We explore the amplification of magnetic seed fields during the formation ofthe first stars and galaxies. During gravitational collapse, turbulence iscreated from accretion shocks, which may act to amplify weak magnetic fields inthe protostellar cloud. Numerical simulations showed that such turbulence issub-sonic in the first star-forming minihalos, and highly supersonic in thefirst galaxies with virial temperatures larger than 10^4 K. We investigate themagnetic field amplification during the collapse both for Kolmogorov andBurgers-type turbulence with a semi-analytic model that incorporates theeffects of gravitational compression and small-scale dynamo amplification. Wefind that the magnetic field may be substantially amplified before theformation of a disk. On scales of 1/10 of the Jeans length, saturation occursafter ~10^8 yr. Although the saturation behaviour of the small-scale dynamo isstill somewhat uncertain, we expect a saturation field strength of the order~10^{-7} n^{0.5} G in the first star-forming halos, with n the number densityin cgs units. In the first galaxies with higher turbulent velocities, themagnetic field strength may be increased by an order of magnitude, andsaturation may occur after 10^6 to 10^7 yr. In the Kolmogorov case, themagnetic field strength on the integral scale (i.e. the scale with mostmagnetic power) is higher due to the characteristic power-law indices, but thedifference is less than a factor of 2 in the saturated phase. Our results thusindicate that the precise scaling of the turbulent velocity with length scaleis of minor importance. They further imply that magnetic fields will besignificantly enhanced before the formation of a protostellar disk, where theymay change the fragmentation properties of the gas and the accretion rate.
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