Constraints on the Strength of Primordial Magnetic Fields from Big Bang Nucleosynthesis Revisited

In this paper, we revisit in detail the effects of primordial magnetic fields on big bang nucleosynthesis (BBN) including a discussion of the magnetic field geometry and the anomalous magnetic moment. The presence of magnetic fields affects BBN by (1) increasing the weak reaction rates; (2) increasing the electron density due to changes to the electron phase space; and (3) by increasing the expansion rate of the universe, due both to the magnetic field energy density and to the modified electron energy density. Of the effects considered, the increase in the expansion rate due to the magnetic field energy is the most significant for the interests of BBN. The allowed magnetic field intensity at the end of nucleosynthesis (0.01 MeV) is about $2 \times 10^{9}$G and corresponds to an upper limit on the magnetic field energy density of about 28\% of the neutrino energy density ($\rho_B \le 0.28 \rho_\nu$).