STRUCTURE FORMATION - A SPHERICAL MODEL FOR THE EVOLUTION OF THE DENSITY DISTRIBUTION

Within the framework of hierarchical clustering we show that a simple Press-Schechter-like approximation, based on spherical dynamics, provi des a good estimate of the evolution of the density field in the quasi -linear regime up to Sigma similar to 1. Moreover, it allows one to re cover the exact series of the cumulants of the probability distributio n of the density contrast in the limit Sigma --> 0 which sheds some li ght on the rigorous result and on ''filtering''. We also obtain simila r results for the divergence of the velocity field. Next, we extend th is prescription to the highly non-linear regime, using a stable-cluste ring approximation. Then we recover a specific scaling of the counts-i n-cells which is indeed seen in numerical simulations, over a well-def ined range. To this order we also introduce an explicit treatment of t he behaviour of underdensities, which takes care of the normalization and is linked to the low-density bubbles and the walls one can see in numerical simulations. We compare this to a 1-dimensional adhesion mod el, and we present the consequences of our prescription for the power- law tail and the cutoff of the density distribution.