Magnetic fields in the formation of the first stars - I. Theory versus simulation

While magnetic fields are important in contemporary star formation, their role in primordial star formation is unknown. Magnetic fields of the order of 10(-16) G are produced by the Biermann battery due to the curved shocks and turbulence associated with the infall of gas into the dark matter minihaloes that are the sites of formation of the first stars. These fields are rapidly amplified by a small-scale dynamo until they saturate at or near equipartition with the turbulence in the central region of the gas. Analytical results are given for the outcome of the dynamo, including the effect of compression in the collapsing gas. The mass-to-flux ratio in this gas is two to three times the critical value, comparable to that in contemporary star formation. Predictions of the outcomes of simulations using smooth particle hydrodynamics (SPH) and grid-based adaptive mesh refinement are given. Because the numerical viscosity and resistivity for the standard resolution of 64 cells per Jeans length are several orders of magnitude greater than the physical values, dynamically significant magnetic fields affect a much smaller fraction of the mass in simulations than in reality. An appendix gives an analytical treatment of free-fall collapse, including that in a constant-density background. Another appendix presents a new method of estimating the numerical viscosity; results are given for both SPH and grid-based codes.