  We study in detail a recently proposed mechanism for producing non-thermal
dark photon dark matter at the end of inflation in the mass range $\mu\,{\rm
eV} \lesssim m \lesssim 10\,{\rm TeV}$. A tachyonic instability induced by a
rolling inflaton leads to the coherent production of dark (abelian) gauge
bosons with a peak in the power spectrum corresponding to the Hubble scale at
the end of inflation. As the Universe expands after inflation the dark photons
redshift and, at some point in their cosmic evolution, they obtain a mass. We
focus in particular on the case where the dark photons are relativistic at the
time their mass is generated and examine the associated cosmic evolution to
compute the relic abundance today. We also examine the late time power spectrum
demonstrating explicitly that it preserves the peak generated at the end of
inflation. We show that the peak corresponds to small physical scales today,
$\ell_{\rm today} \sim {\rm cm} - 100\,{\rm km}$, with large density
fluctuations at $\ell_{\rm today}$ leading to a clumpy nature for the dark
photon dark matter. We also discuss potential phenomenology and future
directions, briefly commenting on the non-relativistic case.
