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Chemical separation of primordial Li$^+$ during structure formation caused by nanogauss magnetic field

During the structure formation, charged and neutral chemical species may have separated from each other at the gravitational contraction in primordial magnetic field (PMF). A gradient in the PMF in a direction perpendicular to the field direction leads to the Lorentz force on the charged species. Resultantly, an ambipolar diffusion occurs, and charged species can move differently from neutral species, which collapses gravitationally during the structure formation. We assume a gravitational contraction of neutral matter in a spherically symmetric structure, and calculate fluid motions of charged and neutral species. It is shown that the charged fluid, i.e., proton, electron and $^7$Li$^+$, can significantly decouple from the neutral fluid depending on the field amplitude. The charged species can, therefore, escape from the gravitational collapse. We take the structure mass, the epoch of the gravitational collapse, and the comoving Lorenz force as parameters. We then identify a parameter region for an effective chemical separation. This type of chemical separation can reduce the abundance ratio of Li/H in early structures because of inefficient contraction of $^7$Li$^+$ ion. Therefore, it may explain Li abundances of Galactic metal-poor stars which are smaller than the prediction in standard big bang nucleosynthesis model. Amplitudes of the PMFs are controlled by a magneto-hydrodynamic turbulence. The upper limit on the field amplitude derived from the turbulence effect is close to the value required for the chemical separation.