Revisiting Soltan's argument based on a semi-analytical model for galaxy
  and black hole evolution
  We show the significance of the super-Eddington accretion for the cosmic
growth of supermassive black holes (SMBHs) with a semi-analytical model for
galaxy and black hole evolution. The model explains various observed properties
of galaxies and active galactic nuclei at a wide redshift range. By tracing the
growth history of individual SMBHs, we find that the fraction of the SMBH mass
acquired during the super-Eddington accretion phases to the total SMBH mass
becomes larger for less massive black holes and at higher redshift. Even at z =
0, SMBHs with > 1e+9 Msun have acquired more than 50% of their mass by
super-Eddington accretions, which is apparently inconsistent with classical
Soltan's argument. However, the mass-weighted radiation efficiency of SMBHs
with > 1e+8 Msun obtained with our model, is about 0.08 at z = 0, which is
consistent with Soltan's argument within the observational uncertainties. We,
therefore, conclude that Soltan's argument cannot reject the possibility that
SMBHs are grown mainly by super-Eddington accretions.
