Optimal neighbourhood to nurture giants: a fundamental link between star-forming galaxies and direct collapse black holes

Massive 10^4-5 M_⊙ black hole seeds resulting from the direct collapse of pristine gas require a metal-free atomic cooling halo with extremely low H₂ fraction, allowing the gas to cool isothermally in the presence of atomic hydrogen. In order to achieve this chemo-thermodynamical state, the gas needs to be irradiated by both Lyman-Werner (LW) photons in the energy range of 11.2-13.6 eV capable of photodissociating H₂ and 0.76 eV photons capable of photodetaching H^-. Employing cosmological simulations capable of creating the first galaxies in high resolution, we explore if there exists a subset of galaxies that favour direct collapse black hole (DCBH) formation in their vicinity. We find a fundamental relation between the maximum distance at which a galaxy can cause DCBH formation and its star formation rate (SFR), which automatically folds in the chemo-thermodynamical effects of both H₂ photodissociation and H^- photodetachment. This is in contrast to the approximately three order of magnitude scatter seen in the LW flux parameter computed at the maximum distance, which is synonymous with a scatter in `J_crit'. Thus, computing the rates and/or the LW flux from a galaxy is no longer necessary to identify neighbouring sites of DCBH formation, as our relation allows one to distinguish regions where DCBH formation could be triggered in the vicinity of a galaxy of a given SFR.