Merge or survive: Number of Population III stars per minihalo
  The formation process of Population III (PopIII) stars in the mass accretion
phase is investigated by numerical experiments. The barotropic relation of
primordial gas and artificial stiffening of the equation of state in very dense
regions ($> 10^{15}~{\rm cm}^{-3}$) enables us to follow the fragmentation of
PopIII circumstellar disks and the merging processes of the fragments. The disk
becomes gravitationally unstable to fragmentation , followed by a rapid merger
process typically within 100 yrs, which roughly corresponds to one orbital time
of the circumstellar disk. We also find that the fragmentation of the gas disk
around a multiple system, a circumbinary disk, is rare; however, it is frequent
in the disk around an individual protostar. We also perform a simulation with
standard sink particles, where the number and total mass of sink particles are
in rough agreement with those of the stiff equation of state runs. Based on the
results of these numerical results, we model the evolution of the number of
fragments with a simple phenomenological equation. We find that the average
number of fragments is roughly proportional to $t^{0.3}$, where $t$ is the
elapsed time since the formation of the first protostar. Next, we compare this
trend with a number of published numerical studies by scaling the elapsed time
according to the scale-free nature of the system. As a result, we find most of
the results in the literature agree well with the relation. Present results
combined with the previous studies in the literature imply that the PopIII
stars tend to be born not as single stars, but in multiple systems.
