E/B decomposition of finite pixelized CMB maps

Separation of the E and B components of a microwave background polarization map or a weak lensing map is an essential step in extracting science from it, but when the map covers only part of the sky and/or is pixelized, this decomposition cannot be done perfectly. We present a method for decomposing an arbitrary sky map into a sum of three orthogonal components that we term ``pure E,'' ``pure B,'' and ``ambiguous.'' The fluctuations in the pure E and B maps are due only to the E and B power spectra, respectively, whereas the source of those in the ambiguous map is completely indeterminate. This method is useful both for providing intuition for experimental design and for analyzing data sets in practice. We show how to find orthonormal bases for all three components in terms of bi-Laplacian eigenfunctions, thus providing a type of polarized signal-to-noise eigenmodes that simultaneously separate both angular scale and polarization type. The number of pure and ambiguous modes probing a characteristic angular scale \ensuremath{\theta} scales as the map area over ${\ensuremath{\theta}}^{2}$ and as the map boundary length over \ensuremath{\theta}, respectively. This implies that fairly round maps (with short perimeters for a given area) will yield the most efficient $E/B$ decomposition and also that the fraction of the information lost to ambiguous modes grows towards larger angular scales. For real-world data analysis, we present a simple matrix eigenvalue method for calculating nearly pure E and B modes in pixelized maps. We find that the dominant source of leakage between E and B is aliasing of small-scale power caused by the pixelization, essentially since derivatives are involved. This problem can be eliminated by heavily oversampling the map, but is exacerbated by the fact that the E power spectrum is expected to be much larger than the B power spectrum and by the extremely blue power spectrum that cosmic microwave background polarization is expected to have. We found that a factor of 2 to 3 more pixels are needed in a polarization map to achieve the same level of contamination by aliased power than in a temperature map. Oversampling is therefore much more important for the polarized case than for the unpolarized case, which should be reflected in experimental design.