In this review, we have focussed on three main topics: how the first star-forming minihalos come into existence and why they have the properties that they do; how gas within a representative minihalo cools, collapses and forms a protostar; and how this protostar and the massive clump of gas surrounding it subsequently evolve.
We have seen that on large scales, we now have a relatively good understanding of the physical processes involved in the formation of the first star-forming minihalos. In order for gas to accumulate within a dark matter minihalo, it must be able to overcome the effects of both gas pressure and also the large-scale streaming motion of the gas relative to the dark matter. Since this streaming motion is typically supersonic, the latter effect generally dominates, and the result is that gas is prevented from accumulating in large quantities within minihalos with masses of less than around 105 M. Within more massive minihalos, the gravitational force exerted by the dark matter is strong enough to overcome the effects of the gas pressure and the coherent streaming, and the gas begins to undergo gravitational collapse, reaching densities that are several hundred times higher than the cosmological background density. As the gas collapses, however, it is heated by compression and shocks. In order for the collapse to continue, the gas must be able to dissipate this energy, which it does through rotational and vibrational line emission from H2.
In Section 1.2, we saw that the amount of H2 formed within a given minihalo is a strong function of the temperature of the gas, with the final molecular fraction scaling roughly as xH2 T3/2 with the temperature T. The H2 cooling rate is also a strong function of temperature. As a result, one finds that there is a critical minihalo virial temperature, Tcrit ~ 1000 K marking the division between cooler halos that do not dissipate much energy within a Hubble time, and hence which do not form stars, and warmer halos that do manage to cool and form stars. As the virial temperature of a minihalo is a simple function of its mass and redshift, one can derive a critical minihalo mass that must be exceeded in order for the gas to cool effectively. This critical mass scales approximately as Mcrit ~ 1.6 × 106 (1+z / 10)-3/2 M, given standard values for the cosmological parameters. Combining this constraint with that arising from coherent streaming, one finds that at redshifts z > 40, the minimum mass of a star-forming minihalo is set by the need to overcome the effects of the streaming, and is roughly 105 M, while at z < 40, H2 cooling is the limiting factor, and the minimum mass scale is somewhat larger.
On smaller scales, we have also developed an increasingly good understanding of how the gas evolves as it cools, undergoes runaway gravitational collapse, and forms the first protostar. As outlined in Section 2, the gas first passes through a "loitering" phase, during which cold gas accumulates at the centre of the minihalo. The temperature and density of the gas at this point depend on the nature of the dominant coolant. When H2 dominates, we have T ~ 200 K and n ~ 104 cm-3, while if HD dominates, then T ~ 100 K and n ~ 106 cm-3. The loitering phase ends and the collapse of the gas accelerates once the mass of cold gas that has accumulated exceeds the local value of the Bonnor-Ebert mass, which is around 1000 M in the H2-dominated case, but only 40 M in the HD-dominated case. The next major event to occur is the onset of three-body H2 formation at n ~ 108 cm-3 which rapidly converts most of the atomic hydrogen into H2. The associated heat input leads to an increase in the gas temperature to T ~ 1000-2000 K, with the details depending to a significant extent on the rate coefficient chosen for reaction 41, which is poorly constrained at low temperatures. At n ~ 1010 cm-3, the gas becomes optically thick in the main H2 cooling lines, but remains optically thin in the continuum. It can therefore continue to cool reasonably effectively at these densities, with the mean temperature only rising relatively slowly with increasing density. At n ~ 1014 cm-3, a new process, collision-induced emission from H2, begins to dominate the cooling. However, this does not lead to a significant drop in the gas temperature, as the gas quickly becomes optically thick in the continuum. At densities above n ~ 1016 cm-3, further radiative cooling of the gas is ineffective and the only remaining process capable of slowing the temperature rise is collisional dissociation of the H2. While the H2 fraction in the gas remains significant, the temperature is prevented from rising much above 3000 K, but once most of the H2 has been destroyed, the temperature in the core rises steeply, and the internal thermal pressure eventually becomes strong enough to halt the collapse. State-of-the-art simulations have followed the gravitational collapse of the gas up to this point, which we can identify as the moment at which the first true Population III protostar forms.
