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The existence of a primordial magnetic field (PMF) would affect both thetemperature and polarization anisotropies of the cosmic microwave background(CMB). It also provides a plausible explanation for the possible disparitybetween observations and theoretical fits to the CMB power spectrum. Here wereport on calculations of not only the numerical CMB power spectrum from thePMF, but also the correlations between the CMB power spectrum from the PMF andthe primary curvature perturbations. We then deduce a precise estimate of thePMF effect on all modes of perturbations. We find that the PMF affects not onlythe CMB TT and TE modes on small angular scales, but also on large angularscales. The introduction of a PMF leads to a better fit to the CMB powerspectrum for the higher multipoles, and the fit at lowest multipoles can beused to constrain the correlation of the PMF with the density fluctuations forlarge negative values of the spectral index. Our prediction for the BB mode fora PMF average field strength $|B_\lambda| =4.0$ nG is consistent with the upperlimit on the BB mode deduced from the latest CMB observations. We find that theBB mode is dominated by the vector mode of the PMF for higher multipoles. Wealso show that by fitting the complete power spectrum one can break thedegeneracy between the PMF amplitude and its power spectral index.

Effects of a Primordial Magnetic Field on Low and High Multipoles of the CMB