The Effective Field Theory of Large Scale Structure at Two Loops: the apparent scale dependence of the speed of sound

TL;DR

This work advances the EFT of Large Scale Structure by computing the density and momentum power spectra at two-loop order and addressing the apparent scale-dependence of the leading EFT parameter c_s^2. It identifies UV sensitivities and introduces a one-parameter counterterm ansatz that ties the two-loop corrections to the low-k behavior, enabling a close match to simulations up to k ~ 0.3 h/Mpc at z=0. The study also extends the analysis to momentum statistics and demonstrates consistency with IR resummation, suggesting a robust description of the deterministic part of the power spectrum on mildly nonlinear scales. These results quantify the limits of perturbation theory and provide a practical framework for incorporating higher-order EFT corrections in cosmological analyses.

Abstract

We study the Effective Field Theory of Large Scale Structure for cosmic density and momentum fields. We show that the finite part of the two-loop calculation and its counterterms introduce an apparent scale dependence for the leading order parameter $c_\text{s}^2$ of the EFT starting at k=0.1 h/Mpc. These terms limit the range over which one can trust the one-loop EFT calculation at the 1 % level to k<0.1 h/Mpc at redshift z=0. We construct a well motivated one parameter ansatz to fix the relative size of the one- and two-loop counterterms using their high-k sensitivity. Although this one parameter model is a very restrictive choice for the counterterms, it explains the apparent scale dependence of $c_\text{s}^2$ seen in simulations. It is also able to capture the scale dependence of the density power spectrum up to k$\approx$ 0.3 h/Mpc at the 1 % level at redshift $z=0$. Considering a simple scheme for the resummation of large scale motions, we find that the two loop calculation reduces the need for this IR-resummation at k<0.2 h/Mpc. Finally, we extend our calculation to momentum statistics and show that the same one parameter model can also describe density-momentum and momentum-momentum statistics.

Figures (13)

• Figure 1: Diagrams for the tree level, one- and two-loop expressions of the SPT power spectrum.
• Figure 2: Diagrams that are regularized in our approach. The dashed loops are the ones where the momenta is large and are fixed by a counterterm.
• Figure 3: Effect of changing the cut-off from $\Lambda_\text{h}=5\; h\text{Mpc}^{-1}$ to $\Lambda_\text{l}=1\; h\text{Mpc}^{-1}$ for the one and two loop calculations normalized by $k^2 P$. Left panel: Contributions from the low-high and high-high terms (single- and double-hard). The mixed term clearly dominates the $k^2 P$ part and also the deviations from this behavior. Right panel: Contributions from the separate diagrams. At the one loop level $P_{13}$ leads to a $k^2 P$ contributions, whereas the $k^4$ contribution from $P_{22}$ is suppressed. $P_{15}$ dominates the $k^2 P$ part but for the deviations from this scaling, there is a cancellation between $P_{15}$, $P_{33\text{-II}}$ and $P_{24}$. Like $P_{22}$ in the one loop case, the $k^4$ term arising from $P_{33-I}$ is suppressed.
• Figure 4: Comparison between the two-loop counterterm deduced from the divergencies and the one-loop power spectrum weighted by wavenumber squared. We see that the explicit calculation of the two loop counterterms $\bar{P}_\text{ctr,2loop}$ is proportional to the naive estimate $k^2 P_\text{1loop}$.
• Figure 5: Left panel: Estimate of the size of the corrections arising from various contributions to the two-loop calculation. The finite part of the two-loop calculation leads to percent level corrections at $k=0.1 \; h\text{Mpc}^{-1}$. We also show the corrections from the square of the speed of sound term $k^4 P_{11}$, which is suppressed over the range considered here. The size of the three-loop counterterm can be estimated as $\mathcal{O}(1)\times k^2 \bar{P}_\text{2loop}$ and leads to percent level contributions at $k=0.3\; h\text{Mpc}^{-1}$. We also show the estimate for the stochastic part of the total power spectrum from Baldauf:2015tlb which leads to percent level corrections at $k=0.25\; h\text{Mpc}^{-1}$. Right panel: Estimator for the leading EFT coefficient $c_\text{s}^2$. The model is evaluated for $c_\text{s}^2=0.98\; h^{-2}\text{Mpc}^{2}$ and the gray band shows the effect of a $10\%$ change in this value. Note that at $k=0.2\; h\text{Mpc}^{-1}$ the one-loop counterterm and the two-loop correction are of the same order. The two-loop term leads to a considerable scale dependence of $\hat{c}_s^2$ for $k>0.07\; h\text{Mpc}^{-1}$.