Cosmology With a Very Light $L_\mu - L_\tau$ Gauge Boson

In this paper, we explore in detail the cosmological implications of an abelian $L_\mu-L_\tau$ gauge extension of the Standard Model featuring a light and weakly coupled $Z'$. Such a scenario is motivated by the longstanding $\sim \, 4 \sigma$ discrepancy between the measured and predicted values of the muon's anomalous magnetic moment, $(g-2)_\mu$, as well as the tension between late and early time determinations of the Hubble constant. If sufficiently light, the $Z'$ population will decay to neutrinos, increasing the overall energy density of radiation and altering the expansion history of the early universe. We identify two distinct regions of parameter space in this model in which the Hubble tension can be significantly relaxed. The first of these is the previously identified region in which a $\sim \, 10-20$ MeV $Z'$ reaches equilibrium in the early universe and then decays, heating the neutrino population and delaying the process of neutrino decoupling. For a coupling of $g_{\mu-\tau} \simeq (3-8) \times 10^{-4}$, such a particle can also explain the observed $(g-2)_{\mu}$ anomaly. In the second region, the $Z'$ is very light ($m_{Z'} \sim 1\,\text{eV}$ to $\text{MeV}$) and very weakly coupled ($g_{\mu-\tau} \sim 10^{-13}$ to $10^{-9}$). In this case, the $Z'$ population is produced through freeze-in, and decays to neutrinos after neutrino decoupling. Across large regions of parameter space, we predict a contribution to the energy density of radiation that can appreciably relax the reported Hubble tension, $\Delta N_{\rm eff} \simeq 0.2$.