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The Boltzmann equation describes the evolution in time of the
spatial () and angular () distribution of the radiation
under gravity and scattering processes. In the notation of , it
can be written implicitly as
where encapsulates the perturbation
to the temperature and the polarization
(Stokes Q and U parameters) in units of the temperature fluctuation.
The term accounts for collisions, here Compton scattering of the
photons with the electrons,
while the term accounts for gravitational redshifts.