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 greater than or similar to 10 in the primordial gas in the case where H-2 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 x 10(-4) Z(circle dot)) 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 H-2 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 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) similar to 10(-10) cMpc(-3) at z = 10, which is roughly consistent with the observed number density of high-redshift SMBHs n(SMBH) similar to 10(-9) cMpc(-3) at z similar to 6, considering large uncertainties in the estimation.