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It has been proposed that supermassive black holes (SMBHs) are originated from direct-collapse black holes (DCBHs) that are formed at z gtrsim 10 in the primordial gas in the case that H2 cooling is suppressed by strong external radiation. In this work, we study the critical specific intensity Jᶜrit required for DCBH formation for various radiation spectral shapes by a series of one-zone calculations of a collapsing primordial- gas cloud. We calculate the critical specific intensity at the Lyman-Werner (LW) bands JᶜritLW, 21 (in units of 10^-21 erg s^-1 Hz^-1 sr^-1 cm^-2) for realistic spectra of metal-poor galaxies. We find Jᶜrit is not sensitive to the age or metallicity for the constant star formation galaxies with JᶜritLW, 21 = 1300-1400, while Jᶜrit decreases as galaxies become older or more metal-enriched for the instantaneous starburst galaxies. However, such dependence for the instantaneous starburst galaxies is weak for the young or extremely metal-poor galaxies: JᶜritLW, 21 = 1000-1400 for the young galaxies and JᶜritLW, 21 approx 1400 for the extremely metal-poor galaxies. The typical value of Jᶜrit for the realistic spectra is higher than those expected in the literature, which affects the estimated DCBH number density nDCBH. By extrapolating the result of Dijkstra, Ferrara and Mesinger, we obtain nDCBH sim 10^-10 cMpc^-3 at z = 10, although there is still large uncertainty in this estimation.
Sugimura et al. (Mon,) studied this question.