  (abridged) We perform 12 extremely high resolution adaptive mesh refinement
cosmological hydrodynamic simulations of Population III star formation in a
Lambda CDM universe, varying the box size and large-scale structure, to
understand systematic effects in the formation of primordial protostellar
cores. We find results that are qualitatively similar to those observed
previously. We observe that the threshold halo mass for formation of a
Population III protostar does not evolve significantly with time in the
redshift range studied (33 > z > 19) but exhibits substantial scatter due to
different halo assembly histories: Halos which assembled more slowly develop
cooling cores at lower mass than those that assemble more rapidly, in agreement
with Yoshida et al. (2003). We do, however, observe significant evolution in
the accretion rates of Population III protostars with redshift, with objects
that form later having higher maximum accretion rates, with a variation of two
orders of magnitude (10^-4 - 10^-2 Msolar/year). This can be explained by
considering the evolving virial properties of the halos with redshift and the
physics of molecular hydrogen formation at low densities. Our result implies
that the mass distribution of Population III stars inferred from their
accretion rates may be broader than previously thought, and may evolve with
redshift. Finally, we observe that our collapsing protostellar cloud cores do
not fragment, consistent with previous results, which suggests that Population
III stars which form in halos of mass 10^5 - 10^6 Msun always form in
isolation.
