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How realistic UV spectra and X-rays suppress the abundance of direct collapse black holes

Observations of high redshift quasars at $z>6$ indicate that they harbor 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 halos 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 ($J_{21}^{\rm crit}$) for realistic spectra of Pop II stars through three-dimensional cosmological simulations. We derive the dependence of $J_{21}^{\rm crit}$ 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 $J_{21}^{\rm crit}$ is a few times $\rm 10^4$ and only weakly depends on the adopted radiation spectra in the range between $T_{\rm rad}=2 \times 10^4-10^5$ K. For three simulated halos of a few times $\rm 10^{7}$~M$_{\odot}$, $J_{21}^{\rm crit}$ varies from $\rm 2 \times 10^4 - 5 \times 10^4$. The impact of X-ray ionization is almost negligible and within the expected scatter of $J_{21}^{\rm crit}$ for background fluxes of $J_{\rm X,21} \leq 0.1$. The computed estimates of $J_{21}^{\rm crit}$ 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 holes density. However, the sites with moderate amounts of $\rm H_2$ cooling may still form massive objects sufficient to be compatible with observations.