The EFT of Large Scale Structures at All Redshifts: Analytical Predictions for Lensing

TL;DR

This paper extends the Effective Field Theory of Large Scale Structures (EFTofLSS) to all redshifts, showing that the nonlinear matter power spectrum can be predicted with percent-level accuracy using a two-parameter time-dependent framework. By fixing $c_{s(1)}^2(0)$ at a reference redshift and introducing a single time-slope parameter $\beta$, the authors model the redshift evolution of the counterterms and demonstrate that two-loop EFT remains predictive up to high wavenumbers across $z=0$–4. They provide analytical estimates for the range of validity of one-, two-, and three-loop predictions and develop an accurate lensing calculation, computing the lensing potential power spectrum $C_\ell^\psi$ for CMB and galaxy lensing with modest theoretical uncertainties. The results yield substantial gains in the number of modes accessible to analytic control compared to standard perturbation theory and establish a concrete link between the matter power spectrum and gravitational lensing observables, including the impact of baryonic effects. Overall, the work substantiates the EFTofLSS as a practical and powerful tool for interpreting LSS data and guiding future observational tests.

Abstract

We study the prediction of the Effective Field Theory of Large Scale Structures (EFTofLSS) for the matter power spectrum at different redshifts. In previous work, we found that the two-loop prediction can match the nonlinear power spectrum measured from $N$-body simulations at redshift zero within approximately 2% up to $k\sim 0.6\,h\, {\rm Mpc}^{-1}$ after fixing a single free parameter, the so-called "speed of sound". We determine the time evolution of this parameter by matching the EFTofLSS prediction to simulation output at different redshifts, and find that it is well-described by a fitting function that only includes one additional parameter. After the two free parameters are fixed, the prediction agrees with nonlinear data within approximately 2% up to at least $k\sim 1\,h\, {\rm Mpc}^{-1}$ at $z\geq 1$, and also within approximately 5% up to $k\sim 1.2\,h\, {\rm Mpc}^{-1}$ at $z=1$ and $k\sim 2.3\,h\, {\rm Mpc}^{-1}$ at $z=3$, a major improvement with respect to other perturbative techniques. We also develop an accurate way to estimate where the EFTofLSS predictions at different loop orders should fail, based on the sizes of the next-order terms that are neglected, and find agreement with the actual comparisons to data. Finally, we use our matter power spectrum results to perform analytical calculations of lensing potential power spectra corresponding to both CMB and galaxy lensing. This opens the door to future direct applications of the EFTofLSS to observations of gravitational clustering on cosmic scales.

Figures (20)

• Figure 1: Left: effective slope $n_{\rm eff}(k) = d\log P_{\rm nw}(k)/d\log k$, where $P_{\rm nw}$ is the linear matter power spectrum without BAO wiggles. $P_{\rm nw}(k)$ is given by Eq. (\ref{['eq:ehps']}). Center: running of this slope, $dn_{\rm eff}/d\log k$. Right: running of the running, $d^2 n_{\rm eff}/d\log k^2$. The large running of the slope implies that a simple analogy with the case of a pure scaling universe will be insufficient to approximate the behavior of various terms in the power spectrum prediction.
• Figure 2: Left: Linear theory prediction for the power spectrum, normalized to the nonlinear spectrum from the Coyote emulator at different redshifts. The dashed lines show the estimated error on the nonlinear data, while the dotted line at the value $P_{11}/P_\text{Coyote}=1$ is provided as a visual aid. Right: Estimates for when linear theory should fail, based on when the ratio $P_\text{1-loop}^\text{(total)}/P_{11}$ exceeds the error on the nonlinear data. We quote a range of wavenumbers based on dividing and multiplying the estimate from Eq. (\ref{['eq:p1looptotest']}) by a factor of 2. By comparing these estimates with the plots on the left, we find that linear theory fails roughly where it should.
• Figure 3: Schematic depiction of the dominant two-loop diagrams in the power spectrum: $\left[ P_{22}+P_{13} \right] \times P_{13}^{\text{(no-IR)}}/P_{11}$.
• Figure 4: Exact and approximate versions of $P_\text{2-loop}^\text{(total)}/P_{11}$ at $z=0$, the latter plotted using Eq. (\ref{['eq:p2looptotest']}) with $\alpha\simeq3/10$ and $\beta=1$. The bottom panel is a zoomed-in version of the top panel. Overall, the shapes of both curves match very well up to $k\sim 0.6\,h\, {\rm Mpc}^{-1}\,$.
• Figure 5: Approximate versions of $P_\text{2-loop}^\text{(total)}/P_{11}$ at $z=1$, plotted using Eq. (\ref{['eq:p2looptotest']}) with $\alpha\simeq3/10$ (dashed curve) and $-1/6+3/10$ (dotted cruve). Both curves use $\beta=1$. The dashed curve has zero crossings that are not expected from the exact calculation, while the dotted curve has the same slope as the dashed one at low $k$, but reflected across the $k$-axis, eliminating any zero crossings. This shift in the value of $\alpha$ is the minimal way to modify our estimate to remove the zero crossings; we also implement similar shifts at $z=2$ and 3.