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.