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We investigate the effect on the Big Bang Nucleosynthesis (BBN) from thepresence of a stochastic primordial magnetic field (PMF) whose strength isspatially inhomogeneous. We assume a uniform total energy density and agaussian distribution of field strength. In this case, domains of differenttemperatures exist in the BBN epoch due to variations in the local PMF. We showthat in such case, the effective distribution function of particle velocitiesaveraged over domains of different temperatures deviates from theMaxwell-Boltzmann distribution. This deviation is related to the scaleinvariant strength of the PMF energy density $\rho_{\rm Bc}$ and thefluctuation parameter $\sigma_{\rm B}$. We perform BBN network calculationstaking into account the PMF strength distribution, and deduce the elementabundances as functions of the baryon-to-photon ratio $\eta$, $\rho_{\rm Bc}$,and $\sigma_{\rm B}$. We find that the fluctuations of the PMF reduces the$^7$Be production and enhances D production. We analyze the averagedthermonuclear reaction rates compared with those of a single temperature, andfind that the averaged charged-particle reaction rates are very different.Finally, we constrain the parameters $\rho_{\rm Bc}$ and $\sigma_{\rm B}$ fromobserved abundances of $^4$He and D, and find that the $^7$Li abundance issignificantly reduced. We also find that if the $\eta$ value during BBN waslarger than the present-day value due to a dissipation of the PMF or aradiative decay of exotic particles after BBN or if the stellar depletion of$^7$Li occurred, abundances of all light elements can be consistent withobservational constraints.

Big Bang Nucleosynthesis with an Inhomogeneous Primordial Magnetic Field Strength