Second order Boltzmann equation : gauge dependence and gauge invariance
  In the context of cosmological perturbation theory, we derive the second
order Boltzmann equation describing the evolution of the distribution function
of radiation without a specific gauge choice. The essential steps in deriving
the Boltzmann equation are revisited and extended given this more general
framework: i) the polarisation of light is incorporated in this formalism by
using a tensor-valued distribution function; ii) the importance of a choice of
the tetrad field to define the local inertial frame in the description of the
distribution function is emphasized; iii) we perform a separation between
temperature and spectral distortion, both for the intensity and for
polarisation for the first time; iv) the gauge dependence of all perturbed
quantities that enter the Boltzmann equation is derived, and this enables us to
check the correctness of the perturbed Boltzmann equation by explicitly showing
its gauge-invariance for both intensity and polarization. We finally discuss
several implications of the gauge dependence for the observed temperature.
