A mild correlation exists in active galaxies between the mean black hole accretion, as traced by the mean X-ray luminosity and the host galaxy stellar mass M∗, characterised by a normalization 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 - M∗ relation of X-ray-detected active galaxies. We find that the normalization of the - 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 MBH - 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 MBH - 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 - M∗ relation and independent measurements of the mean Eddington ratios are characterised by MBH - M∗ relations lower than those derived from dynamically measured local black holes. Our results point to the - M∗ relation as a powerful diagnostic to: (1) probe black hole-galaxy scaling relations and the level of accretion on to black holes; (2) efficiently break the degeneracies between duty cycles and accretion rates in cosmological models of black holes.
