Imprints of Dark Photons on Gravitational Wave Polarizations

We study conversion processes between gravitons and dark photons and reveal the effects of dark photons on the polarization of gravitational waves. Considering cosmological dark magnetic fields, we investigate the evolution of the intensity and polarization of gravitational waves through the conversion. Specifically, we demonstrate that for minimal coupling between gravitons and dark photons, the intensity, circular polarization, and linear polarization evolve separately. We derive explicit formulas for the statistical mean and variance of the intensity and polarization when the gravitational waves pass through magnetic fields with random orientation. The formulas capture how the initial polarization of dark photons will be imprinted on the observed gravitational wave background.