An Upper Limit to the Degree of Evolution Between Supermassive Black   Holes and their Host Galaxies
  We describe a model-independent integral constraint which 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_BH(min) 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 M_BH/M_host relation) in all cases, and provides tighter constraints than have generally been obtained previously, ruling out at >6 sigma observational and theoretical estimates suggesting that M_BH/M_host was significantly larger at high redshifts than locally, although relatively weak (factor ~2 by z=2) evolution is still allowed. 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. 
Philip F. Hopkins, Brant Robertson, Elisabeth Krause, Lars Hernquist,   Thomas J. Cox (Harvard/CfA)