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 direct-collapse 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 this work, 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_LW,21^crit (in units of 10^-21 erg s^-1 Hz^-1 sr^-1 cm^-2) for realistic spectra of metal-poor galaxies. We find that J^crit is not sensitive to the age or metallicity for the constant star formation galaxies with J_LW,21^crit=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_LW,21^crit= 1000-1400. We also find that J^crit is solely determined by the ratio between the H^- and H₂ photodissociation rate coefficients, k_H^-,pd/k_H_2,pd, with which we develop a formula to estimate J^crit for a given spectrum. The higher value of J^crit 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.