  In this paper we examine aspects of primordial star formation in a gravitino
warm dark matter universe with a cosmological constant. We compare a set of
simulations using a single cosmological realization but with a wide range of
warm dark matter particle masses which have not yet been conclusively ruled out
by observations. The addition of a warm dark matter component to the initial
power spectrum results in a delay in the collapse of high density gas at the
center of the most massive halo in the simulation and, as a result, an increase
in the virial mass of this halo at the onset of baryon collapse. Both of these
effects become more pronounced as the warm dark matter particle mass becomes
smaller. A cosmology using a gravitino warm dark matter power spectrum assuming
a particle mass of m_{WDM} ~ 40keV is effectively indistinguishable from the
cold dark matter case, whereas the m_{WDM} ~ 15 keV case delays star formation
by approx. 10^8 years. There is remarkably little scatter between simulations
in the final properties of the primordial protostar which forms at the center
of the halo, possibly due to the overall low rate of halo mergers which is a
result of the WDM power spectrum. The detailed evolution of the collapsing halo
core in two representative WDM cosmologies is described. At low densities
(n_{b} <= 10^5 cm^{-3}), the evolution of the two calculations is qualitatively
similar, but occurs on significantly different timescales, with the halo in the
lower particle mass calculation taking much longer to evolve over the same
density range and reach runaway collapse. Once the gas in the center of the
halo reaches relatively high densities (n_{b} >= 10^5 cm^{-3}) the overall
evolution is essentially identical in the two calculations.
