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

TL;DR

This work delivers a full second-order treatment of the Boltzmann equation with polarization in cosmology, providing a gauge-invariant framework that handles both radiation and baryons. It develops the gauge-invariant Liouville operator for radiation and baryons, computes the collision term including Thomson scattering and Kompaneets-like effects, and systematically converts PSTF multipoles to normal-mode components for numerical implementation. By integrating frame transformations, tetrad perturbations, and a double perturbative expansion, the paper lays the groundwork to compute the cosmic microwave background bispectrum arising from second-order evolution, enabling clearer disentanglement of primordial non-Gaussianity from nonlinear evolution. The results supply a rigorous, ready-to-use formalism for advancing second-order CMB transfer calculations and guide future numerical implementations.

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.