Implications of Gravitational-wave Production from Dark Photon Resonance to Pulsar-timing Observations and Effective Number of Relativistic Species

The coherent oscillation of axionic fields naturally drives copious production of dark photon particles in the early universe, due to resonance and tachyonic enhancement. During the process, energy is abruptly transferred from the former to the latter, sourcing gravitational wave generation. The resulting gravitational waves are eventually to be observed as stochastic background today. We report analytical results of this production and connect them to the recent pulsar-timing results by the NANOGrav collaboration. We show an available parameter space, around the mass $m_\phi \sim 10^{-13} \, {\rm eV}$ and the decay constant $f_\phi \sim 10^{16} \, {\rm GeV}$ with a dimensionless coupling of ${\cal O}(1)$, for our mechanism to account for the signal. A mechanism to avoid the axion over-dominating the universe is a necessary ingredient of this model, and we discuss a possibility to recover a symmetry and render the axion massless after the production. We also comment on potential implications of the required effective number of relativistic species to the determination of the present Hubble constant.