Notes. Our calculated values of the critical LW radiation field J_21,crit required for DCBH formation, for two different background radiation temperatures, with and without accounting for the impact of photodetachment of H⁻ by Lyman α cooling radiation, which is assumed to be either uniform (ℬ = 1) or strongly centrally concentrated (ℬ = 10). While the photodetachment rate due to the background radiation is relatively high already when T_rad is relatively low, the detachment rate due to Lyman α feedback comes to dominate the rate due to the background when T_rad is high, in particular at high gas densities (see Fig. 3).