Dynamics of gravitational clustering IV. The probability distribution of rare events

TL;DR

This paper investigates the probability distribution of cosmic overdensities, showing how non-perturbative methods align with perturbative results for low densities and explaining the dynamics of high-density regions, including halo formation and virialization.

Contribution

It introduces a non-perturbative approach to analyze the tails of the overdensity distribution, clarifies the limitations of perturbative methods, and connects spherical models with stable-clustering in gravitational clustering.

Findings

Low-density tail agrees with perturbative results for certain power spectra.

High-density regions evolve towards isotropic equilibrium, supporting stable-clustering.

Results justify the mass cutoff in the Press-Schechter formalism.

Abstract

Using a non-perturbative method developed in a previous article (paper II) we investigate the tails of the probability distribution $P(\rho_R)$ of the overdensity within spherical cells. We show that our results for the low-density tail of the pdf agree with perturbative results when the latter are finite (up to the first subleading term), that is for power-spectra with $-3<n<-1$. Over the range $-1<n<1$ some shell-crossing occurs (which leads to the break-up of perturbative approaches) but this does not invalidate our approach. In particular, we explain that we can still obtain an approximation for the low-density tail of the pdf. This feature also clearly shows that perturbative results should be viewed with caution (even when they are finite). We point out that our results can be recovered by a simple spherical model but they cannot be derived from the stable-clustering ansatz in the regime $\sigma \gg 1$ since they involve underdense regions which are still expanding. Second, turning to high-density regions we explain that a naive study of the radial spherical dynamics fails. Indeed, a violent radial-orbit instability leads to a fast relaxation of collapsed halos (over one dynamical time) towards a roughly isotropic equilibrium velocity distribution. Then, the transverse velocity dispersion stabilizes the density profile so that almost spherical halos obey the stable-clustering ansatz for $-3<n<1$. We again find that our results for the high-density tail of the pdf agree with a simple spherical model (which takes into account virialization). Moreover, they are consistent with the stable-clustering ansatz in the non-linear regime. Besides, our approach justifies the large-mass cutoff of the Press-Schechter mass function (although the various normalization parameters should be modified).