Frequency-Dependent Constraints on Cosmic Birefringence from the LFI and
  HFI Planck Data Release 4
  We present new constraints on the frequency dependence of the cosmic
birefringence angle from the Planck data release 4 polarization maps. An axion
field coupled to electromagnetism predicts a nearly frequency-independent
birefringence angle, $\beta_\nu = \beta$, while Faraday rotation from local
magnetic fields and Lorentz violating theories predict a cosmic birefringence
angle that is proportional to the frequency, $\nu$, to the power of some
integer $n$, $\beta_\nu \propto \nu^n$. In this work, we first sample
$\beta_\nu$ individually for each polarized HFI frequency band in addition to
the 70 GHz channel from the LFI. We also constrain a power-law formula for the
birefringence angle, $\beta_\nu=\beta_0(\nu/\nu_0)^n$, with $\nu_0 = 150$ GHz.
For a nearly full-sky measurement, $f_{\text{sky}}=0.93$, we find $\beta_0 =
0.26^{\circ}\pm0.11^\circ$ $(68\% \text{ C.L.})$ and $n=-0.45^{+0.61}_{-0.82}$
when we ignore the intrinsic $EB$ correlations of the polarized foreground
emission, and $\beta_0 = 0.33^\circ \pm 0.12^\circ$ and
$n=-0.37^{+0.49}_{-0.64}$ when we use a filamentary dust model for the
foreground $EB$. Next, we use all the polarized Planck maps, including the 30
and 44 GHz frequency bands. These bands have a negligible foreground
contribution from polarized dust emission. We, therefore, treat them
separately. Without any modeling of the intrinsic $EB$ of the foreground, we
generally find that the inclusion of the 30 and 44 GHz frequency bands raises
the measured values of $\beta_\nu$ and tightens $n$. At nearly full-sky, we
measure $\beta_0=0.29^{\circ+0.10^\circ}_{\phantom{\circ}-0.11^\circ}$ and
$n=-0.35^{+0.48}_{-0.47}$. Assuming no frequency dependence, we measure
$\beta=0.33^\circ \pm 0.10^\circ$. If our measurements have effectively
mitigated the $EB$ of the foreground, our constraints are consistent with a
mostly frequency-independent signal of cosmic birefringence.
