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While magnetic fields are important in contemporary star formation, theirrole in primordial star formation is unknown. Magnetic fields of order 10^-16 Gare produced by the Biermann battery due to the curved shocks and turbulenceassociated with the infall of gas into the dark matter minihalos that are thesites of formation of the first stars. These fields are rapidly amplified by asmall-scale dynamo until they saturate at or near equipartition with theturbulence in the central region of the gas. Analytic results are given for theoutcome of the dynamo, including the effect of compression in the collapsinggas. The mass-to-flux ratio in this gas is 2-3 times the critical value,comparable to that in contemporary star formation. Predictions of the outcomesof simulations using smooth particle hydrodynamics (SPH) and grid-basedadaptive mesh refinement (AMR) are given. Because the numerical viscosity andresistivity for the standard resolution of 64 cells per Jeans length areseveral orders of magnitude greater than the physical values, dynamicallysignificant magnetic fields affect a much smaller fraction of the mass insimulations than in reality. An appendix gives an analytic treatment offree-fall collapse, including that in a constant density background. Anotherappendix presents a new method of estimating the numerical viscosity; resultsare given for both SPH and grid-based codes.

Magnetic Fields in the Formation of the First Stars. I. Theory vs. Simulation