Rapid cosmological inference with the two-loop matter power spectrum

TL;DR

This work extends the EFT of large-scale structure to two-loop order using the COBRA framework, achieving fast and accurate predictions for the nonlinear matter power spectrum with sub-permille precision. COBRA decomposes the linear power spectrum into a small basis, precomputes cosmology-independent loop tensors, and projects cosmologies to obtain loop corrections in about 1 ms on a single CPU. When validated against the Dark Sky simulation, the two-loop EFT with IR resummation provides unbiased constraints on $\Omega_m$, $H_0$, and $A_s$ up to $k_{\max}\approx0.26\,h\mathrm{Mpc}^{-1}$, outperforming the one-loop EFT and increasing the Figure of Merit by roughly a factor of 2.6. The results imply a substantial gain in information content from mildly nonlinear scales for Stage IV surveys, while also illustrating the method’s scalability and potential extensions to biased tracers and redshift-space distortions.

Abstract

We compute the two-loop effective field theory (EFT) power spectrum of dark matter density fluctuations in $Λ$CDM using the recently proposed COBRA method (Bakx. et al, 2025). With COBRA, we are able to evaluate the two-loop matter power spectrum in $\sim 1$ millisecond at $ \sim 0.1 \%$ precision on one CPU for arbitrary redshifts and on scales where perturbation theory applies. As an application, we use the nonlinear matter power spectrum from the Dark Sky simulation to assess the performance of the two-loop EFT power spectrum compared to the one-loop EFT power spectrum at $z=0$. We find that, for volumes typical for Stage IV galaxy surveys, $V = 25 \,(\text{Gpc}/h)^3$, the two-loop EFT can provide unbiased cosmological constraints on $Ω_m,H_0$ and $A_s$ using scales up to $k_\text{max}=0.26\, h/\text{Mpc}$, thereby outperforming the constraints from the one-loop EFT ($k_\text{max}=0.11\, h/\text{Mpc}$). The Figure of Merit on these three parameters increases by a factor $\sim 2.6$ and the one-dimensional marginalized constraints improve by $\sim35\%$ for $Ω_m$, $\sim20\%$ for $H_0$ and $\sim 15\%$ for $A_s$.

Figures (8)

• Figure 1: Error on individual contributions to the two-loop EFT power spectrum using the COBRA expansion relative to the non-expanded numerical result. They are calculated as the 95th percentiles of a total of 200 random test cosmologies within the COBRA prior range. We display the full and no-wiggle spectra in red and blue, respectively. The linear spectra, the one-loop SPT term, the quadratic counterterm and the two-loop contributions are shown in different rows. In each subplot, we show two separate choices of $N_b$ in different line styles, where the solid lines correspond to Eqs. \ref{['eq:nbnw']} and \ref{['eq:nbtot']}. The spikes in the second and fourth panels are largely due to the integrals taking values close to zero, which are suppressed when considering the full spectra (see Fig. \ref{['fig:total_result']}). The 'quad' term (third panel) scales as $k^4$ on large scales and thus presents larger relative errors at low $k$. The black dashed lines indicate the $10^{-3}$ required precision of the numerical integrals.
• Figure 2: The 95th percentile of errors on the total two-loop spectrum (cf. Eq. \ref{['eq:twoloops']}) of the test set of 200 random cosmologies calculated using COBRA, compared to the direct numerical result. We obtain the counterterms by fitting the two-loop spectra to the HaloFit at three different redshifts, $z=0,1,2$ using two different values of $k_\text{max}$ at all redshifts (vertical dashed lines). We include the IR resummation and take $N_b^\text{nw} = \{12,12,12,6\}$ basis functions for no-wiggle contributions and $N_b^\text{full} = \{12,12,12,9\}$ for the full contributions, respectively, as in Eqs. \ref{['eq:nbnw']} and \ref{['eq:nbtot']}.
• Figure 3: Illustration of the UV-sensitivity of the two-loop SPT prediction [cf. Eq. \ref{['eq:twoloopspt']}] and how it can be absorbed by counterterms. We compute the two-loop matter power spectrum at two different cutoffs $\Lambda_1 = 2\,h/\text{Mpc}$ and $\Lambda_2 = 4.5\,h/\text{Mpc}$. Their ratio is the red line, which clearly exhibits running beyond the error we can tolerate [gray shades which represent $1\sigma$ and $2\sigma$ errors for a volume of $V_\text{fit} = 25 \,(\text{Gpc}/h)^3$, see Section \ref{['sec:simulation']}]. The blue line indicates the error on the ratio after adding the counterterms, determined by fitting up to $k=0.4\,h/\text{Mpc}$. The vertical dotted line indicates the maximum scale we consider for the two-loop matter power spectrum in this work (see Section \ref{['sec:results']}).
• Figure 4: Running of the one and two-loop counterterms as a function of $k_{\rm max}$. The first two panels show the $\propto k^2 P_L(k)$ counterterm (which we label as $c_\text{s}^2$), which is $c_{\text{s},1}^2$ for the one-loop case [in red, cf. Eq. \ref{['eq:oneloop']}] and $c_{\text{s},1}^2 +c_{\text{s},2}^2$ for the two-loop case [in blue, cf. Eq. \ref{['eq:twoloops']}]. The remaining panels show the other two-loop counterterms. The vertical lines at $k_\text{max} = 0.11\,h/\text{Mpc}$ and $k_\text{max} = 0.26\,h/\text{Mpc}$ indicate our fiducial choice of maximum wavenumber for the one-loop and two-loop case (see text). For visual purposes, points are slightly displaced horizontally.
• Figure 5: Top panel: the FoB defined in Eq. \ref{['eq:FoB']} for the one-loop (red) and two-loop (blue) EFT models, where the shaded band indicates the region satisfying Eq. \ref{['eq:fobcrit']}. Second panel: the $\chi^2_\text{red}$ from Eq. \ref{['eq:chi2red']} statistic (computed with the full volume covariance), where the shaded regions indicate the $1\sigma$ and $2\sigma$ confidence regions for the volume $V_\text{full}$. Third panel: FoM from Eq. \ref{['eq:fom']} as a function of the maximum wavenumber.