Constraining the supermassive black holes evolution through the
  continuity equation
  The population of supermassive black holes (SMBHs) is composed by quiescent
SMBHs, such as those seen in local galaxies including the Milky Way's, and
active ones, resulting in quasars and active galactic nuclei (AGN). Outside our
neighbourhood, all the information we have on SMBHs is derived from quasars and
AGN, giving us a partial view. We study the evolution of the SMBH population,
total and active, by the continuity equation, backwards in time from z=0 to
z=4. Type-1 and type-2 AGN are differentiated in the model on the basis of the
Eddington ratio distribution, chosen on the basis of observational estimates.
The duty cycle is obtained by matching the luminosity function of quasars, and
the average radiative efficiency is the only free parameter in the model. For
higher radiative efficiencies (>~0.07) a large fraction of the SMBH population,
most of them quiescent, must already be in place by z=4. For lower radiative
efficiencies (~0.05), the duty cycle increases with the redshift and the SMBH
population evolves dramatically since z=4. The mass function of active SMBHs
does not depend on the choice of the radiative efficiency or of the local SMBH
mass function, but it is mainly determined by the quasar luminosity function,
once the Eddington ratio distribution is fixed. Only a direct measurement of
the total BHMF at redshifts z>~2 could break these degeneracies giving
important constraints on the average radiative efficiency. Focusing on type-1
AGN, for which observational estimates of the mass function and Eddington ratio
distribution exist at various redshift, models with lower radiative
efficiencies reproduce better the high-mass end of the mass function at high-z,
but they tend to over-predict it at low-z, and vice-versa for models with
higher radiative efficiencies.
