Dynamics of gravitational clustering II. Steepest-descent method for the quasi-linear regime

Abstract

We develop a non-perturbative method to derive the probability distribution $P(δ_R)$ of the density contrast within spherical cells in the quasi-linear regime. Indeed, since this corresponds to a rare-event limit a steepest-descent approximation can yield asymptotically exact results. We check that this is the case for Gaussian initial density fluctuations, where we recover most of the results obtained by perturbative methods from a hydrodynamical description. Moreover, we correct an error which was introduced in previous works for the high-density tail of the pdf. This feature, which appears for power-spectra with a slope $n<0$, points out the limitations of perturbative approaches which cannot describe the pdf $P(δ_R)$ for $δ_R \ga 3$ even in the limit $σ\to 0$. This break-up does not involve shell-crossing and it is naturally explained within our framework. Thus, our approach provides a rigorous treatment of the quasi-linear regime, which does not rely on the hydrodynamical approximation for the equations of motion. Besides, it is actually simpler and more intuitive than previous methods. Our approach can also be applied to non-Gaussian initial conditions.