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

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.

24 pages, 13 figures. ApJ accepted

References in corpus (2)

• E/B decomposition of finite pixelized CMB maps
• A general solution to the E-B mixing problem

Cited by in corpus (12)

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• Lensing Simulation and Power Spectrum Estimation for High Resolution CMB Polarization Maps
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• Probing the statistical properties of CMB $B$-mode polarization through Minkowski Functionals
• Wiener filtering and pure E/B decomposition of CMB maps with anisotropic correlated noise
• Pure E and B polarization maps via Wiener filtering
• Optimal and fast E/B separation with a dual messenger field
• Fast scalar quadratic maximum likelihood estimators for the CMB B-mode power spectrum
• Scalar quadratic maximum likelihood estimators for the CMB cross power spectrum
• Statistical imprints of CMB B-type polarization leakage in an incomplete sky survey analysis
• Smoothing methods comparison for CMB E- and B-mode separation
• Fisher Matrix Optimization of Cosmic Microwave Background Interferometers