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The existence of a primordial magnetic field (PMF) would affect both the temperature and polarization anisotropies of the cosmic microwave background (CMB). It also provides a plausible explanation for the possible disparity between observations and theoretical fits to the CMB power spectrum. Here we report on calculations of not only the numerical CMB power spectrum from the PMF, but also the correlations between the CMB power spectrum from the PMF and the primary curvature perturbations. We then deduce a precise estimate of the PMF effect on all modes of perturbations. We find that the PMF affects not only the CMB TT (temperature fluctuation), TE (cross correlation as temperature---E-type polarization) modes on small angular scales, but also on large angular scales. The introduction of a PMF leads to a better fit to the CMB power spectrum for the higher multipoles, and the fit at lowest multipoles can be used to constrain the correlation of the PMF with the density fluctuations for large negative values of the spectral index. Our prediction for the BB (B-type polarization) mode for a PMF average field strength |B_|=4. 0 nG is consistent with the upper limit on the BB mode deduced from the latest CMB observations. We find that the BB mode is dominated by the vector mode of the PMF for higher multipoles. We also show that by fitting the complete power spectrum one can break the degeneracy between the PMF amplitude and its power spectral index.
Yamazaki et al. (Fri,) studied this question.