Empirical constraints on the evolution of the relationship between black hole and galaxy mass: scatter matters

I investigate whether useful constraints on the evolution of the relationship between galaxy mass (m(gal)) and black hole (BH) mass (m(BH)) can be obtained from recent measurements of galaxy stellar mass functions and quasi-stellar object (QSO) bolometric luminosity functions at high redshift. I assume a simple power-law relationship between m(gal) and m(BH), as implied by BH mass measurements at low redshift, and consider only evolution in the zero-point of the relation. I argue that one can obtain a lower limit on the zero-point evolution by assuming that every galaxy hosts a BH, shining at its Eddington rate. One can obtain an upper limit by requiring that the number of massive BH at high redshift does not exceed that observed locally. I find that, under these assumptions, and neglecting scatter in the m(gal)-m(BH) relation, BH must have been a factor of similar to 2 larger at z similar to 1 and five to six times more massive relative to their host galaxies at z similar to 2. However, accounting for intrinsic scatter in m(gal)-m(BH) considerably relaxes these constraints. With a logarithmic scatter of 0.3-0.5 dex in m(BH) at fixed m(gal), similar to estimates of the intrinsic scatter in the observed relation today, there are enough massive BH to produce the observed population of luminous QSOs at z similar to 2 even in the absence of any zero-point evolution. Adopting more realistic estimates for the fraction of galaxies that host active BH and the Eddington ratios of the associated quasars, I find that the zero-point of the m(gal)-m(BH) relation at z similar to 2 cannot be much more than a factor of 2 times larger than the present-day value, as the number of luminous quasars predicted would exceed the observed population.