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 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 \sim 2 larger at z \sim 1 and 5–6 times more massive relative to their host galaxies at z \sim 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 \sim 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 \sim 2 can not 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 .