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. 
Atsuhisa Ota
astro-ph.CO gr-qc hep-ph hep-th