  A dark photon may kinetically mix with the ordinary photon, inducing
oscillations with observable imprints on cosmology. Oscillations are resonantly
enhanced if the dark photon mass equals the ordinary photon plasma mass, which
tracks the free electron number density. Previous studies have assumed a
homogeneous Universe; in this Letter, we introduce for the first time an
analytic formalism for treating resonant oscillations in the presence of
inhomogeneities of the photon plasma mass. We apply our formalism to determine
constraints from Cosmic Microwave Background photons oscillating into dark
photons, and from heating of the primordial plasma due to dark photon dark
matter converting into low-energy photons. Including the effect of
inhomogeneities demonstrates that prior homogeneous constraints are not
conservative, and simultaneously extends current experimental limits into a
vast new parameter space.
