Efficient evaluation of the dark-matter two-loop power spectrum in the EFT of LSS

Abstract

Rapid progress in cosmological Large Scale Structure (LSS) surveys motivates precise theoretical predictions. The Effective Field Theory of Large-Scale Structure (EFTofLSS) is routinely applied to data, and requires fast computation of its predictions when sampling the large space of cosmological parameters. Going beyond existing one-loop techniques, we present a method to rapidly evaluate the two-loop power spectrum. Our method decomposes the typically small difference between a given linear power spectrum and a reference power spectrum into a cosmology-independent basis of functions resembling massive scalar propagators in Quantum Field Theory. By taking the leading terms in such a small difference, we numerically evaluate the cosmology-independent loop integrals where in the integrand only the relevant combinations of basis functions appear. We achieve an efficient numerical evaluation via physically motivated local ultraviolet subtractions and by arranging the cancellation of infrared singularities locally in the integrands. Final predictions are obtained by contracting these precomputed integrals with the cosmology-dependent coordinates of the expansion in the fixed basis. We present and publicly release the precomputed integrals for the renormalized two-loop dark-matter power spectrum in the EFTofLSS. These require eight EFT counterterms, which include the effect of generated vorticity, and are sufficient to analyze the lensing galaxy signal in LSS surveys at this order.

Figures (14)

• Figure 1: The fractions $\Delta^{\rm no-CT}_{\rm 1-loop}$ (blue) and $\Delta^{\rm no-CT}_{\rm 2-loop}$ (red), computed using no$-$CT (often called SPT) kernels and the Planck linear power spectrum up to $k=10^4$$h$/Mpc. The UV contribution to the loop corrections is significant, especially at two-loops. The sharp peaks at $k \sim 0.082 \, h/\text{Mpc}$ and $k\sim 0.48 \, h/\text{Mpc}$ are due to the loop correction vanishing.
• Figure 2: $\Delta^{{\rm no}-{\rm CT}}_{\rm 1-loop}(\nu)$ (\ref{['fig:Delta_1L_noCT_nu']}) and $\Delta^{{\rm no}-{\rm CT}}_{\rm 2-loop}(\nu)$ (\ref{['fig:Delta_2L_noCT_nu']}) at $k=0.25 \, h/\text{Mpc}$. We show that the infrared part of the loop corrections, without the counterterms, are sensitive to the tilt $\nu$ of the linear power spectrum in the UV.
• Figure 3: The fractions $\Delta^{\rm UV-reg.}_{\rm 1-loop}$ (blue) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}$ (red), computed using UV-reg. kernels and the Planck linear power spectrum up to $k=10^4$$h$/Mpc. Compared to the no-CT case, the UV contribution is significantly lower. The sharp peak for the two-loop at $k \sim 0.3 \, h/\text{Mpc}$ is due to the loop correction vanishing.
• Figure 4: $\Delta^{\rm UV-reg.}_{\rm 1-loop}(\nu)$ (\ref{['fig:Delta_1L_UVreg_nu']}) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}(\nu)$ (\ref{['fig:Delta_2L_UVreg_nu']}) at ${k=0.25~h/\text{Mpc}}$. We show that UV contribution to the loop corrections is now much less sensitive, compared to no-CT, to the tilt $\nu$ of the linear power spectrum. Furthermore, we are able to compute the loop integrands at $\nu=1$, where the no-CT expressions were singular.
• Figure 5: The power spectrum (i) ${\cal P}_{\rm LO}^{\rm Planck}$ computed at leading order (linear contribution) (blue), (ii) ${\cal P}_{\rm NLO}^{\rm Planck}$ computed at NLO which includes the one-loop correction (red), and (iii) ${\cal P}_{\rm NNLO}^{\rm Planck}$ computed at NNLO which includes both the one-loop and two-loop corrections (green). In Fig. (a) we show the scaling of the full power spectrum across the range $[10^{-3}, 1]$$h/$Mpc, while in Fig. (b) we focus on the $k$ range $[0.1, 0.6]$$h/$Mpc, where the effect of the loop correction becomes more significant. Both figures are at redshift $z = 0.57$.