Direct collapse to supermassive black hole seeds: the critical conditions for suppression of H2 cooling

Observations of high-redshift quasars imply the presence of supermassive black holes (SMBHs) already at z ∼ 7.5. An appealing and promising pathway to their formation is the direct collapse scenario of a primordial gas in atomic-cooling haloes at z ∼ 10-20, when the H₂ formation is inhibited by a strong background radiation field, whose intensity exceeds a critical value, J_crit. To estimate J_crit, typically, studies have assumed idealized spectra, with a fixed ratio of H₂ photodissociation rate k_H_2 to the H^- photodetachment rate k_H^-. This assumption, however, could be too narrow in scope as the nature of the background radiation field is not known precisely. In this work we argue that the critical condition for suppressing the H₂ cooling in the collapsing gas could be described in a more general way by a combination of k_H_2 and k_H^- parameters, without any additional assumptions about the shape of the underlying radiation spectrum. By performing a series of cosmological zoom-in simulations covering a wide range of relevant k_H_2 and k_H^- parameters, we examine the gas flow by following evolution of basic parameters of the accretion flow. We test under what conditions the gas evolution is dominated by H₂ and/or atomic cooling. We confirm the existence of a critical curve in the k_H_2-k_H^- plane and provide an analytical fit to it. This curve depends on the conditions in the direct collapse, and reveals domains where the atomic cooling dominates over the molecular cooling. Furthermore, we have considered the effect of H₂ self-shielding on the critical curve, by adopting three methods for the effective column density approximation in H₂. We find that the estimate of the characteristic length scale for shielding can be improved by using λ_Jeans25, which is 0.25 times that of the local Jeans length, which is consistent with previous one-zone modelling.