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

TL;DR

This paper introduces a non-perturbative steepest-descent method to accurately derive the probability distribution of density contrasts in the quasi-linear regime of gravitational clustering, improving upon previous perturbative approaches.

Contribution

It develops a rigorous, simpler, and more intuitive method that corrects previous errors and extends applicability to non-Gaussian initial conditions in the quasi-linear regime.

Findings

Recovers known results for Gaussian initial conditions.

Corrects high-density tail errors in previous models.

Applicable to non-Gaussian initial conditions.

Abstract

We develop a non-perturbative method to derive the probability distribution $P(\delta_R)$ 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 $P(\delta_R)$ for $\delta_R \ga 3$ even in the limit $\sigma \to 0$. 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.