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This article investigates the full Boltzmann equation up to second order inthe cosmological perturbations. Describing the distribution of polarizedradiation by a tensor valued distribution function, we study the gaugedependence of the distribution function and summarize the construction of thegauge-invariant distribution function. The Liouville operator which describesthe free streaming of electrons, and the collision term which describes thescattering of photons on free electrons are computed up to second order.Finally, the remaining dependence in the direction of the photon momentum ishandled by expanding in projected symmetric trace-free multipoles and also inthe more commonly used normal modes components. The results obtained remain tobe used for computing numerically the contribution in the cosmic microwavebackground bispectrum which arises from the evolution of second orderperturbations, in order to disentangle the primordial non-Gaussianity from theone generated by the subsequent non-linear evolution.

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