Stringent Limit on Primordial Magnetic Fields from the Cosmic Microwave Background Radiation

Primordial magnetic fields (PMFs), being present before the epoch of cosmic recombination, induce small-scale baryonic density fluctuations. These inhomogeneities lead to an inhomogeneous recombination process that alters the peaks and heights of the large-scale anisotropies of the cosmic microwave backround (CMB) radiation. Utilizing numerical compressible MHD calculations and a Monte Carlo Markov chain analysis, which compares calculated CMB anisotropies with those observed by the \textit{WMAP} and \textit{Planck} satellites, we derive limits on the magnitude of putative PMFs. We find that the {\it total} remaining present day field, integrated over all scales, cannot exceed 47 pG for scale-invariant PMFs and 8.9 pG for PMFs with a violet Batchelor spectrum at 95\% confidence level. These limits are more than one order of magnitude more stringent than any prior stated limits on PMFs from the CMB which have not accounted for this effect.