A critical look at cosmological perturbation theory techniques

TL;DR

The paper benchmarks multiple analytic perturbation theories for the non-linear matter power spectrum against high-resolution N-body simulations in two CDM cosmologies, highlighting that while these methods capture the onset of non-linearity and improve large-scale predictions, they fail on small, quasi-linear scales, especially at low redshift. It shows that 2-loop SPT can improve predictions at higher redshift but breaks down at late times, and that other resummation approaches offer mixed success. A key contribution is the construction of a reference spectrum that blends simulations with perturbation theory to quantify deviations across models, illustrating the limited but valuable regime where perturbation theory is reliable. The work underlines the need for non-perturbative methods with error control and provides public data and tools to advance future analyses.

Abstract

Recently a number of analytic prescriptions for computing the non-linear matter power spectrum have appeared in the literature. These typically involve resummation or closure prescriptions which do not have a rigorous error control, thus they must be compared with numerical simulations to assess their range of validity. We present a direct side-by-side comparison of several of these analytic approaches, using a suite of high-resolution N-body simulations as a reference, and discuss some general trends. All of the analytic results correctly predict the behavior of the power spectrum at the onset of non-linearity, and improve upon a pure linear theory description at very large scales. All of these theories fail at sufficiently small scales. At low redshift the dynamic range in scale where perturbation theory is both relevant and reliable can be quite small. We also compute for the first time the 2-loop contribution to standard perturbation theory for CDM models, finding improved agreement with simulations at large redshift. At low redshifts however the 2-loop term is larger than the 1-loop term on quasi-linear scales, indicating a breakdown of the perturbation expansion. Finally, we comment on possible implications of our results for future studies.

Figures (10)

• Figure 1: SPT power spectrum at linear (black; dotted), 1-loop (red; solid), and 2-loop (blue; dashed) order. The squares with error bars show the mean and error from our N-body simulations. The four panels show $\Lambda$CDM (left) and $c$CDM (right) at redshifts 1 (top) and 0 (bottom). Each curve has been divided by the no-wiggle power spectrum of Eisenstein98 to reduce the dynamic range. We also indicate the domain of validity of 1-loop SPT according to the heuristic prescription of Jeong06 ($\Delta^2<0.4$), and according to the criterion $P^{(3)} < \alpha\, P_L$ for $\alpha = 0.01, 0.03$.
• Figure 2: Comparison of the tree-level, 1-loop and 2-loop power spectrum from RPT and closure theory, for $\Lambda$CDM (left) and $c$CDM (right). Each curve has been divided by the no-wiggle power spectrum of Eisenstein98 to reduce the dynamic range. The (black) dotted line is linear theory, the (red) solid line is tree-level RPT, the (green) dashed line is 1-loop RPT, the (blue) long-dashed line is 2-loop RPT, the thick (yellow) short-long dashed line is tree-level closure, the (magenta) dot-long dashed line is 1-loop closure, and the (cyan) dot-dashed line is 2-loop closure.
• Figure 3: Comparison of the power spectrum for the remaining theories. Each curve has been divided by the no-wiggle power spectrum of Eisenstein98 to reduce the dynamic range. The (red) solid line is 1-loop SPT, the (magenta) dot-long dashed line is large-$N$ theory, the (green) dashed line is Lagrangian resummation, the thick (yellow) short-long dashed line is time-RG theory, and the (cyan) dot-dashed line is RGPT.
• Figure 4: Comparison between analytic models for $P(k;z=0)$ and the reference spectrum (Section \ref{['sec:compare']}) for model $c$CDM, focusing on large scales. Each curve has been divided by the no-wiggle power spectrum of Eisenstein98 to reduce the dynamic range. The points with error bars are the 'reference spectrum' defined in the text. The (black) dotted line is linear theory, the (red) solid line is 2-loop SPT, the (blue) long-dashed line is 2-loop RPT, the (green) short-dashed line is Lagrangian resummation, the (cyan) dot-dashed line is 2-loop closure theory, the thick (magenta) dot-long dashed line is the large-N expansion, and the thick (yellow) short-long dashed line is time-RG theory.
• Figure 5: The fractional deviation of each method from the reference spectrum, for $\Lambda$CDM at $z=0$ (left) and $z=1$ (right). This figure focuses on the region $k<0.15\,h\,\text{Mpc}^{-1}$ where linear theory is inadequate but higher order methods are still viable. As in Figure \ref{['fig:previr']} the (black) dotted line is linear theory, the (red) solid line is 2-loop SPT, the (blue) long-dashed line is 2-loop RPT, the (green) short-dashed line is Lagrangian resummation, the thick (cyan) dot-short dashed line is 2-loop closure theory the thick (magenta) dot-long dashed line is the large-$N$ expansion, and the thick (yellow) short-long dashed line is time-RG theory.