How to make a clean separation between CMB E and B modes with proper
  foreground masking
  We investigate the E/B decomposition of CMB polarization on a masked sky. In
real space, operators of E and B mode decomposition involve only differentials
of CMB polarization. We may, therefore in principle, perform a clean E/B
decomposition from incomplete sky data. Since it is impractical to apply second
derivatives to observation data, we usually rely on spherical harmonic
transformation and inverse transformation, instead of using real-space
operators. In spherical harmonic representation, jump discontinuities in a cut
sky produces Gibbs phenomenon, unless a spherical harmonic expansion is made up
to an infinitely high multipole. By smoothing a foreground mask, we may
suppress the Gibbs phenomenon effectively in a similar manner to apodization of
a foreground mask discussed in other works. However, we incur foreground
contamination by smoothing a foreground mask, because zero-value pixels in the
original mask may be rendered non-zero by the smoothing process. In this work,
we investigate an optimal foreground mask, which ensures proper foreground
masking and suppresses Gibbs phenomenon. We apply our method to a simulated map
of the pixel resolution comparable to the Planck satellite. The simulation
shows that the leakage power is lower than unlensed CMB B mode power spectrum
of tensor-to-scalar ratio $r\sim 1\times10^{-7}$. We compare the result with
that of the original mask. We find that the leakage power is reduced by a
factor of $10^{6} \sim 10^{9}$ at the cost of a sky fraction $0.07$, and that
the enhancement is highest at lowest multipoles. We confirm that all the
zero-value pixels in the original mask remain zero in our mask. The application
of this method to the Planck data will improve the detectability of primordial
tensor perturbation.
