  The cosmological birefringence caused by the coupling between the cosmic
scalar field and the cosmic microwave background (CMB) photons through the
Chern-Simons term can rotate the polarization planes of the photons, and mix
the CMB E-mode and B-mode polarizations. The rotation angle induced by the
dynamical scalar field can be separated into the isotropic background part and
the anisotropic fluctuations. The effect of the background part has been be
studied in the previous work (Zhao & Li, arXiv:1402.4324). In this paper, we
focus on the influence of the anisotropies of the rotation angle. We first
assume that the cosmic scalar field is massless, consistent with other works,
we find that the rotation spectrum can be quite large, which may be detected by
the potential CMB observations. However, if the scalar field is identified as
the quintessence field, by detailed discussing both the entropy and adiabatic
perturbation modes for the first time, we find that the anisotropies of the
rotation angle are always too small to be detectable.In addition, as the main
goal of this paper, we investigate the effect of rotated polarization power
spectrum on the detection of relic gravitational waves. we find that, the
rotated B-mode polarization could be fairly large, and comparable with those
generated by the gravitational waves. This forms a new contamination for the
detection of relic gravitational waves in the CMB. In particular, we also
propose the method to reconstruct and subtract the rotated B-mode polarization,
by which the residuals become negligible for the gravitational-wave detection.
