Population III star formation: multiple gas phases prevent the use of an equation of state at high densities

Advanced primordial chemistry networks have been developed to model the collapse of metal-free baryonic gas within the gravitational well of dark matter (DM) halos and its subsequent collapse into Population III stars. At the low densities of 10$^{-26}$-10$^{-21}$ g cm$^{-3}$ (10$^{-3}$-10$^2$ cm$^{-3}$) the collapse is dependent on H$_2$ production, which is a function of the compressional heating provided by the DM potential. Once the gas decouples from the DM, the temperature-density relationship follows a well established path dictated by various chemical reactions until the formation of the protostar at 10$^{-4}$ g cm$^{-3}$ (10$^{19}$ cm$^{-3}$). Here we explore the feasibility of replacing the chemical network (CN) with a barotropic equation of state (EoS) just before the formation of the first protostar, to reduce the computational load of simulating the further fragmentation, evolution and characteristics of the very high density gas. We find a significant reduction in fragmentation when using the EoS. The EoS method produces a protostellar mass distribution that peaks at higher masses when compared to CN runs. The change in fragmentation behaviour is due to a lack of cold gas falling in through the disc around the first protostar when using an EoS. Despite this, the total mass accreted across all sinks was invariant to the switch to an EoS, hence the star formation rate (M$_\odot$ yr$^{-1}$) is accurately predicted using an EoS. The EoS routine is approximately 4000 times faster than the CN, however this numerical gain is offset by the lack of accuracy in modelling secondary protostar formation and hence its use must be considered carefully.