Relativistic corrections to the multiple scattering effect on the Sunyaev-Zel'dovich effect in the isotropic approximation

Abstract

We extend the formalism for the calculation of the relativistic corrections to the Sunyaev-Zel'dovich effect for clusters of galaxies and include the multiple scattering effects in the isotropic approximation. We present the results of the calculations by the Fokker-Planck expansion method as well as by the direct numerical integration of the collision term of the Boltzmann equation. The multiple scattering contribution is found to be very small compared with the single scattering contribution. For high-temperature galaxy clusters of kT=15keV, the ratio of the both contributions is -0.2% in the Wien region. In the Rayleigh--Jeans region the ratio is -0.03%. Therefore the multiple scattering contribution is safely neglected for the observed galaxy clusters.