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The polytropic equation of state (EOS) of primordial gas clouds with modestenrichment is computed, motivated by the recent observations of veryFe-deficient stars, [Fe/H]~10^{-3.5}-10^{-5}, such as HE 0107-5240 and CS29498-043. These stars are overabundant, relative to Fe, in C and O. We assumethat the observed abundances of species like C, O, Si and Fe are representativeof the gas from which the currently observed metal-deficient stars formed.Under this assumption, we find that this primordial metal abundance pattern hasprofound consequences for the thermal balance and chemical composition of thegas, and hence for the EOS of the parental cloud. The polytopic EOS is soft forlow, [O/H]<10^{-3}, oxygen abundances, but stiffens to a polytropic index gammalarge than unity for [O/H]>10^{-2} due to the large opacity in the CO and H2Ocooling lines. It is further found that a regulating role is played by thepresence and temperature of the dust, even when the overall carbon abundance isonly [C/H]~10^{-2}. When the dust is warmer than the gas, a region withgamma~1.2 results around a density of ~10^4 cm^{-3}. When the dust is colderthan the gas, a region with gamma~0.8 is found for a density of ~10^6 cm^{-3}.Implications for the primordial initial mass function (IMF) as well as the IMFin starburst galaxies, where the metallicity is super-solar, are explored andrelated to processes that influence the temperature of the ambient dust.Comment: Version accepted by Ap

The Polytropic Equation of State of Primordial Gas Clouds