Boltzmann hierarchy for the cosmic microwave background at second order including photon polarization

TL;DR

This work derives a complete second-order Boltzmann hierarchy for the cosmic microwave background photon polarization, extending the polarization-density-matrix formalism to include all relevant propagation and collision terms. By working in conformal Newtonian gauge and assuming no first-order vector or tensor perturbations, the authors provide explicit expressions for the second-order propagation and Thomson-scattering collision terms, including their Fourier and multipole decompositions into the $I,V,E,B$ basis via spin-weighted harmonics. A key finding is that new sources of $B$-mode polarization arise at second order, notably from the collision term, which can generate $B$-mode polarization directly from intensity perturbations, as well as from time-delay and lensing-like effects in propagation; in the tight-coupling limit, polarization vanishes while a second-order intensity quadrupole persists. These results lay the groundwork for numerical evaluation of the second-order polarized CMB signal, to be presented in a follow-up study, and provide a framework to assess whether observed non-Gaussianity and $B$-mode signals could have significant second-order contributions.

Abstract

Non-gaussianity and B-mode polarization are particularly interesting features of the cosmic microwave background, as -- at least in the standard model of cosmology -- their only sources to first order in cosmological perturbation theory are primordial, possibly generated during inflation. If the primordial sources are small, the question arises how large is the non-gaussianity and B-mode background induced in second-order from the initially gaussian and scalar perturbations. In this paper we derive the Boltzmann hierarchy for the microwave background photon phase-space distributions at second order in cosmological perturbation theory including the complete polarization information, providing the basis for further numerical studies. As an aside we note that the second-order collision term contains new sources of B-mode polarization and that no polarization persists in the tight-coupling limit.