Relativistic Corrections to the Sunyaev-Zeldovich Effect for Clusters of Galaxies. III. Polarization Effect

We extend the formalism of relativistic thermal and kinematic Sunyaev-Zeldovich effects and include the polarization of the cosmic microwave background photons. We consider the situation of a cluster of galaxies moving with a velocity β≡v/c with respect to the cosmic microwave background radiation. In the present formalism, polarization of the scattered cosmic microwave background radiation caused by the proper motion of a cluster of galaxies is naturally derived as a special case of the kinematic Sunyaev-Zeldovich effect. The relativistic corrections are also included in a natural way. Our results are in complete agreement with the recent results of relativistic corrections obtained by Challinor, Ford, & Lasenby with an entirely different method, as well as the nonrelativistic limit obtained by Sunyaev & Zeldovich. The relativistic correction becomes significant in the Wien region.