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 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 $A^{\alpha \alpha }<0.104[deg$^2$]$ and $A^{\alpha T}={1.50}_{-4.10}^{+2.41}[$/mu$K$/cdot$deg]bothat95\%C.L.$. 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 $A^{\alpha \alpha }<0.085[deg$^2$]$ at 95$\%C.L.$. 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 $B_{1Mpc}<26.9$ nG and $B_{1Mpc}<24.3$ nG at 95$\%$ C.L. for the marginalised and sliced case respectively.