Analysis of CMB polarization on an incomplete sky

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Abstract
The full sky cosmic microwave background polarization field can be decomposed into “electric” and “magnetic” components. Working in harmonic space we construct window functions that allow clean separation of the electric and magnetic modes from observations over only a portion of the sky. We explicitly demonstrate the method for azimuthally symmetric patches, but also present it in a form in principle applicable to arbitrarily shaped patches. From the window functions we obtain variables that allow for robust estimation of the magnetic component without risk of contamination from the probably much larger electric signal. The variables have a very simple noise properties, and further analysis using them should be no harder than analyzing the temperature field. For an azimuthally symmetric patch, such as that obtained from survey missions when the galactic region is removed, the exactly separated variables are fast to compute. We estimate the magnetic signal that could be detected by the Planck satellite in the absence of extra-galactic foregrounds. We also discuss the sensitivity of future experiments to tensor modes in the presence of a magnetic signal generated by weak lensing, and give lossless methods for analyzing the electric polarization field in the case that the magnetic component is negligible. A series of Appendixes review the spin weight formalism and give recursion relations for fast computation of the spin-weighted spherical harmonics and their inner products over azimuthally symmetric patches of the sphere. A further Appendix discusses the statistics of weak signal detection.