Analysis of CMB polarization on an incomplete sky

TL;DR

The paper tackles the problem of extracting the magnetic polarization $B$ of the CMB from partial-sky data, where the $E$/$B$ decomposition is non-local and susceptible to leakage from the much stronger $E$ signal. It develops a harmonic-space, spin-weighted window-function framework to achieve (primarily exact for azimuthally symmetric patches) separation of $E$ and $B$, and to construct a finite, non-redundant set of observables $E_W$ and $B_W$ that preserve information while minimizing leakage. A practical algorithm using singular-value decompositions of the coupling matrices and boundary-term elimination yields clean $B$-mode estimators with well-behaved, diagonal noise properties under isotropic, uncorrelated noise, enabling robust Planck-scale magnetic signal estimation and meaningful assessment of tensor-mode sensitivity with lensing contamination. The approach provides lossless or near-lossless estimators for polarization power spectra on partial skies and offers a principled path toward robustly detecting primordial gravitational waves via $B$-modes in future CMB experiments.

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. Our construction is exact for azimuthally symmetric patches, but should continue to perform well for arbitrary 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. For isotropic, uncorrelated noise the variables have a very simple diagonal noise correlation, and further analysis using them should be no harder than analysing 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 allowing us to estimate the magnetic signal that could be detected by the Planck satellite in the absence of non-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 analysing the electric polarization field in the case that the magnetic component is negligible.

Figures (2)

• Figure 1: The window functions $W_{+(l'm)(lm)}$ (solid lines) and $W_{-(l'm)(lm)}$ (dashed lines) for $l'=100$ and various $m$ for an azimuthally symmetric patch with $\theta < 10^\circ$. The dashed lines show the $E_{lm}$ contamination of $\tilde{B}_{l'm}$ as a function of $l$. For $m=0$ there is no contamination, and as $m$ increases the functions decrease in amplitude as the corresponding harmonics become more localized outside of the patch.
• Figure 2: The distribution of eigenvalues of $\bm{W}_+$ for two azimuthally-symmetric sky cuts with $l_{\text{max}}= \{250,1000\}$. The distribution is approximately bimodal, and the fraction of the eigenvalues corresponding to well-determined modes (eigenvalue significantly non-zero) is given by the fraction of the sky area in the patch in the limit $l_{\text{max}}\rightarrow\infty$.