Dynamics of gravitational clustering V. Subleading corrections in the quasi-linear regime

TL;DR

This paper analyzes the limitations of standard perturbative expansions in modeling early gravitational clustering, revealing divergence issues at finite orders and proposing alternative approaches for accurate large-scale predictions.

Contribution

It demonstrates the breakdown of perturbation theory beyond a certain order in hierarchical scenarios and connects these findings to Burgers and Zel'dovich dynamics, suggesting new non-perturbative methods.

Findings

Perturbation theory diverges beyond a finite order q_+ in hierarchical scenarios.

Leading-order results remain valid despite divergence of subleading terms.

Large-scale correlation functions involve non-perturbative terms beyond simple power expansions.

Abstract

We investigate the properties of the standard perturbative expansions which describe the early stages of the dynamics of gravitational clustering. We show that for hierarchical scenarios with no small-scale cutoff perturbation theory always breaks down beyond a finite order $q_+$. Besides, the degree of divergence increases with the order of the perturbative terms so that renormalization procedures cannot be applied. Nevertheless, we explain that despite the divergence of these subleading terms the results of perturbation theory are correct at leading order because they can be recovered through a steepest-descent method which does not use such perturbative expansions. Finally, we investigate the simpler cases of the Zel'dovich and Burgers dynamics. In particular, we show that the standard Burgers equation exhibits similar properties. This analogy suggests that the results of the standard perturbative expansions are valid up to the order $q_+$ (i.e. until they are finite). Moreover, the first ``non-regular'' term of a large-scale expansion of the two-point correlation function should be of the form $R^{-2} \sigma^2(R)$. At higher orders the large-scale expansion should no longer be over powers of $\sigma^2$ but over a different combination of powers of 1/R. However, its calculation requires new non-perturbative methods.