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 rate, ρ̇?, in cold dark matter universes. Based on our analysis, we formulate a model to characterise the redshift dependence of ρ̇? and compare it to results obtained from a set of hydrodynamic simulations which include star formation and feedback. We find that the cosmic star formation rate is described by two regimes. 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, ρ̇? 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 as ρ̇? ∝ H(z), in proportion to the cooling rate within halos. We demonstrate that the existence of these two regimes leads to a peak in the star formation rate at an intermediate redshift z = zpeak. We discuss how the location of this peak depends on our model parameters. Only star formation efficiencies that are unrealistically low would delay the peak to z ' 3 or below, and we show that the peak cannot occur above a limiting redshift of z ≈ 8.7. For the star formation efficiency adopted in our numerical simulations, zpeak ≈ 5− 6. 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 for ρ̇? 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 (GRB) rates observable on Earth, the metal enrichment history of the Universe, and the density of compact objects. We also briefly discuss the expected dependence of the star formation history on cosmological parameters and the physics of the gas.