  In this work, we present a lattice study of an axion - dark photon system in
the early Universe and show that the stochastic gravitational wave (GW)
background produced by this system may be probed by future GW experiments
across a vast range of frequencies. The numerical simulation on the lattice
allows us to take into account non-linear backreaction effects and enables us
to accurately predict the final relic abundance of the axion or axion-like
particle (ALP) as well as its inhomogeneities, and gives a more precise
prediction of the GW spectrum. Importantly, we find that the GW spectrum has
more power at high momenta due to $2\rightarrow1$ processes. Furthermore, we
find the degree of polarization of the peak of the GW spectrum depends on the
ALP-dark photon coupling and that the polarization can be washed out or even
flipped for large values thereof. In line with recent results in the
literature, we find the ALP relic abundance may be suppressed by two orders of
magnitude and discuss possible extensions of the model that expand the viable
parameter space. Finally, we discuss the possibility to probe ultralight ALP
dark matter via spectral distortions of the CMB.
