Nonlinear Power Spectrum from Resummed Perturbation Theory: a Leap Beyond the BAO Scale

TL;DR

This paper presents a fast, resummed perturbation theory scheme to compute the nonlinear matter power spectrum beyond the BAO scale by recasting cosmological perturbation theory as time-evolution equations for the PS and propagator. It introduces an interpolation strategy that smoothly connects the low-$k$ (1-loop-inspired) regime with the high-$k$ (eikonal) regime, using a combination of nonlinear inputs, renormalized propagators, and a carefully filtered mode-coupling term $\tilde{\Phi}_{ab}$. The key contributions include the derivation of the evolution equations, practical small-$k$ resummations ($P^{R1}$, $P^{R2}$), a large-$k$ eikonal limit, and an intermediate regime interpolation (eqs. for $\tilde{\Phi}$ and the filter), all validated against N-body simulations to percent-level accuracy in the BAO range and up to $k\sim1\,h\mathrm{Mpc}^{-1}$ at higher redshifts. The method offers substantial speed and cosmology flexibility, enabling parameter inference and extensions toward weak lensing, while acknowledging intrinsic limitations from multi-streaming not captured by the single-stream Eulerian framework.

Abstract

A new computational scheme for the nonlinear cosmological matter power spectrum (PS) is presented. Our method is based on evolution equations in time, which can be cast in a form extremely convenient for fast numerical evaluations. A nonlinear PS is obtained in a time comparable to that needed for a simple 1-loop computation, and the numerical implementation is very simple. Our results agree with N-body simulations at the percent level in the BAO range of scales, and at the few-percent level up to $k ~ 1$ h/Mpc at $z >= 0.5$, thereby opening the possibility of applying this tool to scales interesting for weak lensing. We clarify the approximations inherent to this approach as well as its relations to previous ones, such as the Time Renormalization Group, and the multi-point propagator expansion. We discuss possible lines of improvements of the method and its intrinsic limitations by multi streaming at small scales and low redshifts.

Figures (19)

• Figure 1: (left) The nonlinear PS from the N-body simulations of Sato:2011qr, divided by the no-wiggle PS of eisensteinhu, plotted against the variable $y$ defined in eq. (\ref{['ly']}). The color-code is the following: green for $z=0$, black for $z=0.5$, red for $z=1$, and purple for $z=2$. Each PS has been truncated at $k=1\,\mathrm{h/Mpc}$. (right) The relation between $y$ and $k$ at different redshifts.
• Figure 2: The Feynman Rules for cosmological perturbation theory
• Figure 3: The two 1PI functions $\Sigma^{(1)}_{ab}$ and $\Phi^{(1)}_{ab}$ at 1-loop, given explicitly by eqs. (\ref{['SIGMA1L']}) and (\ref{['phicomp']}).
• Figure 4: The 2-loop contributions to $\Phi_{ab}$ giving leading contributions in the large $k$ limit, with no vertex renormalization.
• Figure 5: The dominant lowest order contribution to $\Phi_{ab}$ with a vertex renormalization.