  We investigate the formation of metal-free, Population III (Pop III), stars
within a minihalo at z ~ 20 with a smoothed particle hydrodynamics (SPH)
simulation, starting from cosmological initial conditions. Employing a
hierarchical, zoom-in procedure, we achieve sufficient numerical resolution to
follow the collapsing gas in the center of the minihalo up to number densities
of 10^12 cm^-3. This allows us to study the protostellar accretion onto the
initial hydrostatic core, which we represent as a growing sink particle, in
improved physical detail. The accretion process, and in particular its
termination, governs the final masses that were reached by the first stars. The
primordial initial mass function (IMF), in turn, played an important role in
determining to what extent the first stars drove early cosmic evolution. We
continue our simulation for 5000 yr after the first sink particle has formed.
During this time period, a disk-like configuration is assembled around the
first protostar. The disk is gravitationally unstable, develops a pronounced
spiral structure, and fragments into several other protostellar seeds. At the
end of the simulation, a small multiple system has formed, dominated by a
binary with masses ~ 40 M_Sun and ~ 10 M_Sun. If Pop III stars were to form
typically in binaries or small multiples, the standard model of primordial star
formation, where single, isolated stars are predicted to form in minihaloes,
would have to be modified. This would have crucial consequences for the
observational signature of the first stars, such as their nucleosynthetic
pattern, and the gravitational-wave emission from possible Pop III black-hole
binaries.
