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Effects of power law primordial magnetic field on big bang nucleosynthesis

Big bang nucleosynthesis (BBN) is affected by the energy density of a primordial magnetic field (PMF). For an easy derivation of constraints on models for PMF generations, we assume a PMF with a power law (PL) distribution in wave number defined with a field strength, a PL index, and maximum and minimum scales at a generation epoch. We then show a relation between PL-PMF parameters and the scale invariant (SI) strength of PMF for the first time. We perform a BBN calculation including PMF effects, and show abundances as a function of baryon to photon ratio $η$. The SI strength of the PMF is constrained from observational constraints on abundances of $^4$He and D. The minimum abundance of $^7$Li/H as a function of $η$ slightly moves to a higher $^7$Li/H value at a larger $η$ value when a PMF exists during BBN. We then discuss degeneracies between the PL-PMF parameters in the PMF effect. In addition, we assume a general case in which both the existence and the dissipation of PMF are possible. It is then found that an upper limit on the SI strength of the PMF can be derived from a constraint on $^4$He abundance, and that a lower limit on the allowed $^7$Li abundance is significantly higher than those observed in metal-poor stars.