An All-Sky Analysis of Polarization in the Microwave Background

TL;DR

The paper develops an all-sky polarization formalism for the CMB using spin-weighted spherical harmonics, enabling a rotationally invariant decomposition into E- and B-mode components. It derives exact line-of-sight integral expressions for scalar and tensor perturbations and shows that the full polarization statistics require four power spectra: $C_{Tl}$, $C_{El}$, $C_{Bl}$, and $C_{Cl}$, with $B$-mode parity preventing cross-correlation with $T$ and $E$. Scalar modes contribute only to $E$ (with $B^{(S)}=0$), while tensor modes generate all four spectra; the work also outlines all-sky map generation, optimal estimators, and variance calculations. These results provide a robust framework for interpreting future all-sky polarization data, improving constraints on reionization and primordial gravitational waves through exact, large-scale predictions.

Abstract

Using the formalism of spin-weighted functions we present an all-sky analysis of polarization in the Cosmic Microwave Background (CMB). Linear polarization is a second-rank symmetric and traceless tensor, which can be decomposed on a sphere into spin $\pm 2$ spherical harmonics. These are the analog of the spherical harmonics used in the temperature maps and obey the same completeness and orthogonality relations. We show that there exist two linear combinations of spin $\pm 2$ multipole moments which have opposite parities and can be used to fully characterize the statistical properties of polarization in the CMB. Magnetic-type parity combination does not receieve contributions from scalar modes and does not cross-correlate with either temperature or electric-type parity combination, so there are four different power spectra that fully characterize statistical properties of CMB. We present their explicit expressions for scalar and tensor modes in the form of line of sight integral solution and numerically evaluate them for a representative set of models. These general solutions differ from the expressions obtained previously in the small scale limit both for scalar and tensor modes. A method to generate and analyze all sky maps of temperature and polarization is given and the optimal estimators for various power spectra and their corresponding variances are discussed.

Figures (1)

• Figure 1: Comparison between exact calculation (solid lines) and small scale approximation (dashed lines) for standard CDM model with and without reionization. In the latter case we use optical depth of 0.2. The reionized models are the upper curves on large scales. The comparison is for scalar $E$ (a) and tensor $E$ (b) and $B$ (c) polarization power spectra. The spectra are in units of $T_0^2=(2.729K)^2$ and are normalized to COBE. While the predictions agree for large $l$ there are significant discrepancies in certain models for small $l$.