Population III Star Formation and IMF

We review recent 3D cosmological hydrodynamic simulations of primordial star formation from cosmological initial conditions (Pop III.1) and from initial conditions that have been altered by radiative feedback from stellar sources (Pop III.2). We concentrate on simulations that resolve the formation of the gravitationally unstable cloud cores in mini-halos over the mass range $10^5 < M/\Msun < 10^7 $ and follow their evolution to densities of at least $10^{10} \cmm3$ and length scales of $<10^{-2}$ pc such that accretion rates can be estimated. The advent of ensembles of such simulations exploring a variety of conditions permits us to assess the robustness of the standard model for Pop III.1 star formation and investigate scatter in their formation redshifts and accretion rates, thereby providing much needed information about the Pop III IMF. The simulations confirm the prediction that Pop III.1 stars were massive ($\sim 100 \Msun$), and form in isolation in primordial mini-halos. Simulations of Pop III.2 star forming in relic HII regions suggest somewhat lower masses ($\sim 30 \Msun$) which may help explain the chemical abundances of extremely metal poor stars. We note that no 3D simulation at present has achieved stellar density let alone followed the entire accretion history of the star in any scenario, and thus the IMF of Pop III stars remains poorly determined theoretically.