Relativistic corrections to the Sunyaev-Zeldovich effect for clusters of galaxies

We study the generalized Kompaneets equation (the kinetic equation for the photon distribution function taking into account Compton scattering by electrons) using a relativistically covariant formalism. Using the generalized Kompaneets equation, we derive an analytic expression for the Sunyaev-Zeldovich effect, which takes into account up to O(theta(e)(5)) terms, where theta(e) = k(B) T-e/mc(2) is the relativistic expansion parameter and T-e is the electron temperature. We carefully study the applicable region of the obtained analytic expression by comparing it with the result of the direct numerical integration. We conclude that the present analytic expression can be reliably applied to the calculation of the Sunyaev-Zeldovich effect for k(B) T-e less than or equal to 15 keV, which is the highest electron temperature in the clusters of galaxies presently known. Therefore, the present analytic expression can be applied to all known clusters of galaxies.