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Title:
A Wavelet-Galerkin Algorithm of the E/B Decomposition of Cosmic Microwave Background Polarization Maps
Authors:
Cao, Liang; Fang, Li-Zhi
Affiliation:
AA(Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China ; Graduate University of the Chinese Academy of Science, 19A Yuquan Road, Beijing 100049, China ), AB(Department of Physics, University of Arizona, Tucson, AZ 85721, USA)
Publication:
The Astrophysical Journal, Volume 706, Issue 2, pp. 1545-1555 (2009). (ApJ Homepage)
Publication Date:
12/2009
Origin:
IOP
ApJ Keywords:
cosmology: theory, large-scale structure of universe
DOI:
10.1088/0004-637X/706/2/1545
Bibliographic Code:
2009ApJ...706.1545C

Abstract

We develop an algorithm of separating the E and B modes of the cosmic microwave background (CMB) polarization from the noisy and discretized maps of Stokes parameters 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 discrete wavelet transform 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 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 102, 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.
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