Updated constraint on a primordial magnetic field during big bang nucleosynthesis and a formulation of field effects

Abstract

A new upper limit on the amplitude of primordial magnetic field (PMF) is derived by a comparison between a calculation of elemental abundances in big bang nucleosynthesis model and the latest observational constraints on the abundances. Updated nuclear reaction rates are adopted in the calculation. Effects of PMF on the abundances are consistently taken into account in the numerical calculation with the precise formulation of changes in physical variables. We find that abundances of $^{3}\mathrm{He}$ and $^{6}\mathrm{Li}$ increase while that of $^{7}\mathrm{Li}$ decreases when the PMF amplitude increases, in the case of the baryon-to-photon ratio determined from the measurement of cosmic microwave background radiation. We derive a constraint on the present amplitude of PMF, i.e., $B(0)<1.5\text{ }\text{ }\ensuremath{\mu}\mathrm{G}$ [corresponding to the amplitude less than $2.0\ifmmode\times\else\texttimes\fi{}{10}^{11}\text{ }\text{ }\mathrm{G}$ at big bang nucleosynthesis temperature of $T={10}^{9}\text{ }\text{ }\mathrm{K}$], based on the rigorous calculation.