An analytical model for the history of cosmic star formation

We use simple analytic reasoning to identify physical processes that drive the evolution of the cosmic star formation density in cold dark matter universes. Based on our analysis, we formulate a model to characterise the redshift dependence of the star formation history and compare it to results obtained from a set of hydrodynamic simulations which include star formation and feedback. At early times, densities are sufficiently high and cooling times sufficiently short that abundant quantities of star-forming gas are present in all dark matter halos that can cool by atomic processes. Consequently, the star formation density generically rises exponentially as z decreases, independent of the details of the physical model for star formation, but dependent on the normalisation and shape of the cosmological power spectrum. This part of the evolution is dominated by gravitationally driven growth of the halo mass function. At low redshifts, densities decline as the universe expands to the point that cooling is inhibited, limiting the amount of star-forming gas available. We find that in this regime the star formation rate scales approximately in proportion to the cooling rate within halos. We derive analytic expressions for the full star formation history and show that they match our simulation results to better than ~10%. Using various approximations, we reduce the expressions to a simple analytic fitting function that can be used to compute global cosmological quantities that are directly related to the star formation history. As examples, we consider the integrated stellar density, the supernova and gamma-ray burst rates observable on Earth, the metal enrichment history of the Universe, and the density of compact objects. (abridged)