CMB spectral distortions as solutions to the Boltzmann equations

Understand

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.