de Broglie-Proca and Bopp-Podolsky massive photon gases in cosmology

We investigate the influence of massive photons on the evolution of the expanding universe. Two particular models for generalized electrodynamics are considered, namely de Broglie-Proca and Bopp-Podolsky electrodynamics. We obtain the equation of state (EOS) \(P=P(\varepsilon)\) for each case using dispersion relations derived from both theories. The EOS are inputted into the Friedmann equations of a homogeneous and isotropic space-time to determine the cosmic scale factor \(a(t)\). It is shown that the photon non-null mass does not significantly alter the result \(a\propto t^{1/2}\) valid for a massless photon gas; this is true either in de Broglie-Proca's case (where the photon mass \(m\) is extremely small) or in Bopp-Podolsky theory (for which \(m\) is extremely large).