# Modeling The Pollution Of Pristine Gas In The Early Universe

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We conduct a comprehensive theoretical and numerical investigation of the pollution of pristine gas in turbulent flows, designed to provide useful new tools for modeling the evolution of the first generation of stars. The properties of such Population III (Pop III) stars are thought to be very different than those of later stellar generations, because cooling is dramatically different in gas with a metallicity below a critical value Z(c), which lies between similar to 10(-6) and similar to 10(-3) Z(circle dot). The critical value is much smaller than the typical overall average metallicity, < Z >, and therefore the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in determining the transition from Pop III to normal star formation. The small critical value, Z(c), corresponds to the far left tail of the probability distribution function (PDF) of the metal abundance. Based on closure models for the PDF formulation of turbulent mixing, we derive evolution equations for the fraction of gas, P, lying below Z(c), in statistically homogeneous compressible turbulence. Our simulation data show that the evolution of the pristine fraction P can be well approximated by a generalized "self-convolution" model, which predicts that (P) over dot = -(n/tau(con))P(1 - P-1/n), where n is a measure of the locality of the mixing or PDF convolution events and the convolution timescale tau(con) is determined by the rate at which turbulence stretches the pollutants. Carrying out a suite of numerical simulations with turbulent Mach numbers ranging from M = 0.9 to 6.2, we are able to provide accurate fits to n and tau(con) as a function of M, Z(c)/< Z >, and the length scale, L-p, at which pollutants are added to the flow. For pristine fractions above P = 0.9, mixing occurs only in the regions surrounding blobs of pollutants, such that n = 1. For smaller values of P, n is larger as the mixing process becomes more global. We show how these results can be used to construct one-zone models for the evolution of Pop III stars in a single high-redshift galaxy, as well as subgrid models for tracking the evolution of the first stars in large cosmological numerical simulations.