The critical radiation intensity for direct collapse black hole formation: Dependence on the radiation spectral shape

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 J^crit 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 J^crit_LW,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 J^crit_LW,21 = 1300-1400, while J^crit 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 J^crit_LW,21 = 1000-1400. We also find that Jcrit is solely determined by the ratio between the H^- and H₂ photodissociation rate coefficients, k_H-,pd/k_H2,pd, with which we develop a formula to estimate J^crit 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 n_DCBH. By extrapolating the result of Dijkstra, Ferrara & Mesinger, we obtain n_DCBH ~ 10^-10 cMpc^-3 at z = 10, which is roughly consistent with the observed number density of high-redshift SMBHs n_SMBH ~ 10^-9 cMpc^-3 at z ~ 6, considering large uncertainties in the estimation.