Simulating the formation of massive seed black holes in the early Universe. III: The influence of X-rays

The direct collapse black hole (DCBH) model attempts to explain the observed number density of supermassive black holes in the early Universe by positing that they grew from seed black holes with masses of $10^{4}$-$10^{5} \: {\rm M_{\odot}}$ that formed by the quasi-isothermal collapse of gas in metal-free protogalaxies cooled by atomic hydrogen emission. For this model to work, H$_{2}$ formation must be suppressed in at least some of these systems by a strong extragalactic radiation field. The predicted number density of DCBH seeds is highly sensitive to the minimum value of the ultraviolet (UV) flux required to suppress H$_{2}$ formation, $J_{\rm crit}$. In this paper, we examine how the value of $J_{\rm crit}$ varies as we vary the strength of a hypothetical high-redshift X-ray background. We confirm earlier findings that when the X-ray flux $J_{\rm X}$ is large, the critical UV flux scales as $J_{\rm crit} \propto J_{\rm X}^{1/2}$. We also carefully explore possible sources of uncertainty arising from how the X-rays are modelled. We use a reaction-based reduction technique to analyze the chemistry of H$_{2}$ in the X-ray illuminated gas and identify a critical subset of 35 chemical reactions that must be included in our chemical model in order to predict $J_{\rm crit}$ accurately. We further show that $J_{\rm crit}$ is insensitive to the details of how secondary ionization or He$^{+}$ recombination are modelled, but does depend strongly on the assumptions made regarding the column density of the collapsing gas.