Following a cosmological gaseous collapse over many orders of
magnitude in length makes such a simulation technically difficult, but
with improvements in algorithms and physical models, several groups
have been able to make substantial progress. In the late 1990's and
early 2000's, two independent groups used three-dimensional
simulations to study Population III star formation. Using smoothed
particle hydrodynamics (SPH) simulations of isolated and virialized
dark matter halos with masses 2 × 106
M
at
z = 30, one group found that the object cooled to 300 K through
H2 formation, using the chemical network of
[30],
and fragmented into a
filamentary structure with a Jeans mass of 103
M
[13,
14].
In the second paper, they followed the collapse to
higher gas densities of 108 cm-3 and studied the
continued formation of dense clumps with the same 103
M
characteristic mass. The other group used cosmological adaptive mesh
refinement (AMR) simulations to focus on the formation of a molecular cloud
hosted by a 7 × 105
M DM
halo
[1,
3,
4].
In each successive paper, the H2 chemistry model
[2,
8]
was improved to include more processes, such as the
three-body H2 formation process, to follow the central
collapse to gas densities up to 3 × 1013
cm-3. Both groups came to
the conclusion that further fragmentation was suppressed because of a
lack of cooling below 300 K and that Population III stars were
very massive in the range 30-300
M.
These simulations only represented a limited sample of collapses and
could not provide much insight to the Population III initial mass
function (IMF). Twelve additional AMR simulations were conducted to
look at any variations in primordial gas collapses
[45].
They found that the collapses occurred in DM halos with masses in the
range 1.5-7 × 105
M with
the scatter caused by differing
halo formation histories. The mass accretion rate onto the central
molecular cloud was higher at 10-4
M
yr-1 at z ~ 30, and it can increase by two orders of
magnitude at z ~ 20 in some halos, agreeing with
[4].
Following the collapse to densities higher than 1013
cm-3
required the inclusion of collisionally induced emission, chemical
heating from H2 formation, and gas opacity above
1018 cm-3.
[77]
found with SPH simulations that the initial
collapsing region did not fragment as it condensed to protostellar
densities n ~ 1021 cm-3, forming a
protostellar shock in the process. The inner 10
M
had an accretion rate varying between 0.01 and 0.1
M
yr-1, possibly growing to 10
M within
1000 yr.
In the past few years, multiple groups have been focusing on the
subsequent growth of these protostars over several dynamical times,
improving upon the earlier works that stopped at the first collapse.
This has proved to be challenging because of the ever-decreasing
Courant factors at higher densities. One workaround is the creation
of "sink particles" that accrete nearby gravitationally-bound gas,
allowing the simulation to progress past the first collapse; however,
one loses all hydrodynamical information above some density threshold.
In one out of five realizations in AMR calculations without sink
particles,
[65]
found that the collapsing core fragmented at a density of
1011 cm-3 into two clumps that are separated by
800 AU with 100
M of gas
within a sphere with radius twice their separation. At the same time,
[56]
also found that
disk instabilities causes fragmentation into a binary system with a
40 M
and 10 M.
This was later confirmed by simulations of a
collapse of an isolated Jeans-unstable primordial gas cloud that
fragmented into many multiple systems with some very tightly bound to
separations less than an AU
[19].
Utilizing a new moving mesh code,
[29]
studied the collapse in
five different primordial DM halos, and they evolved them for 1000 yr
after the first protostar forms. By evolving these protostars
further, they included the effects of protostellar radiative feedback
in the infrared in the optically-thin limit. In all cases, the
molecular cloud fragments into ~ 10 sink particles, some of which
later merge to form more massive protostars. The mass function from
these simulations is relatively flat, i.e. a top-heavy IMF.
After the protostar has reached ~ 10
M,
radiative feedback from
ionizing radiation will begin to suppress further accretion. Only
recently has this been incorporated into numerical simulations of
Population III star formation. Starting from initial conditions
extracted from a cosmological simulation
[77],
two-dimensional axi-symmetric radiation hydrodynamical simulations
showed that an accretion disk forms around a new protostar with the
ionizing radiation preferentially escaping through the polar regions
[32].
The disk itself is slowly photo-evaporated,
halting accretion after 70,000 yr. At this point, the final mass of
the Population III star is 43
M.
Without any radiative feedback, the protostar would have continued to
grow to ~ 100
M. In a
cosmological setting,
[57]
found a binary system still forms in the presence of radiative
feedback. Without feedback, the primary star grows to 28
M
over 5,000 yr. With feedback, the
primary and secondary stars only grow to 19 and 10
M,
respectively. An extrapolation of the mass accretion history shows that
both stellar masses will asymptote to 30
M,
creating an equal-mass binary. Once
the stars have entered the main sequence, they will start to ionize
and heat their cosmic neighborhood, which I will review next.