Polarization of the Sunyaev-Zel'dovich effect: relativistic imprint of
  thermal and non-thermal plasma
  [Abridged] Inverse Compton scattering of CMB fluctuations off cosmic electron
plasma generates a polarization of the associated Sunyaev-Zel'dovich (SZ)
effect. This signal has been studied so far mostly in the non-relativistic
regime and for a thermal electron population and, as such, has limited
astrophysical applications. Partial attempts to extend this calculation for a
thermal electron plasma in the relativistic regime have been done but cannot be
applied to a general relativistic electron distribution. Here we derive a
general form of the SZ effect polarization valid in the full relativistic
approach for both thermal and non-thermal electron plasmas, as well as for a
generic combination of various electron population co-spatially distributed in
the environments of galaxy clusters or radiogalaxy lobes. We derive the
spectral shape of the Stokes parameters induced by the IC scattering of every
CMB multipole, focusing on the CMB quadrupole and octupole that provide the
largest detectable signals in galaxy clusters. We found that the CMB quadrupole
induced Stoke parameter Q is always positive with a maximum amplitude at 216
GHz which increases slightly with increasing cluster temperature. The CMB
octupole induced Q spectrum shows, instead, a cross-over frequency which
depends on the cluster electron temperature, or on the minimum momentum p_1 as
well as on the power-law spectral index of a non-thermal electron population.
We discuss some possibilities to disentangle the quadrupole-induced Q spectrum
from the octupole-induced one which allow to measure these quantities through
the SZ effect polarization. We finally apply our model to the realistic case of
the Bullet cluster and derive the visibility windows of the total,
quandrupole-induced and octupole-induced Stoke parameter Q in the frequency
ranges accessible to SKA, ALMA, MILLIMETRON and CORE++ experiments.
