Scaling laws in gravitational clustering for counts-in-cells and mass functions

We present an analysis of some of the properties of the density field reali zed in numerical simulations for power-law initial power spectra in the cas e of a critical density universe. We study the non-linear regime, which is the most difficult to handle analytically, and we compare our numerical res ults with the predictions of a specific hierarchical clustering scaling mod el that have been made recently, focusing specifically on its much wider ra nge of applicability, which is one of its main advantages over the standard Press-Schechter approximation. We first check that the two-point correlati on functions, measured from both counts-in-cells and neighbour counts, agre e with the known analytically exact scaling requirement (i.e., depend only on sigma(2)), and we also find the stable-clustering hypothesis to hold. Ne xt, we show that the statistics of the counts-in-cells obey the scaling law predicted by the above scaling model. Then we turn to mass functions of overdense and underdense regions, which w e obtain numerically from 'spherical overdensity' and 'friends-of-friends' algorithms. We first consider the mass function of 'just-collapsed' objects defined by a density threshold Delta = 177, and we note, as was found by p revious studies, that the usual Press-Schechter prescription agrees reasona bly well with the simulations (although there are some discrepancies). On t he other hand, the numerical results are also consistent with the predictio ns of the scaling model. Next, we consider more general mass functions (nee ded to describe for instance galaxies or Lyman-alpha absorbers) defined by different density thresholds, which can even be negative. The scaling model is especially suited to account for such cases, which are out of reach of the Press-Schechter approach, and it still shows reasonably good agreement with the numerical results. Finally, we show that mass functions defined by a condition on the radius of the objects also satisfy the theoretical scal ing predictions. Thus we find that the scaling model provides a reasonable description of th e density field in the highly non-linear regime, for the cosmologies we hav e considered, for both the counts-in-cells statistics and the mass function s. The advantages of this approach are that it clarifies the links between several statistical tools and it allows one to study many different classes of objects, for any density threshold, provided one is in the fully non-li near regime.