An Upper Limit to the Degree of Evolution between Supermassive Black Holes and Their Host Galaxies

We describe a model-independent integral constraint that defines an upper limit to the allowed degree of evolution in the ratio of black hole (BH) mass to host galaxy luminosity or mass, as a function of redshift. Essentially, if the BH/host ratio is excessive at redshift z, then it would imply that the total mass density in BHs above some M is larger at that redshift than at z = 0, which is impossible. This argument requires no knowledge of host or BH properties, only a lower limit to the observed luminosity density in the brightest galaxies at some z. We calculate this upper limit from a variety of luminosity and mass functions in different bands from redshifts z = 0-2. We show that it is consistent with passive evolution of spheroid populations (with a fixed MBH/Mhost relation) in all cases, and provides tighter constraints than have generally been obtained previously, ruling out at ≳6 σ observational and theoretical estimates suggesting that MBH/Mhost was significantly larger at high redshifts than locally. We discuss a variety of possible "loopholes" or changes in the BH/host populations and correlations, and show that they typically lower the upper limits and strengthen our conclusions.