Big Bang Nucleosynthesis with an Inhomogeneous Primordial Magnetic Field Strength

We investigate the effect on the Big Bang Nucleosynthesis (BBN) from the presence of a stochastic primordial magnetic field (PMF) whose strength is spatially inhomogeneous. We assume a uniform total energy density and a gaussian distribution of field strength. In this case, domains of different temperatures exist in the BBN epoch due to variations in the local PMF. We show that in such case, the effective distribution function of particle velocities averaged over domains of different temperatures deviates from the Maxwell-Boltzmann distribution. This deviation is related to the scale invariant strength of the PMF energy density $\rho_{\rm Bc}$ and the fluctuation parameter $\sigma_{\rm B}$. We perform BBN network calculations taking into account the PMF strength distribution, and deduce the element abundances 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 averaged thermonuclear reaction rates compared with those of a single temperature, and find that the averaged charged-particle reaction rates are very different. Finally, we constrain the parameters $\rho_{\rm Bc}$ and $\sigma_{\rm B}$ from observed abundances of $^4$He and D, and find that the $^7$Li abundance is significantly reduced. We also find that if the $\eta$ value during BBN was larger than the present-day value due to a dissipation of the PMF or a radiative decay of exotic particles after BBN or if the stellar depletion of $^7$Li occurred, abundances of all light elements can be consistent with observational constraints.