On the non-linear scale of cosmological perturbation theory

TL;DR

This paper proves that non-linear corrections from soft modes do not cause polynomial enhancement in equal-time cosmological correlators, suggesting that their impact is limited to logarithmic corrections, which refines understanding of perturbation theory convergence.

Contribution

The paper demonstrates the absence of polynomial enhancement from soft modes in equal-time correlators through resummation, eikonal approximation, and explicit two-loop calculations.

Findings

No polynomial enhancement from soft modes in the power spectrum.

Non-linear corrections are at most logarithmic in soft mode effects.

Provides a detailed analysis of asymptotic behaviors in different momentum regimes.

Abstract

We discuss the convergence of cosmological perturbation theory. We prove that the polynomial enhancement of the non-linear corrections expected from the effects of soft modes is absent in equal-time correlators like the power or bispectrum. We first show this at leading order by resumming the most important corrections of soft modes to an arbitrary skeleton of hard fluctuations. We derive the same result in the eikonal approximation, which also allows us to show the absence of enhancement at any order. We complement the proof by an explicit calculation of the power spectrum at two-loop order, and by further numerical checks at higher orders. Using these insights, we argue that the modification of the power spectrum from soft modes corresponds at most to logarithmic corrections. Finally, we discuss the asymptotic behavior in the large and small momentum regimes and identify the expansion parameter pertinent to non-linear corrections.