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 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 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 J^crit 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, such dependence for the instantaneous starburst galaxies is weak for the young or extremely metal-poor galaxies: J^crit_LW,21 = 1000-1400 for the young galaxies and J^crit_LW,21 approx 1400 for the extremely metal-poor galaxies. The typical value of J^crit for the realistic spectra is higher than those expected in the literature, which affects the estimated DCBH number density n_DCBH. By extrapolating the result of Dijkstra, Ferrara and Mesinger, we obtain n_DCBH sim 10^-10 cMpc^-3 at z = 10, although there is still large uncertainty in this estimation.