Two-loop power spectrum with full time- and scale-dependence and EFT corrections: impact of massive neutrinos and going beyond EdS

We compute the density and velocity power spectra at next-to-next-to-leading order taking into account the effect of time- and scale-dependent growth of massive neutrino perturbations as well as the departure from Einstein-de-Sitter (EdS) dynamics at late times non-linearly. We determine the impact of these effects by comparing to the commonly adopted approximate treatment where they are not included. For the bare cold dark matter (CDM)+baryon spectrum, we find percent deviations for k ≳ 0.17h Mpc-1, mainly due to the departure from EdS. For the velocity and cross power spectrum the main difference arises due to time- and scale-dependence in presence of massive neutrinos yielding percent deviation above k ≃ 0.08, 0.13, 0.16h Mpc-1 for ∑mν = 0.4, 0.2, 0.1 eV, respectively. We use an effective field theory (EFT) framework at two-loop valid for wavenumbers k ≫ k FS, where k FS is the neutrino free-streaming scale. Comparing to Quijote N-body simulations, we find that for the CDM+baryon density power spectrum the effect of neutrino perturbations and exact time-dependent dynamics at late times can be accounted for by a shift in the one-loop EFT counterterm, Δγ̅1 ≃ - 0.2 Mpc2/h 2. We find percent agreement between the perturbative and N-body results up to k ≲ 0.12h Mpc-1 and k ≲ 0.16h Mpc-1 at one- and two-loop order, respectively, for all considered neutrino masses ∑mν ≤ 0.4 eV.