Delta-map method to remove CMB foregrounds with spatially varying
  spectra
  We extend the internal template foreground removal method by accounting for
spatially varying spectral parameters such as the spectral indices of
synchrotron and dust emission and the dust temperature. As the previous
algorithm had to assume that the spectral parameters are uniform over the full
sky (or some significant fraction of the sky), it resulted in a bias in the
tensor-to-scalar ratio parameter $r$ estimated from foreground-cleaned
polarization maps of the cosmic microwave background (CMB). The new algorithm,
"Delta-map method", accounts for spatially varying spectra to first order in
perturbation. The only free parameters are the cosmological parameters such as
$r$ and the sky-averaged foreground parameters. We show that a cleaned CMB map
is the maximum likelihood solution to first order in perturbation, and derive
the posterior distribution of $r$ and the sky-averaged foreground parameters
using Bayesian statistics. Applying this to realistic simulated sky maps
including polarized CMB, synchrotron and thermal dust emission, we find that
the new algorithm removes the bias in $r$ down to undetected level in our
noiseless simulation ($r\lesssim 4\times 10^{-4}$). We show that the frequency
decorrelation of polarized foregrounds due to averaging of spatially varying
spectra can be accounted for to first order in perturbation by using slightly
different spectral parameters for the Stokes $Q$ and $U$ maps. Finally, we show
that the effect of polarized anomalous microwave emission on foreground removal
can be absorbed into the curvature parameter of the synchrotron spectrum.
