Planck 2018 constraints on anisotropic birefringence and its cross-correlation with CMB anisotropy

Parity-violating extensions of standard electromagnetism produce cosmic birefringence, the in vacuo rotation of the linear polarisation direction of a photon during propagation. We employ {it Planck} 2018 CMB polarised data to constrain anisotropic birefringence, modeled by its angular power spectrum $C_{ell}^{alpha alpha}$, and the cross-correlation with CMB temperature maps, $C_{ell}^{alpha T}$, at scales larger than $sim$15 degrees. We present joint limits on the scale invariant quantity, $A^{alpha alpha} equiv ell (ell +1) , C_{ell}^{alpha alpha} / 2 pi$, and on the analogous amplitude for the cross-correlation, $A^{alpha T} equiv ell (ell +1) , C_{ell}^{alpha T} / 2 pi$. We find no evidence of birefringence within the error budget and obtain $A^{alpha alpha} < 0.104 , mbox{[deg$^2$]}$ and $A^{alpha T}=1.50^{+2.41}_{-4.10} , mbox{[$mu$K$cdot$deg] both at } 95 % mbox{ C.L.}$. The latter bound appears competitive in constraining a few early dark energy models recently proposed to alleviate the $H_{0}$ tension. Slicing the joint likelihood at $A^{alpha T}=0$, the bound on $A^{alpha alpha}$ becomes tighter at $A^{alpha alpha} < 0.085 , mbox{[deg$^2$]}$ at 95$% mbox{ C.L.}$. In addition we recast the constraints on $A^{alpha alpha}$ as a bound on the amplitude of primordial magnetic fields responsible for Faraday rotation, finding $B_{1 {tiny mbox{Mpc}}} < 26.9$ nG and $B_{1 {tiny mbox{Mpc}}} < 24.3$ nG at 95$%$ C.L. for the marginalised and sliced case respectively.