Cosmological Constraint on Dark Photon from $N_{\rm eff}$

A new U(1) gauge symmetry is the simplest extension of the Standard Model and has various theoretical and phenomenological motivations. In this paper, we study the cosmological constraint on the MeV scale dark photon. After the neutrino decoupling era at $T = \mathcal{O}(1)\,$MeV, the decay and annihilation of the dark photon heats up the electron and photon plasma and accordingly decreases the effective number of neutrino $N_{\mathrm{eff}}$ in the recombination era. We derive a conservative lower-limit of the dark photon mass around 8.5 MeV from the current Planck data if the mixing between the dark photon and ordinary photon is larger than $\mathcal{O}(10^{-9})$. We also find that the future CMB stage-$\rm I\! V$ experiments can probe up to 17 MeV dark photon.