  The potential importance of magnetic fields during structure formation and
gravitational collapse in the early Universe has been shown in several studies.
In particular, magnetic field amplification by the small-scale dynamo plays an
important role in addition to the pure amplification expected from
gravitational collapse. In this paper, we study the small-scale dynamo for
halos of $\gtrsim10^7$ M$_\odot$ collapsing at $z\gtrsim12$, under different
ambient conditions due to the strength of the Lyman-Werner background.
Additionally, we estimate the approximate saturation level by varying the
initial magnetic field strength. We performed cosmological
magnetohydrodynamical simulations for three distinct halos of $\sim10^7$
M$_{\odot}$ at $z\geq13$ by varying the Jeans resolution from $32-256$ cells
and employed Lyman Werner background flux of strengths $10^2-10^5$ in units of
$J_{21}$, where $J_{21}=10^{-21}$ erg$/$cm$^2/$sr$/$s$/$Hz. To follow the
chemical and thermal evolution of the gas we made use of the KROME package. In
addition to the compression by collapse, we find magnetic field amplification
via the dynamo both in the regimes of atomic and molecular hydrogen cooling.
Moreover, we find a lower saturation level in the molecular hydrogen cooling
regime. This behaviour can be understood due to the generally reduced radial
infall velocities and vorticities in this regime, as well as the higher Mach
numbers of the gas, which give rise to a smaller saturation ratio. Our results
overall suggest that the dynamo operates over a large range of conditions in
the collapsing gas.
