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

Abstract

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 $k\gtrsim 0.17h{\mathrm{Mpc}}^{-1}$, 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 $k\simeq 0.08,0.13,0.16h{\mathrm{Mpc}}^{-1}$ for $\sum m_{\nu }=0.4,0.2,0.1\mathrm{eV}$, respectively. We use an effective field theory (EFT) framework at two-loop valid for wavenumbers $k\gg k_{\mathrm{FS}}$, where $k_{\mathrm{FS}}$ 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, $\Delta {\bar{\gamma }}_1\simeq -0.2{\mathrm{Mpc}}^2/h^2$. We find percent agreement between the perturbative and N-body results up to $k\lesssim 0.12h{\mathrm{Mpc}}^{-1}$ and $k\lesssim 0.16h{\mathrm{Mpc}}^{-1}$ at one- and two-loop order, respectively, for all considered neutrino masses $\sum m_{\nu }\le 0.4\mathrm{eV}$.

30 pages, 14 figures

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