Polarization of the CMB photons crossing the galaxy clusters

In this paper, I investigate a local effect of polarization of the Cosmic Microwave Background (CMB) in clusters of galaxies, due to the Thomson scattering of the anisotropic radiation. A local anisotropy of the CMB is produced by some gravitational and scattering effects, as, for instance, the Sunyaev Zel'dovich effect, the Doppler shift due to the cluster motion and the gravitational lensing. The resulting anisotropy $\Delta I / I$ depends on the physical properties of the clusters, in particular on their emissivity in the X band, their size, their gravitational potential and on the peculiar conditions characterizing the cluster gas. By solving the Boltzmann radiative transfer equation in presence of such anisotropies I calculate the average polarization at the centre of some clusters, namely A2218, A576 and A2163, whose properties are quite well known. I prove the gravitational effects due to the contraction or to the expansion of the cluster have some importance and dominate the SZ effect, particularly for high density stuctures.