DEMNUni: comparing nonlinear power spectra prescriptions in the presence of massive neutrinos and dynamical dark energy

We provide an accurate comparison, against large cosmological $N$-body simulations, of different prescriptions for modelling nonlinear matter power spectra in the presence of massive neutrinos and dynamical dark energy. We test the current most widely used approaches: fitting functions (HALOFIT and HMcode), the halo-model reaction (ReACT) and emulators (baccoemu and EuclidEmulator2). Focussing on redshifts $z\leq2$ and scales $k\lesssim 1 \ h/$Mpc (where the simulation mass resolution provides $\sim 1\%$ accuracy), we find that HMcode and ReACT considerably improve over the HALOFIT prescriptions of Smith and Takahashi (both combined with the Bird correction), with an overall agreement of 2\% for all the cosmological scenarios considered. Concerning emulators, we find that, especially at low redshifts, EuclidEmulator2 remarkably agrees with the simulated spectra at $\lesssim 1\%$ level in scenarios with dynamical dark energy and massless neutrinos, reaching a maximum difference of $\sim 2\%$ at $z=2$. baccoemu has a similar behaviour as EuclidEmulator2, except for a couple of dark energy models. In cosmologies with massive neutrinos, at $z=0$ all the nonlinear prescriptions improve their agreement with respect to the massless neutrino case, except for the Bird and TakaBird models which, however, are not tailored to $w_0$--$w_a$ models. At $z>0$ we do not find a similar improvement when including massive neutrinos, probably due to the lower impact of neutrino free-streaming at higher redshifts; rather at $z=2$ EuclidEmulator2 exceeds $2\%$ agreement for some dark energy equation of state. When considering ratios between the matter power spectrum computed in a given cosmological model and its $\Lambda$CDM counterpart, all the tested prescriptions agree with simulated data, at sub-percent or percent level, depending on $z$. [ABRIDGED]