  Dark matter is poorly constrained by direct detection experiments at masses
below 1 MeV. This is an important target for the next generation of
experiments, and several methods have been proposed to probe this mass range.
One class of such experiments will search for dark matter--electron recoils.
However, simplified models with new light degrees of freedom coupled to
electrons face significant pressure from cosmology, and the extent of these
restrictions more generally is poorly understood. Here, we perform a systematic
study of cosmological constraints on models with a heavy mediator in the
context of an effective field theory. We include constraints from (i)
disruption of primordial nucleosynthesis, (ii) overproduction of dark matter,
and (iii) the effective number of neutrino species at recombination. We
demonstrate the implications of our results for proposed electron recoil
experiments, and highlight scenarios which may be amenable to direct detection.
