Formation of the first stars

From studying the cosmic microwave background, we know our Universe started out very simple. It was by and large homogeneous and isotropic, with small fluctuations that can be described by linear perturbation theory. In stark contrast, the Universe today is highly structured on a vast range of length and mass scales. In the evolution towards increasing complexity, the formation of the first stars marks a primary transition event. The first generation of stars, the so-called Population III (or Pop. III) build up from truly metal-free primordial gas. They have long been thought to live short, solitary lives, with only one massive star forming per halo. However, in recent years this simple picture has undergone substantial revision, and we now understand that stellar birth in the early Universe is subject to the same complexity as star formation at present days. In this chapter, I review the current state of the field. I begin by introducing the basics concepts of star-formation theory and by discussing the typical environment in which Pop. III stars are thought to form. Then I argue that the accretion disk that builds up in the center of a halo is likely to fragment, resulting in the formation of a cluster of stars with a wide range of masses, and I speculate about how this process may be influenced by stellar feedback, the presence of magnetic fields, the energy input from dark matter annihilation, and the occurrence of large- scale streaming velocities between baryons and dark matter. Finally, I discuss direct and indirect constraints on Pop. III star formation from high-redshift observations and from the search for extremely metal-poor stars in the Milky Way and its satellites.