In the previous sections of this review I have concentrated on the Sunyaev-Zel'dovich effects in the specific intensity, the Stokes I parameter. Any effects in the polarized intensity, the Stokes Q, U, and V terms, will be of smaller order by factors e, or v / c. An early reference was made to polarization terms in the paper by Sunyaev & Zel'dovich (1980b), with particular reference to their use to measure the velocities of clusters of galaxies across the line of sight. A more thorough discussion of polarization effects in inverse-Compton scattering is given by Nagirner & Poutanen (1994). All the polarization terms depend on higher powers of e, vxy, or vz than the thermal (or non-thermal) and kinematic effects discussed earlier, and therefore are not detectable with the current generation of experiments, although they may be measured in the future.
The simplest polarization term arises from multiple scatterings of photons within a scattering atmosphere. If a plasma distribution lies in front of a radio source, then the scattering of unpolarized radiation from that source by the surrounding atmosphere will produce a polarized halo, with the fractional polarization proportional to e but depending on the detailed geometry of the scattering process. For a radio source located centrally behind a spherical atmosphere, the pattern of polarization is circumferential. Similarly, scattering of the thermal (or non-thermal) Sunyaev-Zel'dovich effect by the same plasma producing the effect will produce a polarization, which may be circumferential (at high frequencies, where the Sunyaev-Zel'dovich effect appears as a source) or radial (at low frequencies, where it appears as a ``hole''). The peak polarization in this case will be less than a fraction e of the Sunyaev-Zel'dovich effect itself, or less than e y relative to the overall CMBR. For the particularly prominent cluster CL 0016+16 (Sec. 4), this factor is 2 x 10-6 h100-1, so that polarized signals of order 1 µK are the most that might be expected.
The motion of the plasma cloud across the line of sight also introduces polarization effects, from the Thomson scattering of the anisotropic radiation field in the frame of the moving cluster. The two largest contributions to the polarized intensity in this case were identified by Sunyaev & Zel'dovich (1980b) as a component of about 0.1 e (vxy / c)2 of the CMBR intensity, due to single scatterings of the quadrupolar term in the anisotropic radiation field seen in the frame of the moving cluster, and a component of about 0.025 e2 (vxy / c) from repeated scatterings of the dipolar term in the radiation field. Taking CL 0016+16 as an example again, the first of these polarizations is roughly a fraction 3 x 10-9 h100-1/2 (vxy / 1000 km s-1)2 of the intensity of the CMBR, while the second is of order 3 x 10-8 h100-1 (vxy / 1000 km s-1) of the CMBR intensity. Neither signal is likely to be measurable in the near future.
Similar effects will arise in the case of the non-thermal Sunyaev-Zel'dovich effect, but here the anisotropy of the electron distribution function is likely to be more significant. Polarized synchrotron radiation is also likely to be a bad contaminating signal for observational studies of the Sunyaev-Zel'dovich effect from relativistic populations of electrons.
No useful observational limits have yet been set on these polarization terms, and considerable development of observational techniques would be needed to make possible the measurement of even the largest of these effects.