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Cosmic Birefringence (CB) is a phenomenon in which the polarization of theCosmic Microwave Background (CMB) radiation is rotated as it travels throughspace due to the coupling between photons and an axion-like field. We look fora solution able to explain the result obtained from the \textit{Planck} PublicRelease 4 (PR4), which has provided a hint of detection of the CB angle,$\alpha=(0.30\pm0.11)^{\circ}$. In addition to the solutions, already presentin the literature, which need a non-negligible evolution in time of theaxion-like field during recombination, we find a new region of the parameterspace which allows for a nearly constant time evolution of such a field in thesame epoch. The latter reinforces the possibility to employ the commonly usedrelations connecting the observed CMB spectra with the unrotated ones, throughtrigonometric functions of the CB angle. However, if the homogeneous axionfield sourcing isotropic birefringence is almost constant in time during thematter-dominated era, this does not automatically implies that the same holdstrue also for the associated inhomogeneous perturbations. For this reason, inthis paper we present a full generalized Boltzmann treatment of thisphenomenon, that is able, for the first time to our knowledge to deal with thetime evolution of anisotropic cosmic birefringence (ACB). We employ thisapproach to provide predictions of ACB, in particular for the set of best-fitparameters found in the new solution of the isotropic case. If the latter isthe correct model, we expect an ACB spectrum of the order of$(10^{-15}\div10^{-32})$ deg$^2$ for the auto-correlation, and$(10^{-7}\div10^{-17})$ $\mu $K$\cdot\,$deg for the cross-correlations with theCMB $T$ and $E$ fields, depending on the angular scale.

A New Solution for the Observed Isotropic Cosmic Birefringence Angle and  its Implications for the Anisotropic Counterpart through a Boltzmann Approach