  Models in which the dark matter is produced at extremely low rates from the
annihilation of Standard Model particles in the early Universe allow us to
explain the current dark matter relic density while easily evading the
traditional experimental constraints. In scenarios where the dark matter
interacts with the Standard Model via a new physics mediator, the early
Universe dynamics of the dark sector can be particularly complex, as the dark
matter and the mediator could be in thermal and chemical equilibrium with each
other. This equilibration takes place via number-changing processes such as
double Compton scattering and bremsstrahlung, whose amplitudes are cumbersome
to calculate. In this paper, we show that in large regions of the parameter
space, these equilibration mechanisms do not significantly affect the final
dark matter relic density. In particular, for a model with a light dark photon
mediator, the relic density can be reasonably estimated by considering that the
dark matter is solely produced through the annihilation of Standard Model
particles. This result considerably simplifies the treatment of a large class
of dark matter theories, facilitating in particular the superimposition of the
relic density constraints on the current and future experimental bounds.
