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

Abstract: Cosmic Birefringence (CB) is a phenomenon in which the polarization of the Cosmic Microwave Background (CMB) radiation is rotated as it travels through space due to the coupling between photons and an axion-like field. We look for a solution able to explain the result obtained from the \textit{Planck} Public Release 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 present in the literature, which need a non-negligible evolution in time of the axion-like field during recombination, we find a new region of the parameter space which allows for a nearly constant time evolution of such a field in the same epoch. The latter reinforces the possibility to employ the commonly used relations connecting the observed CMB spectra with the unrotated ones, through trigonometric functions of the CB angle. However, if the homogeneous axion field sourcing isotropic birefringence is almost constant in time during the matter-dominated era, this does not automatically implies that the same holds true also for the associated inhomogeneous perturbations. For this reason, in this paper we present a full generalized Boltzmann treatment of this phenomenon, that is able, for the first time to our knowledge to deal with the time evolution of anisotropic cosmic birefringence (ACB). We employ this approach to provide predictions of ACB, in particular for the set of best-fit parameters found in the new solution of the isotropic case. If the latter is the 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 the CMB $T$ and $E$ fields, depending on the angular scale.