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

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.

💡 Research Summary

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The paper presents a novel, highly efficient numerical framework for evaluating the two‑loop dark‑matter power spectrum within the Effective Field Theory of Large‑Scale Structure (EFTofLSS). As upcoming galaxy surveys such as DESI, Euclid, and LSST demand sub‑percent theoretical precision across a vast cosmological parameter space, fast and accurate predictions at higher perturbative orders become essential.

The authors’ strategy hinges on two key ideas: (i) a cosmology‑independent decomposition of the linear power spectrum and (ii) a local treatment of ultraviolet (UV) and infrared (IR) divergences directly at the integrand level. First, a reference linear spectrum (P_{0}(k)) (chosen to be a CMB‑compatible ΛCDM model) is fixed. For any target cosmology (\theta), the difference (\Delta P_{\rm lin}(k;\theta)=P_{\rm lin}(k;\theta)-P_{0}(k)) is typically a few percent and can be accurately expanded onto a small set of basis functions (B_{n}(k)) that resemble massive scalar propagators, i.e. (B_{n}(k)\sim (k^{2}+m_{n}^{2})^{-1}). The expansion coefficients (c_{n}(\theta)) encapsulate all cosmology dependence.

Because the two‑loop diagrams are polynomial in the linear spectrum, after the expansion the full two‑loop contribution reduces to a linear combination of a finite set (≈300) of “universal” loop integrals (I_{m}) that involve only products of the basis functions. These integrals are completely independent of the cosmological parameters and can be pre‑computed with very high numerical accuracy. The final power spectrum is then obtained by contracting the pre‑computed (I_{m}) with the cosmology‑dependent coefficients (c_{n}(\theta)).

UV divergences are handled by constructing, for each diagram, a local UV approximation (U_{m}(p,q,\dots)) that reproduces the exact high‑momentum behavior. Subtracting (U_{m}) at the integrand level yields a UV‑finite remainder; the subtracted pieces match the structure of the EFT counterterms, guaranteeing a consistent renormalization.

IR singularities, which cancel only after summing many diagrams, are made manifestly local by mapping all diagrams onto a common integration domain and explicitly removing the (1/q^{n}) (or (1/|k-q|^{n})) pieces that generate the divergences. This “local IR cancellation” leaves a smooth integrand that can be evaluated reliably with stochastic integration methods.

The renormalized two‑loop spectrum requires eight EFT counterterms, one of which accounts for generated vorticity (a short‑distance effect that first appears at two loops). The authors provide explicit expressions for all counterterms and demonstrate that the resulting spectrum matches existing one‑loop EFT results and high‑resolution N‑body simulations to better than 0.5 % over the range (k\lesssim0.3,h,{\rm Mpc}^{-1}).

Numerically, the universal integrals (I_{m}) are computed using quasi‑Monte‑Carlo and multilevel Monte‑Carlo techniques, achieving relative errors of order (10^{-4}) with modest computational resources. Once the database of (I_{m}) is built (a one‑time cost of a few CPU‑hours), evaluating the full two‑loop power spectrum for any cosmology takes less than a second on a standard desktop.

The authors release the pre‑computed integral tables together with a Python interface that takes cosmological parameters as input and returns the renormalized two‑loop power spectrum, including the eight counterterms. This public code enables immediate application to lensing‑galaxy cross‑correlations, galaxy clustering, and other observables where two‑loop accuracy is desirable.

In summary, the paper delivers a practical, scalable solution to the long‑standing bottleneck of two‑loop EFT calculations, opening the door to full‑shape analyses of next‑generation LSS data at unprecedented precision. Future extensions could incorporate biased tracers, redshift‑space distortions, and baryonic effects within the same framework.