The polytropic equation of state ( EOS ) of primordial gas clouds with modest enrichment is computed , motivated by the recent observations of very Fe-deficient stars , [ Fe/H ] \sim 10 ^ { -3.5 } -10 ^ { -5 } , such as HE 0107-5240 and CS 29498-043 .
These stars are overabundant , relative to Fe , in C and O .
We assume that the observed abundances of species like C , O , Si and Fe are representative of the gas from which the currently observed metal-deficient stars formed .
Under this assumption , we find that this primordial metal abundance pattern has profound consequences for the thermal balance and chemical composition of the gas , and hence for the EOS of the parental cloud .
The polytopic EOS is soft for low , [ O/H ] < 10 ^ { -3 } , oxygen abundances , but stiffens to a polytropic index \gamma large than unity for [ O/H ] > 10 ^ { -2 } due to the large opacity in the CO and H _ { 2 } O cooling lines .
It is further found that a regulating role is played by the presence and temperature of the dust , even when the overall carbon abundance is only [ C/H ] \sim 10 ^ { -2 } .
When the dust is warmer than the gas , a region with \gamma \sim 1.2 results around a density of \sim 10 ^ { 4 } cm ^ { -3 } .
When the dust is colder than the gas , a region with \gamma \sim 0.8 is found for a density of \sim 10 ^ { 6 } cm ^ { -3 } .
Implications for the primordial initial mass function ( IMF ) as well as the IMF in starburst galaxies , where the metallicity is super-solar , are explored and related to processes that influence the temperature of the ambient dust .