On the non-linear scale of cosmological perturbation theory
Diego Blas, Mathias Garny, Thomas Konstandin
TL;DR
This work analyzes the convergence of cosmological perturbation theory with respect to soft-mode couplings. By comparing standard perturbation theory, renormalized perturbation theory, and the eikonal resummation, the authors prove that leading and subleading soft corrections cancel for equal-time observables such as the power spectrum and bispectrum, leaving at most logarithmic nonlinear corrections controlled by $σ_l^2$ (and $σ_d^2$ in certain limits). They provide explicit two-loop calculations and general all-orders arguments, showing that the nonlinear growth at large $k$ is governed by logarithms rather than polynomial enhancements. The results clarify which resummation strategies are meaningful for equal-time correlators and guide accurate modeling of the nonlinear matter power spectrum and BAO in large-scale structure surveys.
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.