The qualitative discussion of the previous section can be turned into a
quantitative estimate for *M*_{BH} as follows. The quasar
population produces an integrated comoving energy density of

where (*L, z*) is the comoving
density of quasars of luminosity *L* at
redshift *z* and *t* is cosmic time. For a radiative energy
conversion efficiency of
, the equivalent
present-day mass density is
_{u} =
*u* / (
*c*^{2}) = 2.2 x
10^{4} ^{-1}
*M*_{} Mpc^{-3}.
Comparison of _{u} with the overall galaxy luminosity density,
_{g}
1.4 x 10^{8} *h*
*L*_{}
Mpc^{-3}, where the Hubble constant is
*H*_{0} = 100 *h* km s^{-1} Mpc^{-1},
implies that a typical nearby bright
galaxy (luminosity *L**
10^{10} *h*^{-2}
*L*_{}) should
contain a dead quasar of mass *M*_{BH} ~ 1.6 x 10^{6}
^{-1}
*h*^{-3}
*M*_{}.
Accretion onto a BH is expected to produce energy with an efficiency of
~ 0.1, and the best
estimate of *h* is 0.71 ± 0.06. Therefore
the typical BH should have a mass of ~ 10^{7.7}
*M*_{}. BHs in dwarf
ellipticals should have masses of ~ 10^{6}
*M*_{}.

In fact, the brightest quasars must have had much higher masses. A BH
cannot accrete arbitrarily large amounts of mass to produce arbitrarily high
luminosities. For a given *M*_{BH}, there is a maximum
accretion rate above
which the radiation pressure from the resulting high luminosity blows away the
accreting matter. This ``Eddington limit'' is discussed in the preceeding
article. Eddington luminosities of *L* ~ 10^{47} erg
s^{-1} ~ 10^{14}
*L*_{} require BHs
of mass *M*_{BH}
10^{9}
*M*_{}. These
arguments define the parameter range of interest: *M*_{BH}
~ 10^{6} to 10^{9.5}
*M*_{}. The
highest-mass BHs are likely to be rare, but low-mass
objects should be ubiquitous. Are they?