Small-scale dynamo action during the formation of the first stars and
  galaxies. I. The ideal MHD limit
  We explore the amplification of magnetic seed fields during the formation of
the first stars and galaxies. During gravitational collapse, turbulence is
created from accretion shocks, which may act to amplify weak magnetic fields in
the protostellar cloud. Numerical simulations showed that such turbulence is
sub-sonic in the first star-forming minihalos, and highly supersonic in the
first galaxies with virial temperatures larger than 10^4 K. We investigate the
magnetic field amplification during the collapse both for Kolmogorov and
Burgers-type turbulence with a semi-analytic model that incorporates the
effects of gravitational compression and small-scale dynamo amplification. We
find that the magnetic field may be substantially amplified before the
formation of a disk. On scales of 1/10 of the Jeans length, saturation occurs
after ~10^8 yr. Although the saturation behaviour of the small-scale dynamo is
still 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 density
in cgs units. In the first galaxies with higher turbulent velocities, the
magnetic field strength may be increased by an order of magnitude, and
saturation may occur after 10^6 to 10^7 yr. In the Kolmogorov case, the
magnetic field strength on the integral scale (i.e. the scale with most
magnetic power) is higher due to the characteristic power-law indices, but the
difference is less than a factor of 2 in the saturated phase. Our results thus
indicate that the precise scaling of the turbulent velocity with length scale
is of minor importance. They further imply that magnetic fields will be
significantly enhanced before the formation of a protostellar disk, where they
may change the fragmentation properties of the gas and the accretion rate.
