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

This paper 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 bi-spectrum which arises from the evolution of second-order perturbations, in order to disentangle the primordial non-Gaussianity from the one generated by the subsequent nonlinear evolution.