  In the standard model of cosmology (SMC) the B-mode polarization of the CMB
can be explained by the gravitational effects in the inflation epoch. However,
this is not the only way to explain the B-mode polarization for the CMB. It can
be shown that the Compton scattering in presence of a background besides
generating a circularly polarized microwave, can leads to a B-mode polarization
for the CMB. Here we consider the non-commutative (NC) space time as a
background to explore the CMB polarization at the last scattering surface. We
obtain the B-mode spectrum of the CMB radiation by scalar perturbation of
metric via a correction on the Compton scattering in NC-space-time in terms of
the circular polarization power spectrum and the non-commutative energy scale.
It can be shown that even for the NC-scale as large as $10TeV$ the NC-effects
on the CMB polarization and the r-parameter is significant. We show that the
V-mode power spectrum can be obtained in terms of linearly polarized power
spectrum in the range Micro to Nano-Kelvin squared for the NC-scale about
$1TeV$ to $10TeV$, respectively.
