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To constrain the nature of the very first stars, we investigate the collapse and fragmentation of primordial, metal-free gas clouds. We explore the physics of primordial star formation by means of threedimensional simulations of the dark matter and gas components, using smoothed particle hydrodynamics, under a wide range of initial conditions, including the initial spin, the total mass of the halo, the redshift of virialization, the power spectrum of the DM fluctuations, the presence of HD cooling, and the number of particles employed in the simulation. We find characteristic values for the temperature, T ∼ a few 100 K, and the density, n ∼ 10 3 − 10 4 cm −3 , characterising the gas at the end of the initial free-fall phase. These values are rather insensitive to the initial conditions. The corresponding Jeans mass is M J ∼ 10 3 M ⊙ . The existence of these characteristic values has a robust explanation in the microphysics of H 2 cooling, connected to the minimum temperature that can be reached with the H 2 coolant, and to the critical density at which the transition takes place between levels being populated according to NLTE, and according to LTE.In all cases, the gas dissipatively settles into an irregular, central configuration which has a filamentary and knotty appearance. The fluid regions with the highest densities are the first to undergo runaway collapse due to gravitational instability, and to form clumps with initial masses ∼ 10 3 M ⊙ , close to the characteristic Jeans scale. These results suggest that the first stars might have been quite massive, possibly even very massive with M ⋆ ∼ > 100 M ⊙ .After a gas element has undergone runaway collapse, and has reached densities in excess of 10 8 cm −3 , a sink particle is created. This procedure allows us to follow the evolution of the overall system beyond the point where the first nonlinear region would otherwise force the calculation to a halt. These later evolutionary stages, during which the clumps grow in mass due to accretion and merging with other clumps, are quite sensitive to the initial conditions. The key process in building up very massive clumps, with masses up to a few times 10 4 M ⊙ , is merging between clumps. Since the merging rate sensitively depends on the density of the gas, halos with the highest degree of central concentration are able to assemble the most massive clumps. Among these are halos with a low spin (λ ≃ 0.01), and with DM fluctuations imprinted according to a white-noise spectrum.

We review recent theoretical results on the formation of the first stars in the universe, and emphasize related open questions. In particular, we discuss the initial conditions for Population III star formation, as given by variants of the cold dark matter cosmology. Numerical simulations have investigated the collapse and the fragmentation of metal-free gas, showing that the first stars were predominantly very massive. The exact determination of the stellar masses, and the precise form of the primordial initial mass function, is still hampered by our limited understanding of the accretion physics and the protostellar feedback effects. We address the importance of heavy elements in bringing about the transition from an early star formation mode dominated by massive stars, to the familiar mode dominated by low mass stars, at later times. We show how complementary observations, both at high redshifts and in our local cosmic neighborhood, can be utilized to probe the first epoch of star formation.Comment: 38 pages, 10 figures, draft version for 2004 Annual Reviews of Astronomy and Astrophysics, high-resolution version available at http://cfa-www.harvard.edu/~vbromm