Towards a full description of MeV dark matter decoupling: a self-consistent determination of relic abundance and $N_{\rm eff}$

Thermal dark matter at the MeV mass-scale has its abundance set during the highly non-trivial epochs of neutrino decoupling and electron annihilation. The technical obstacles attached to solving Boltzmann equations of multiple interacting sectors being both relativistic and non-relativistic have to-date prevented the full treatment of this problem. Here, for the first time, we calculate the freeze-out of light dark matter, taking into account the energy transfer between the dark sector, neutrinos, and the electromagnetically interacting plasma from annihilation and elastic scattering processes alike. We develop a numerically feasible treatment that allows to track photon and neutrino temperatures across freeze-out and to arrive at a precision prediction of $N_{\rm eff}$ for arbitrary branching ratios of the dark matter annihilation channels. In addition, our treatment resolves for the first time the dark matter temperature evolution across freeze-out involving three sectors. It enters in the efficiency of velocity-dependent annihilation channels and for a flavor-blind $p$-wave annihilation into electron- and neutrino-pairs of all generations, we find the present Planck data excludes a complex scalar dark matter particle of mass of $m_\phi \leq 7$ MeV.