A dark photon may kinetically mix with the Standard Model photon, leading to observable cosmological signatures. The mixing is resonantly enhanced when the dark photon mass matches the primordial plasma frequency, which depends sensitively on the underlying spatial distribution of electrons. Crucially, inhomogeneities in this distribution can have a significant impact on the nature of resonant conversions. We develop and describe, for the first time, a general analytic formalism to treat resonant oscillations in the presence of inhomogeneities. Our formalism follows from the theory of level crossings of random fields and only requires knowledge of the one-point probability distribution function (PDF) of the underlying electron number density fluctuations. We validate our formalism using simulations and illustrate the photon-to-dark photon conversion probability for several different choices of PDFs that are used to characterize the low-redshift Universe.

22+4 pages, 17 figures; v2, clarified comparison with previous work, minor improvements to the text, results and conclusions unchanged