Nevertheless, several uncertainties remain in this picture of Pop. III star formation. As already noted, the uncertainty in the three-body H2 formation rate limits the accuracy with which we can model the chemical and thermal evolution of the collapsing gas. In addition, current three-dimensional collapse models make use of simplified treatments of the effect of opacity on the H2 cooling rate, and the uncertainty that this introduces into the models has not yet been properly quantified. Further uncertainty comes from two additional issues which have only recently begun to be addressed: the role played by magnetic fields, and the influence of dark matter annihilation. Although the strength of any seed magnetic field existing prior to the assembly of the first star-forming minihalos is still poorly constrained, it now seems clear that the small-scale turbulent dynamo acting during the collapse of the gas will rapidly amplify even a very weak initial field up to a point at which it could potentially become dynamically significant. However, neither the final strength of the field nor its correlation length are well constrained at present, and without a better understanding of these values it is difficult to say to what extent the magnetic field will influence the details of the collapse. The role played by heating and ionization due to dark matter annihilation is even less well understood. Simple models suggest that it may be extremely important and may result in the formation of "dark stars" supported by the dark star energy released by dark matter annihilation rather than by nuclear fusion, but the only hydrodynamical study performed to date suggests that the influence on the collapse is small, and that dark stars do not actually form.
Finally, there remains the question of how the gas evolves after the formation of the first protostar. For much of the last decade, the leading model for this phase of the evolution of the gas has been what we have termed the "smooth accretion" model. In this model, it is assumed that the gas surrounding the newly formed protostar does not fragment, but instead simply smoothly accretes onto the protostar, primarily via a protostellar accretion disk. Considerable work has been done within the framework of this model to understand the structure of the protostar during the accretion phase, and the effect of protostellar feedback on the surrounding gas. This work has shown that feedback!radiative any feedback occurring prior to the protostar joining the main sequence is unlikely to significantly reduce the accretion rate, and that the most plausible mechanism for terminating the accretion is photoionization of the accretion disk by ionizing radiation from the central star, implying that it photoionization must already have grown to some tens of solar masses. This model therefore predicts that Pop. III stars will generally be solitary, with only one or two forming in each minihalo, and massive, with masses M ≫ 10 M.
Over the past couple of years, however, several new studies have appeared that have cast considerable doubt on the smooth accretion model. These studies have attempted to directly model the evolution of the gas as it begins to be accreted, and have shown that the accretion disk that builds up around the protostar is unstable to gravitational fragmentation even if the stabilizing effects of accretion luminosity feedback from the central feedback!radiative protostar are taken into account. Once a few fragments have formed, the dynamical interactions between the individual fragments and between the fragments and the gas can lead to further fragmentation, and to the ejection of low-mass fragments from the system. If we assume that all of the gravitationally bound fragments form protostars, then the result of this model is the assembly of a small, extremely dense cluster of Pop. III protostars with a wide range of masses. As discussed in Section 3.2.4, many aspects of the fragmentation scenario remain unclear and much work remains to be done before we can hope to have a good understanding of the final protostellar mass function. Nevertheless, these results suggest that Population III star formation perhaps has far more in common with present-day star formation than has been previously recognised.
The author would like to thank a large number of people with whom he has had interesting and informative discussions about the physics of Population III star formation, including T. Abel, V. Bromm, P. Clark, G. Dopcke, T. Greif, Z. Haiman, T. Hosokawa, R. Klessen, M. Norman, K. Omukai, B. O'Shea, D. Schleicher, B. Smith, R. Smith, A. Stacy, J. Tan, M. Turk, D. Whalen, and N. Yoshida. Special thanks also go to R. Smith and M. Turk for providing some of the data plotted in Figure 3, and to P. Clark for his assistance with the production of Figure 4. Financial support for this work was provided by the Baden-Württemberg-Stiftung via their program International Collaboration II (grant P-LS-SPII/18), from the German Bundesministerium für Bildung und Forschung via the ASTRONET project STAR FORMAT (grant 05A09VHA), and by a Frontier grant of Heidelberg University sponsored by the German Excellence Initiative.