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.