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

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. Since our approach is based on a steepest-descent approximation which should yield exact results in the limit of rare events it applies to all values of the rms linear density fluctuation sigma, from the quasi-linear up to the highly non-linear regime. First, we derive the low-density tail of the pdf. We show that it agrees with perturbative results when the latter are finite up to the first subleading term, that is for linear power-spectra P(k) proportional to k(n) 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 (this is related to the spherical symmetry of our problem). On the other hand, we show that this low-density tail cannot be derived from the stable-clustering ansatz in the regime sigma >> 1 since it involves 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). Finally, we note that for sigma greater than or similar to -1 an intermediate region of moderate density fluctuations appears which calls for new non-perturbative tools.