Thus, in this article we have developed a simple model for hierarchical clustering based on a spherical dynamics. We have shown it provides a reasonable approximation to the density field, and the divergence of the velocity field, in the quasi-linear regime . Moreover, it allows one to recover the exact series of the moments of the probability distribution of the density contrast in the limit (contrary to other models like the Zeldovich approximation for instance), and sheds some light on the rigorous results. Then, we have developed a way to extend this model into the non-linear regime, by taking into account the virialization of high overdensities (as in the PS approach) and also the behaviour of very underdense areas (in a way different from the PS prescription). This implies a particular scaling in x of the counts-in-cells over a well-defined range which is indeed verified by the results of numerical simulations. Thus, our model deals with the evolution of the density field in both linear and non-linear regimes, for any cosmological parameters (open or critical universe) and provides a simple reference to which one could compare the results of a more rigorous treatment. Naturally, our approximation should be tested in more detail with numerical simulations. The density profile of very underdense regions, the evolution with time of the scale and density of typical "voids", the asymptotic behaviour of the scaling function , should allow one to measure the influence of non-spherical corrections and the effects of substructure disruption on the density field, which are not well described by our model, and precise its accuracy.
© European Southern Observatory (ESO) 1998
Online publication: August 27, 1998