Anisotropy data analysis leaves a significant degeneracy between primeval spectral index (n_s) and cosmic opacity to CMB photons (tau). Low--l polarization measures, in principle, can remove it. We perform a likelihood analysis to see how cosmic variance possibly affects such a problem. We find that, for a sufficiently low noise level (sigma_{pix}) and if tau is not negligibly low, the degeneracy is greatly reduced, while the residual impact of cosmic variance on n_s and tau determinations is under control. On the contrary, if sigma_{pix} is too high, cosmic variance effects appear to be magnified. We apply general results to specific experiments and find that, if favorable conditions occur, it is possible that a 2--sigma detection of a lower limit on tau is provided by the SPOrt experiment. Furthermore, if the PLANCK experiment will measure polarization with the expected precision, the error on low--l harmonics is adequate to determine tau, without significant magnification of the cosmic variance. This however indicates that high sensitivity might be more important than high resolution in tau determinations. We also outline that a determination of tau is critical to perform detailed analyses on the nature of dark energy and/or on the presence of primeval gravitational waves.
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