The thermal and kinematic Sunyaev–Zel’dovich effects revisited

Abstract

This paper shows that a simple convolution integral expression based on the mean value of the isotropic frequency distribution corresponding to photon scattering off electrons leads to useful analytical expressions describing the thermal Sunyaev–Zel’dovich effect. The approach, to first order in the Compton parameter is able to reproduce the Kompaneets equation describing the effect. Second order effects in the parameter z = kT e mc2 induce a slight increase in the crossover frequency.