  As experimental searches for WIMP dark matter continue to yield null results,
models beyond the WIMP paradigm have proliferated in order to elude ever
improving observational constraints, among them that of sub-GeV dark matter
mediated by a massive vector portal (a dark photon) associated with a new dark
$U(1)$ gauge symmetry. It has been previously noted that for a significant
range of the parameter space of this class of models, the annihilation of dark
matter particles into a pair of dark photons can dominate the freeze-out
process even when this process is kinematically forbidden for dark matter at
rest -- this is known as the "forbidden dark matter" (FDM) regime. Prior
studies of this regime, however, assume that any "dark Higgs" associated with
breaking the dark $U(1)$ and imparting mass to the dark photon is decoupled
from the dark matter and as such plays no role in the freeze-out process. In
this paper, we explore the effects of a dark Higgs on sub-GeV dark matter
phenomenology in this FDM regime by considering the simplest possible
construction in which there exist non-trivial dark matter-dark Higgs couplings:
a model with a single complex scalar DM candidate coupled directly to the dark
Higgs field. We find that for a wide range of parameter space, the dark Higgs
can alter the resulting relic abundance by many orders of magnitude, and that
this effect can remain significant even for a small dark matter-dark Higgs
coupling constant. Considering measurements from direct detection and
measurements of the CMB, we further find that points in this model's parameter
space which recreate the appropriate dark matter relic abundance suffer only
mild constraints from other sources at present, but may become accessible in
near-future direct detection experiments.
