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.
