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

TL;DR

This paper derives a comprehensive second-order Boltzmann hierarchy for the cosmic microwave background, including polarization, to better understand non-Gaussianity and B-mode polarization generated from primordial scalar perturbations.

Contribution

It introduces the complete second-order Boltzmann hierarchy with polarization, enabling detailed numerical analysis of non-Gaussianity and B-modes in the CMB.

Findings

Second-order collision term includes new B-mode polarization sources.

No polarization persists in the tight-coupling limit.

Framework facilitates future numerical studies of CMB non-Gaussianity.

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.