show that approximations formul ated in Lagrangian space (such as the Zel'dovich approximation and its extensions) are considerably more accurate than those formulated in E ulerian space such as the frozen flow and linear potential approximati ons. In particular we find that the nth-order Lagrangian perturbation approximation correctly reproduces the first n + 1 parameters S-n. We also evaluate the density probability distribution function for the di fferent approximations in the quasi-linear regime and compare our resu lts with an exact analytic treatment in the case of the Zel'dovich app roximation.