The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization

TL;DR

This paper provides a comprehensive second-order treatment of the Boltzmann equation with polarization for cosmological radiation, essential for precise CMB bispectrum analysis and understanding non-Gaussianities.

Contribution

It develops a full second-order Boltzmann equation framework including polarization, gauge-invariant distribution functions, and multipole expansions for cosmological applications.

Findings

Derived second-order Liouville operator and collision term

Constructed gauge-invariant distribution functions

Expanded photon momentum dependence using multipoles

Abstract

This article investigates the full Boltzmann equation up to second order in the cosmological perturbations. Describing the distribution of polarized radiation by a tensor valued distribution function, we study the gauge dependence of the distribution function and summarize the construction of the gauge-invariant distribution function. The Liouville operator which describes the free streaming of electrons, and the collision term which describes the scattering of photons on free electrons are computed up to second order. Finally, the remaining dependence in the direction of the photon momentum is handled by expanding in projected symmetric trace-free multipoles and also in the more commonly used normal modes components. The results obtained remain to be used for computing numerically the contribution in the cosmic microwave background bispectrum which arises from the evolution of second order perturbations, in order to disentangle the primordial non-Gaussianity from the one generated by the subsequent non-linear evolution.