Limits on Polarized Dust Spectral Index Variations for CMB Foreground
  Analysis
  Using Planck polarization data, we search for and constrain spatial
variations of the polarized dust foreground for cosmic microwave background
(CMB) observations, specifically in its spectral index, $\beta_d$. Failure to
account for such variations will cause errors in the foreground cleaning that
propagate into errors on cosmological parameter recovery from the cleaned CMB
map. It is unclear how robust prior studies of the Planck data which
constrained $\beta_d$ variations are due to challenges with noise modeling,
residual systematics, and priors. To clarify constraints on $\beta_d$ and its
variation, we employ two pixel space analyses of the polarized dust foreground
at $>3.7^{\circ}$ scales on $\approx 60\%$ of the sky at high Galactic
latitudes. A template fitting method, which measures $\beta_d$ over three
regions of $\approx 20\%$ of the sky, does not find significant deviations from
an uniform $\beta_d = 1.55$, consistent with prior Planck determinations. An
additional analysis in these regions, based on multifrequency fits to a dust
and CMB model per pixel, puts limits on $\sigma_{\beta_d}$, the Gaussian
spatial variation in $\beta_d$. At the highest latitudes, the data support
$\sigma_{\beta_d}$ up to $0.45$, $0.30$ at mid-latitudes, and $0.15$ at
low-latitudes. We also demonstrate that care must be taken when interpreting
the current Planck constraints, $\beta_d$ maps, and noise simulations. Due to
residual systematics and low dust signal to noise at high latitudes, forecasts
for ongoing and future missions should include the possibility of large values
of $\sigma_{\beta_d}$ as estimated in this paper, based on current polarization
data.
