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 realized in numerical simulations for power-law initial power spectra in the case of a critical density universe. We study the non-linear regime, which is the most difficult to handle analytically, and we compare our numerical results with the predictions of a specific hierarchical clustering scaling model that have been made recently, focusing specifically on its much wider range of applicability, which is one of its main advantages over the standard Press-Schechter approximation. We first check that the two-point correlation functions, measured from both counts-in-cells and neighbour counts, agree with the known analytically exact scaling requirement (i.e., depend only on σ²), and we also find the stable-clustering hypothesis to hold. Next, 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 we 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 Δ=177, and we note, as was found by previous studies, that the usual Press-Schechter prescription agrees reasonably well with the simulations (although there are some discrepancies). On the other hand, the numerical results are also consistent with the predictions of the scaling model. Next, we consider more general mass functions (needed to describe for instance galaxies or Lyman-α 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 scaling predictions. Thus we find that the scaling model provides a reasonable description of the density field in the highly non-linear regime, for the cosmologies we have considered, for both the counts-in-cells statistics and the mass functions. 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-linear regime.