  Axion-like fields heavier than about $10^{-27}$eV are expected to oscillate
in the radiation dominated epoch when the Hubble parameter drops below their
mass. Considering the Chern-Simons coupling with a dark gauge boson, large
amount of dark photons are produced during a short time interval through
tachyonic resonance instability. The produced dark photons then source
gravitational tensor modes leading to chiral gravitational waves. Through this
process, one can indirectly probe a large parameter space of coupled axion-dark
photon models. In this work we first find an analytic expression for the number
density of the dark photons produced during the tachyonic resonance regime.
Second, by using the saddle point approximation we find an analytic expression
for the gravitational wave spectrum in terms of the mass, coupling and
misalignment angle. Our analytic results can be used for the observational
analysis of these types of scenarios.
