Theory of Cosmic Microwave Background Polarization

TL;DR

This paper provides a comprehensive theoretical overview of cosmic microwave background polarization, focusing on its generation by inflationary gravitational waves and the effects of weak gravitational lensing, including mathematical formalisms and potential observational signatures.

Contribution

It develops the detailed theory of CMB polarization from inflationary gravitational waves and cosmic shear, including tensor-harmonic analysis and the derivation of relevant Boltzmann/Einstein equations.

Findings

Gravitational waves produce a curl component in CMB polarization.

Density perturbations do not produce curl polarization at linear order.

Cosmic shear induces a curl component that can be distinguished and subtracted.

Abstract

These lectures introduce some of the basic theory of cosmic microwave background (CMB) polarization with the primary aim of developing the theory of CMB polarization from inflationary gravitational waves, as well as some of the related theory of weak gravitational lensing (cosmic shear) of CMB polarization. We begin with production of polarization by Thomson scattering. We then discuss tensor-harmonic analysis (the ``grad-curl'' or ``E-B'' decomposition) on a flat and full sky in some detail. The Boltzmann/Einstein equations required to predict the CMB temperature/polarization pattern due to primordial gravitational waves are derived. We show that gravitational waves produce a curl component of the CMB polarization while density perturbations (at linear order) do not. We then show how cosmic shear induces a curl component from a curl-free surface of last scattering. We describe, though in less detail, how higher-order correlations can be used to subtract the cosmic-shear--induced curl. Several exercises are provided.