begin with the proper cosmological initial conditions and follow the collapse of the gas all the way to protostellar densities. These simulations confirm that fragmentation during the initial collapse phase is ineffective: in general, only a single protostar forms, although in roughly 20% of cases, the gas fragments into two clumps, forming a wide binary (@tao09; M. Turk, private communication). However, most of these studies were prevented by technical limitations from following the evolution of the infalling gas after the formation of the first protostar. The technical problem involves the hydrodynamical timestep: as the collapse is followed down to protostellar densities, this becomes extremely short ($\Delta t \sim 10^{-3} \: {\rm yr}$), making it computationally infeasible to follow the evolution of the surrounding dense gas over any significant time period.

In studies of contemporary star formation, a similar problem is avoided by the use of sink particles [@bbp95]. When gravitationally bound regions of gas collapse below the scale on which they can be spatially resolved, they are replaced in the simulation by a sink particle of the same mass. This can accrete gas from its surroundings, but otherwise interacts only via gravity. Recent simulations of Population III star formation that 