Redshift evolution of cosmic birefringence in CMB anisotropies

We study the imprints of a cosmological redshift-dependent pseudoscalar field $\phi$ on the rotation of cosmic microwave background (CMB) linear polarization generated by a coupling $ \phi F^{\mu\nu} \tilde F_{\mu \nu}$. We show how either phenomenological or theoretically motivated redshift dependence of the pseudoscalar field, such as those in models of Early Dark Energy, Quintessence or axion-like dark matter, lead to CMB polarization and temperature-polarization power spectra which exhibit a multipole dependence which goes beyond the widely adopted approximation in which the redshift dependence of the linear polarization angle is neglected. Because of this multipole dependence, the isotropic birefringence effect due to a general coupling $\phi F^{\mu\nu} \tilde F_{\mu \nu}$ is not degenerate with a systematic calibration angle uncertainty. By taking this multipole dependence into account, we calculate the parameters of these phenomenological and theoretical redshift dependence of the pseudoscalar field which can be detected by future CMB polarization experiments on the basis of a $\chi^2$ analysis for a Wishart likelihood. As a final example of our approach, we compute by Markov Chain MonteCarlo (MCMC) the minimal coupling $g_\phi$ in Early Dark Energy which could be detected by future experiments, with or without marginalizing on a systematic rotation angle uncertainty.