  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.
