  Magnetic fields play an important role in the dynamics of present-day
molecular clouds. Recent work has shown that magnetic fields are equally
important for primordial clouds, which form the first stars in the Universe.
While the primordial magnetic field strength on cosmic scales is largely
unconstrained, theoretical models strongly suggest that a weak seed field
existed in the early Universe. We study how the amplification of such a weak
field can influence the evolution of accretion discs around first stars, and
thus affect the primordial initial mass function (IMF). We perform a suite of
3D ideal magneto-hydrodynamic (MHD) simulations with different initial field
strengths and numerical resolutions. We find that, in simulations with
sufficient spatial resolution to resolve the Jeans scale during the collapse,
even initially weak magnetic fields grow exponentially to become dynamically
important due to both the so-called 'small-scale turbulent dynamo' and the
'large-scale mean-field dynamo'. Capturing the small-scale dynamo action
depends primarily on how well we resolve the Jeans length, while capturing the
large-scale dynamo depends on the Jeans resolution as well as the maximum
absolute resolution. Provided enough resolution, we find that fragmentation
does not depend strongly on the initial field strength, because even weak
fields grow to become strong. However, fragmentation in runs with magnetic
fields differs significantly from those without magnetic fields. We conclude
that the development of dynamically strong magnetic fields during the formation
of the first stars is likely inevitable, and that these fields had a
significant impact on the primordial IMF.
