Divergence of Perturbation Theory in Large Scale Structures

TL;DR

This paper investigates the limits of perturbation theory in modeling large scale structures, showing convergence for some statistics like the power spectrum but divergence for others such as the correlation function and PDF, highlighting the role of extreme fluctuations.

Contribution

It extends the understanding of perturbation theory's validity in large scale structures, proving convergence for the power spectrum and divergence for real space correlation functions, and discusses non-perturbative effects.

Findings

Perturbation theory converges for the power spectrum.

Perturbation theory diverges for the real space two-point correlation function.

Extreme fluctuations influence the statistical averages even at large scales.

Abstract

We make progress towards an analytical understanding of the regime of validity of perturbation theory for large scale structures and the nature of some non-perturbative corrections. We restrict ourselves to 1D gravitational collapse, for which exact solutions before shell crossing are known. We review the convergence of perturbation theory for the power spectrum, recently proven by McQuinn and White, and extend it to non-Gaussian initial conditions and the bispectrum. In contrast, we prove that perturbation theory diverges for the real space two-point correlation function and for the probability density function (PDF) of the density averaged in cells and all the cumulants derived from it. We attribute these divergences to the statistical averaging intrinsic to cosmological observables, which, even on very large and "perturbative" scales, gives non-vanishing weight to all extreme fluctuations. Finally, we discuss some general properties of non-perturbative effects in real space and Fourier space.