Distribution function of nuclei from $e^\pm$ scattering in the presence of a strong primordial magnetic field

Abstract

The amplitude of the primordial magnetic field (PMF) is constrained from observational limits on primordial nuclear abundances. Within this constraint, it is possible that nuclear motion is regulated by Coulomb scattering with electrons and positrons ($e^\pm$'s), while $e^\pm$'s are affected by a PMF rather than collisions. For example, at a temperature of $10^9$ K, thermal nuclei typically experience $\sim 10^{21}$ scatterings per second that are dominated by very small angle scattering leading to minuscule changes in the nuclear kinetic energy of order $\mathcal{O}$(1) eV. In this paper the upper limit on the effects of a possible discretization of the $e^\pm$ momenta by the PMF on the nuclear momentum distribution is estimated under the extreme assumptions that the momentum of the $e^\pm$ is relaxed before and after Coulomb scattering to Landau levels, and that during Coulomb scattering the PMF is neglected. This assumption explicitly breaks the time reversal invariance of Coulomb scattering, and the Maxwell-Boltzmann distribution is not a trivial steady state solution of the Boltzmann equation under these assumptions. We numerically evaluate the collision terms in the Boltzmann equation, and show that the introduction of a special direction in the $e^\pm$ distribution by the PMF generates no directional dependence of the collisional destruction term of nuclei. Large anisotropies in the nuclear distribution function are then constrained from big bang nucleosynthesis. Ultimately, we conclude that a PMF does not significantly affect the isotropy or BBN.