It has been proposed that supermassive black holes (SMBHs) are originated from directcollapse black holes (DCBHs) that are formed at z≲10 in the primordial gas in the case where H2 cooling is suppressed by strong external radiation. In thiswork,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 that 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, for the young (the age < 100 Myr) and/or extremely metal poor (Z < 5 × 10-4 Z) instantaneous starburst galaxies, such dependence is not strong and JcritLW,21 = 1000-1400. We also find that Jcrit is solely determined by the ratio between the H- and H2 photodissociation rate coefficients, kH-,pd/kH2,pd, with which we develop a formula to estimate Jcrit for a given spectrum. The higher value of Jcrit for the realistic spectra than those expected in the literature significantly reduces the estimated DCBH number density nDCBH. By extrapolating the result of Dijkstra, Ferrara & Mesinger, we obtain nDCBH ~ 10-10 cMpc-3 at z = 10, which is roughly consistent with the observed number density of high-redshift SMBHs nSMBH ~ 10-9 cMpc-3 at z ~ 6, considering large uncertainties in the estimation.
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