  We consider the effect on early Universe cosmology of the dark photon
associated with the gauging of $U(1)_{T3R}$, a symmetry group under which only
right-handed Standard Model fermions transform non-trivially. We find that
cosmological constraints on this scenario are qualitatively much more severe
than on other well-studied cases of a new $U(1)$ gauge group, because the dark
photon couples to chiral fermions. In particular, the dark photon of
$U(1)_{T3R}$ is always produced and equilibrates in the early Universe, no
matter how small the gauge coupling, unless the symmetry-breaking scale is
extremely large. This occurs because, no matter how the weak the coupling, the
Goldstone mode (equivalently, the longitudinal polarization) does not decouple.
As a result, even the limit of an extremely light and weakly-coupled dark
photon of $U(1)_{T3R}$ is effectively ruled out by cosmological constraints,
unless the symmetry-breaking scale is extremely large. We also discuss the
possibility of ameliorating Hubble tension in this model.
