Relativistic Corrections to the Sunyaev-zel’dovich Effect for Clusters of Galaxies. Iii. Polarization Effect

We extend the formalism of the relativistic thermal and kinematic Sunyaev-Zel’dovich effects and include the polarization of the cosmic microwave background photons. We consider the situation that a cluster of galaxies is 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 the galaxies is naturally derived as a special case of the kinematic Sunyaev-Zel’dovich effect. The relativisitic corrections are included also in a natural way. It is found that the polarization is positive and small in the long wavelength region. In the short wavelength region, however, the polarization becomes large with a negative sign, and the relativistic correction is significant. Subject headings: cosmic microwave background — cosmology: theory — galaxies: clusters: general — radiation mechanisms: nonthermal — polarization — relativity