Constraints on direction-dependent cosmic birefringence from Planck
  polarization data
  Cosmic birefringence is the process that rotates the plane of polarization by
an amount, $\alpha$, as photons propagate through free space. Such an effect
arises in parity-violating extensions to the electromagnetic sector, such as
the Chern-Simons term common in axion models, quintessence models, or
Lorentz-violating extensions to the standard model. Most studies consider the
monopole of this rotation, but it is also possible for the effect to have
spatial anisotropies. Paying particular attention to large scales, we implement
a novel pixel-based method to extract the spherical harmonics for $L \le 30$
and a pseudo-$C_L$ method for $L > 30$. Our results are consistent with no
detection and we set 95% upper limits on the amplitude of a scale-invariant
power spectrum of $L(L+1)C_L/2\pi < [2.2\, (\mathrm{stat.})\, \pm 0.7\,
(\mathrm{syst.})]\times10^{-5} = [0.07\, (\mathrm{stat.}) \pm 0.02\,
(\mathrm{syst.})] \,\mathrm{deg}^2$, on par with previous constraints. This
implies specific limits on the dipole and quadrupole amplitudes to be
$\sqrt{C_1/4\pi} < 0.2^\circ$ and $\sqrt{C_2/4\pi} < 0.1^\circ$, at 95% CL,
respectively, improving previous constraints by an order of magnitude. We
further constrain a model independent $M=0$ quadrupole in an arbitrary
direction to be $\alpha_{20} = 0.02^\circ \pm 0.21^\circ$, with an
unconstrained direction. However, we find an excess of dipolar power with an
amplitude $\sqrt{3C_1/4\pi} = 0.32^\circ \pm 0.10^\circ\, (\mathrm{stat.})\,
\pm 0.08^\circ\, (\mathrm{syst.})$, in the direction $(l, b) = (295^\circ,
17^\circ) \pm (22^\circ, 17^\circ)\, (\mathrm{stat.})\, \pm (5^\circ,
16^\circ)\, (\mathrm{syst.})$, larger than 1.4% of simulations with no
birefringence. We attribute part of this signal to the contamination of
residual foregrounds not accounted for in our simulations, although this should
be further investigated.
