Abstract

Observations of high-redshift quasars at z $\gt$ 6 indicate that they harbour supermassive black holes (SMBHs) of a billion solar masses. The direct collapse scenario has emerged as the most plausible way to assemble SMBHs. The nurseries for the direct collapse black holes are massive primordial haloes illuminated with an intense UV flux emitted by Population II (Pop II) stars. In this study, we compute the critical value of such a flux (J\_$\{$21$\}$\^{}crit) for realistic spectra of Pop II stars through three-dimensional cosmological simulations. We derive the dependence of J\_$\{$21$\}$\^{}crit on the radiation spectra, on variations from halo to halo, and on the impact of X-ray ionization. Our findings show that the value of J\_$\{$21$\}$\^{}crit is a few times 10$^{4}$ and only weakly depends on the adopted radiation spectra in the range between T$_{rad}$ = 2 {\times} 10$^{4}$and10$^{5}$ K. For three simulated haloes of a few times 10$^{7}$ M$_{⊙}$, J\_$\{$21$\}$\^{}crit varies from 2 {\times} 10$^{4}$ to 5 {\times} 10$^{4}$. The impact of X-ray ionization is almost negligible and within the expected scatter of J\_$\{$21$\}$\^{}crit for background fluxes of J$_{X, 21}$ {\le} 0.1. The computed estimates of J\_$\{$21$\}$\^{}crit have profound implications for the quasar abundance at z = 10 as it lowers the number density of black holes forming through an isothermal direct collapse by a few orders of magnitude below the observed black hole density. However, the sites with moderate amounts of H$_{2}$ cooling may still form massive objects sufficient to be compatible with observations.