On the road to percent accuracy V: the non-linear power spectrum beyond $\Lambda$CDM with massive neutrinos and baryonic feedback

In the context of forthcoming galaxy surveys, to ensure unbiased constraints on cosmology and gravity when using non-linear structure information, percent-level accuracy is required when modelling the power spectrum. This calls for frameworks that can accurately capture the relevant physical effects, while allowing for deviations from $\Lambda$CDM. Massive neutrino and baryonic physics are two of the most relevant such effects. We present an integration of the halo model reaction frameworks for massive neutrinos and beyond-$\Lambda$CDM cosmologies. The integrated halo model reaction, combined with a pseudo power spectrum modelled by HMCode2020 is then compared against $N$-body simulations that include both massive neutrinos and an $f(R)$ modification to gravity. We find that the framework is 4% accurate down to at least $k\approx 3 \, h/{\rm Mpc}$ for a modification to gravity of $|f_{\rm R0}|\leq 10^{-5}$ and for the total neutrino mass $M_\nu \equiv \sum m_\nu \leq 0.15$ eV. We also find that the framework is 4% consistent with EuclidEmulator2 as well as the Bacco emulator for \B{most of the considered} $\nu w$CDM cosmologies down to at least $k \approx 3 \, h$/Mpc. Finally, we compare against hydrodynamical simulations employing HMCode2020's baryonic feedback modelling on top of the halo model reaction. For $\nu \Lambda$CDM cosmologies we find 2% accuracy for $M_\nu \leq 0.48$eV down to at least $k\approx 5h$/Mpc. Similar accuracy is found when comparing to $\nu w$CDM hydrodynamical simulations with $M_\nu = 0.06$eV. This offers the first non-linear, theoretically general means of accurately including massive neutrinos for beyond-$\Lambda$CDM cosmologies, and further suggests that baryonic, massive neutrino and dark energy physics can be reliably modelled independently.