An Eddington ratio-driven origin for the ${\rm L}_{\rm X}-{\rm M}_{*}$ relation in quiescent and star forming active galaxies

A mild correlation exists in active galaxies between the mean black hole accretion, as traced by the mean X-ray luminosity $\left<{\rm L}_{\rm X}\right>$, and the host galaxy stellar mass M$_*$, characterised by a normalisation steadily decreasing with cosmic time and lower in more quiescent galaxies. We create comprehensive semi-empirical mock catalogues of active black holes to pin down which parameters control the shape and evolution of the $\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation of X-ray detected active galaxies. We find that the normalisation of the $\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation is largely independent of the fraction of active galaxies (the duty cycle), but strongly dependent on the mean Eddington ratio, when adopting a constant underlying M$_{\rm BH}-{\rm M}_*$ relation as suggested by observational studies. The data point to a decreasing mean Eddington ratio with cosmic time and with galaxy stellar mass at fixed redshift. Our data can be reproduced by black holes and galaxies evolving on similar M$_{\rm BH}-{\rm M}_*$ relations but progressively decreasing their average Eddington ratios, mean X-ray luminosities, and specific star formation rates, when moving from the starburst to the quiescent phase. Models consistent with the observed $\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation and independent measurements of the mean Eddington ratios, are characterised by M$_{\rm BH}-{\rm M}_*$ relations lower than those derived from dynamically measured local black holes. Our results point to the $\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation as a powerful diagnostic to: 1) probe black hole-galaxy scaling relations and the level of accretion onto black holes; 2) efficiently break the degeneracies between duty cycles and accretion rates in cosmological models of black holes.