A wavelet-Galerkin algorithm of the E/B decomposition of CMB polarization maps

TL;DR

This paper introduces a wavelet-Galerkin algorithm for accurately separating E and B modes in CMB polarization maps, effectively handling noisy, discretized data and boundary effects, and demonstrating reliable power spectrum recovery on larger scales.

Contribution

The paper presents a novel wavelet-Galerkin discretization method for E/B decomposition that reduces boundary errors and manages noise, improving the accuracy of power spectrum estimation.

Findings

Effective recovery of E and B power spectra on scales larger than four times the finest scale.

The method maintains accuracy even when E-to-B power ratio is as high as 100.

Contamination from noise and mode mixing can be controlled with the proposed approach.

Abstract

We develop an algorithm of separating the $E$ and $B$ modes of the CMB polarization from the noisy and discretized maps of Stokes parameter $Q$ and $U$ in a finite area. A key step of the algorithm is to take a wavelet-Galerkin discretization of the differential relation between the $E$, $B$ and $Q$, $U$ fields. This discretization allows derivative operator to be represented by a matrix, which is exactly diagonal in scale space, and narrowly banded in spatial space. We show that the effect of boundary can be eliminated by dropping a few DWT modes located on or nearby the boundary. This method reveals that the derivative operators will cause large errors in the $E$ and $B$ power spectra on small scales if the $Q$ and $U$ maps contain Gaussian noise. It also reveals that if the $Q$ and $U$ maps are random, these fields lead to the mixing of the $E$ and $B$ modes. Consequently, the $B$ mode will be contaminated if the powers of $E$ modes are much larger than that of $B$ modes. Nevertheless, numerical tests show that the power spectra of both $E$ and $B$ on scales larger than the finest scale by a factor of 4 and higher can reasonably be recovered, even when the power ratio of $E$- to $B$-modes is as large as about 10$^2$, and the signal-to-noise ratio is equal to 10 and higher. This is because the Galerkin discretization is free of false correlations, and keeps the contamination under control. As wavelet variables contain information of both spatial and scale spaces, the developed method is also effective to recover the spatial structures of the $E$ and $B$ mode fields.