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 and black hole (BH) mass can be obtained from recent
measurements of galaxy stellar mass functions and QSO bolometric luminosity
functions at high redshift. I assume a simple power-law relationship between
galaxy mass and BH mass, 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 BH-galaxy mass relation, BH must have been a factor of about 2
times larger at z=1 and 5 to 6 times more massive relative to their host
galaxies at z=2. However, accounting for intrinsic scatter in the BH-galaxy
mass relationship considerably relaxes these constraints. With a logarithmic
scatter of 0.3 to 0.5 dex in black hole mass at fixed galaxy mass, 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=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 BH-galaxy
mass relation at z=2 cannot be much more than a factor of two times larger than
the present-day value, as the number of luminous quasars predicted would exceed
the observed population.
