Loop corrections to the power spectrum for massive neutrino cosmologies with full time- and scale-dependence

Loop corrections to the power spectrum are frequently computed using approximate non-linear kernels adopted from an Einstein de-Sitter (EdS) cosmology. We present an algorithm that allows us to take the full time- and scale-dependence of the underlying fluid dynamics into account, and apply it to assess the impact of neutrino free-streaming on the 1- and 2-loop matter power spectrum. Neutrino perturbations are described by a hybrid Boltzmann/two-fluid model, that we show to agree with the CLASS Boltzmann solution at the linear level when using an appropriate effective neutrino sound velocity. We then use this scheme at 1- and 2-loop to perform a precision comparison of the matter power spectrum with simplified treatments of massive neutrinos. For a commonly adopted scheme using cold dark matter+baryon perturbations and EdS kernels to compute non-linear corrections, we find deviations above 1% for k > 0.15 h/Mpc at z=0, and sub-percent agreement on weakly non-linear scales at z=0.5. We also demonstrate the impact of the EdS approximation on the 2-loop power spectrum for massless neutrinos.