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

Massive $10^{4-5}\rm\ M_{\odot}$ black hole seeds resulting from the \textit{direct} collapse of pristine gas require a metal-free atomic cooling halo with extremely low H$_2$ 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 $11.2-13.6$ eV capable of photodissociating H$_2$, 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 that favours 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 H$_2$ photo-dissociation and H$^-$ photo-detachment. This is in contrast to the scatter in the LW flux parameter seen at the maximum distance. It shows up to a 3 order of magnitude scatter, which can be interpreted as 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.