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We compute the density and velocity power spectra at next-to-next-to-leadingorder taking into account the effect of time- and scale-dependent growth ofmassive neutrino perturbations as well as the departure fromEinstein--de-Sitter (EdS) dynamics at late times non-linearly. We determine theimpact of these effects by comparing to the commonly adopted approximatetreatment where they are not included. For the bare cold dark matter(CDM)+baryon spectrum, we find percent deviations for $k\gtrsim0.17h~\mathrm{Mpc}^{-1}$, mainly due to the departure from EdS. For thevelocity and cross power spectrum the main difference arises due to time- andscale-dependence in presence of massive neutrinos yielding percent deviationabove $k\simeq 0.08, 0.13, 0.16h~\mathrm{Mpc}^{-1}$ 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 QuijoteN-body simulations, we find that for the CDM+baryon density power spectrum theeffect of neutrino perturbations and exact time-dependent dynamics at latetimes 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 percentagreement between the perturbative and N-body results up to $k\lesssim0.12h~\mathrm{Mpc}^{-1}$ and $k\lesssim 0.16h~\mathrm{Mpc}^{-1}$ at one- andtwo-loop order, respectively, for all considered neutrino masses $\sum m_{\nu}\leq 0.4~\mathrm{eV}$.

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