How to make a clean separation between CMB E and B modes with proper foreground masking

TL;DR

This paper presents a method for optimal foreground masking in CMB polarization analysis, significantly reducing E/B mode leakage and Gibbs phenomenon, thereby enhancing the detection of primordial gravitational waves.

Contribution

The work introduces an optimal foreground mask that suppresses Gibbs phenomenon while maintaining proper masking, improving E/B separation in CMB polarization data.

Findings

Leakage power reduced by factors of 10^6 to 10^9

Leakage is lower than the B mode power spectrum for r~10^-7

Zero-value pixels are preserved in the new mask

Abstract

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.