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The qualitative discussion of the previous section can be turned into a quantitative estimate for MBH as follows. The quasar population produces an integrated comoving energy density of

Equation 1 (1)

where Phi(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 epsilon, the equivalent present-day mass density is rhou = u / (epsilon c2) = 2.2 x 104 epsilon-1 Msun Mpc-3. Comparison of rhou with the overall galaxy luminosity density, rhog appeq 1.4 x 108 h Lsun Mpc-3, where the Hubble constant is H0 = 100 h km s-1 Mpc-1, implies that a typical nearby bright galaxy (luminosity L* appeq 1010 h-2 Lsun) should contain a dead quasar of mass MBH ~ 1.6 x 106 epsilon-1 h-3 Msun. Accretion onto a BH is expected to produce energy with an efficiency of epsilon ~ 0.1, and the best estimate of h is 0.71 ± 0.06. Therefore the typical BH should have a mass of ~ 107.7 Msun. BHs in dwarf ellipticals should have masses of ~ 106 Msun.

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 MBH, 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 ~ 1047 erg s-1 ~ 1014 Lsun require BHs of mass MBH gtapprox 109 Msun. These arguments define the parameter range of interest: MBH ~ 106 to 109.5 Msun. The highest-mass BHs are likely to be rare, but low-mass objects should be ubiquitous. Are they?

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