Revisiting Dark Photon Constraints from CMB Spectral Distortions

Spectral distortions of the cosmic microwave background (CMB) provide stringent constraints on energy and entropy production in the post-BBN (Big Bang Nucleosynthesis) era. This has been used to constrain dark photon models with COBE/FIRAS and forecast the potential gains with future CMB spectrometers. Here, we revisit these constraints by carefully considering the photon to dark photon conversion process and evolution of the distortion signal. Previous works only included the effect of CMB energy density changes but neglected the change to the photon number density. We clearly define the dark photon distortion signal and show that in contrast to previous analytic estimates the distortion has an opposite sign and a $\simeq 1.5$ times larger amplitude. We furthermore extend the treatment into the large distortion regime to also cover the redshift range $\simeq 2\times 10^6-4\times 10^7$ between the $\mu$-era and the end of BBN using CosmoTherm. This shows that the CMB distortion constraints for dark photon masses in the range $10^{-4}\,{\rm eV}\lesssim m_{\rm dp}\lesssim 10^{-3}\,{\rm eV}$ were significantly underestimated. We demonstrate that in the small distortion regime the distortion caused by photon to dark photon conversion is extremely close to a $\mu$-type distortion independent of the conversion redshift. This opens the possibility to study dark photon models using CMB distortion anisotropies and the correlations with CMB temperature anisotropies as we highlight here.