Thermal and kinematic corrections to the microwave background polarization induced by galaxy clusters along the line of sight

We derive analytic expressions for the leading-order corrections to the polarization induced in the cosmic microwave background (CMB) owing to scattering of photons off hot electrons in galaxy clusters along the line of sight. For a thermal distribution of electrons with kinetic temperature k(B)T(e)similar to 10 keV and bulk peculiar velocity V similar to 1000 km s(-1), the dominant corrections to the polarization induced by the primordial CMB quadrupole and the cluster peculiar velocity arise from electron thermal motion and are at the level of similar to 10 per cent in each case, near the peak of the polarization signal. When more sensitive measurements become feasible, these effects will be significant for the determination of transverse peculiar velocities, and the value of the CMB quadrupole at the cluster redshift, via the cluster polarization route.