Cosmic birefringence tomography with polarized Sunyaev Zel'dovich effect

We consider the polarized Sunyaev-Zel'dovich (pSZ) effect for a tomographic probe of cosmic birefringence, including all relevant terms of the pSZ effect in the cosmic microwave background (CMB) observables, some of which were ignored in the previous works. The pSZ effect produces late-time polarization signals from the scattering of the local temperature quadrupole seen by an electron. We forecast the expected constraints on cosmic birefringence at the late time of the universe with the pSZ effect. We find that the birefringence angles at $2\lesssim z\lesssim 5$ are constrained at a subdegree level by the cross-correlations between CMB $E$- and $B$-modes or between CMB $B$-modes and remote quadrupole $E$-modes using data from LiteBIRD, CMB-S4, and LSST. In particular, the cross-correlation between large-scale CMB $B$-modes and remote-quadrupole $E$-modes has a much smaller bias from the Galactic foregrounds and is useful to cross-check the results from the $EB$ power spectrum.