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 μ eV≲ m ≲ 10 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

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... . 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, ℓ_ today∼ cm - 100 km, with large density fluctuations at ℓ_ 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.