The Thomson scattering cross section depends on the polarization of the outgoing photon. If its polarization vector lies in the scattering plane, the cross section is proportional to cos2 β, where β denotes the scattering angle. If, however, the outgoing photon is polarized normal to the scattering plane, no such reduction by a factor cos2 β occurs (see Jackson (1975), Section 14.7). If photons come in isotropically from all directions, this does not lead to any net polarization of the outgoing radiation. However, if, for a fixed outgoing direction, the intensity of incoming photons from one direction is different from the intensity of photons coming in at a right angle with respect to the first direction and with respect to the direction of the outgoing photon (see Fig. 5.1), this anisotropy leads to some polarization of the outgoing photon beam. As it is clear from the figure, it is the quadrupole anisotropy in the reference frame of the scattering electron which is responsible for polarization.

In this chapter we discuss the induced polarization in detail. We derive the equations which govern the generation and propagation of polarization and we discuss their implications. This can be done by different methods, most of which are either rather involved or incomplete. Here we employ the so-called ‘total angular momentum method’ which has been developed in Hu & White (1997b) and Hu et al. (1998), based on previous work mainly by Seljak (1996b), Kamionkowski et al. (1997) and Zaldarriaga & Seljak (1997).