Analytical studies on the Sunyaev–Zeldovich effect in the cluster of galaxies for three Lorentz frames

We study the Sunyaev–Zeldovich (SZ) effect for clusters of galaxies. The Boltzmann equations for the cosmic microwave background photon distribution function are studied in three Lorentz frames. We clarify the relations of the SZ effects among the different Lorentz frames. We derive analytic expressions for the photon redistribution functions. These formulas are applicable to the non-thermal electron distributions as well as the standard thermal distribution. We show that the Fokker–Planck expansion of the Boltzmann equation can be expanded by the power series of the diffusion operator of the original Kompaneets equation.