CMB spectral distortions as solutions to the Boltzmann equations
  Distortions to the cosmic microwave background~(CMB) blackbody spectrum are
calculated in the framework of cosmological perturbation theory. The second
order Boltzmann equation is explicitly solved, with the spectral $y$ distortion
and the frequency independent second order temperature perturbation. We also
solve higher order Boltzmann equations systematically and find new type
spectral distortions in the low energy limit of electrons. As an example, we
concretely construct a solution to the cubic order Boltzmann equation and show
that it can be characterized by three parameters: a cubic order temperature
perturbation and two different types of cubic order spectral distortions. A new
linear Sunyaev-Zel'dovich effect whose frequency dependence is different from
the usual $y$ distortion is also discussed in the presence of the
next-to-leading order Kompaneets terms, and we show that higher order spectral
distortions are also generated as a result of the diffusion process in a
framework of the higher order cosmological perturbation theory. We also comment
that generation of the spectral $\mu$ distortion cannot be explained in this
framework even at second order.
