The kinematic Sunyaev-Zel'dovich effect and transverse cluster velocities

The polarization of the cosmic microwave background radiation (CMBR) scattered by galaxy clusters in the kinematic Sunyaev-Zel'dovich effect depends on the transverse velocity of the cluster. This polarizing effect is proportional to the transverse velocity squared, and so weaker than the change in intensity resulting from the radial motion in the kinematic effect. The value given by Sunyaev and Zeldovich, which is frequently cited, underestimates the polarizing effect by a factor of 10. We show furthermore that the polarization has a strong frequency dependence. At high frequencies the polarization should be detectable with the new,generation of CMBR probes. Thus this effect offers, almost uniquely, a method of obtaining the vectorial velocity of clusters.