Rapid cosmological inference with the two-loop matter power spectrum

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 $A_s$.

💡 Research Summary

This paper presents a breakthrough in the practical application of the Effective Field Theory of Large‑Scale Structure (EFT of LSS) by delivering a rapid, high‑precision computation of the two‑loop matter power spectrum for ΛCDM cosmologies. The authors build on the recently introduced COBRA (Cosmology with Optimally factorized Bases for Rapid Approximation) framework, which decomposes the linear power spectrum P_L(k) into a small set of basis functions v_i(k) multiplied by cosmology‑dependent weights w_i(Θ). Using a singular‑value decomposition (SVD) over a broad ΛCDM prior (ω_b, ω_c, n_s, h), they find that as few as nine (full spectrum) or twelve (including wiggles) basis functions suffice to achieve sub‑percent accuracy across the relevant k‑range (0.0008 h⁎ Mpc⁻¹ < k < 20 h⁎ Mpc⁻¹).

Once the basis is fixed, all loop integrals required by the EFT can be expressed as contractions of pre‑computed tensors S_{ij…} with the weights w_i(Θ). For the one‑loop term a rank‑2 tensor S_{ij} is needed, while the two‑loop term requires a rank‑3 tensor S_{ijk}. These tensors are evaluated once using high‑precision Monte‑Carlo integration (CUBA library with the SUAVE algorithm), targeting a relative error of 10⁻³ and a maximum of 8×10⁷ samples per integral. The pre‑computation is the only expensive step; during parameter inference the evaluation reduces to simple linear algebra, allowing the full two‑loop EFT power spectrum to be computed in roughly one millisecond on a single CPU core with ~0.1 % precision.

The two‑loop EFT expression includes the linear spectrum, the standard one‑loop SPT contribution, the full two‑loop SPT term (P_{33}^I, P_{33}^{II}, P_{24}, P_{15}), and four counterterms: c_{s,1}² and c_{s,2}² (∇²δ operators), c_{quad} (∇²δ²), and c_4 (∇⁴δ). The stochastic term ε is set to zero following previous findings. Infrared (IR) resummation is performed via a wiggle‑no‑wiggle split, damping the BAO wiggles with a scale‑dependent factor Σ²(Θ). The authors also test a Lagrangian‑based IR‑resummation scheme and find negligible differences, confirming robustness against the choice of IR treatment.

To validate COBRA’s accuracy, the authors compare the COBRA‑generated two‑loop spectra against direct numerical two‑loop calculations for 200 randomly sampled cosmologies within the prior. The 95th‑percentile relative error never exceeds 0.1 % for both the full and no‑wiggle components, demonstrating that the basis and tensor truncation are sufficient. They further benchmark against the Dark Sky N‑body simulation’s nonlinear matter power spectrum at z = 0, showing that the two‑loop EFT remains accurate up to k ≈ 0.26 h Mpc⁻¹, whereas the one‑loop EFT deviates already beyond k ≈ 0.11 h Mpc⁻¹.

Armed with this fast model, the authors conduct a Markov Chain Monte Carlo analysis mimicking a Stage IV galaxy survey with a volume V = 25 (Gpc/h)³. Using the two‑loop EFT up to k_max = 0.26 h Mpc⁻¹, they obtain unbiased constraints on Ω_m, H_0, and A_s, improving the one‑dimensional marginalized uncertainties by roughly 35 % (Ω_m), 20 % (H_0), and 15 % (A_s) relative to the one‑loop analysis limited to k_max = 0.11 h Mpc⁻¹. The combined Figure of Merit (FoM) for these three parameters increases by a factor of ~2.6, highlighting the substantial gain in cosmological information when extending perturbation theory to two loops with an efficient computational pipeline.

In conclusion, the paper demonstrates that COBRA enables the two‑loop EFT power spectrum to be evaluated with millisecond latency and sub‑percent accuracy, making it feasible to incorporate full two‑loop perturbative predictions into real‑time cosmological inference pipelines. This opens the door to exploiting mildly non‑linear scales in upcoming large‑scale structure surveys, potentially tightening constraints on fundamental cosmological parameters and testing extensions beyond ΛCDM. Future work will likely incorporate galaxy bias operators, redshift‑space distortions, and explore the method’s applicability to alternative cosmologies such as massive neutrinos or dynamical dark energy.