On the occupation fraction of seed black holes in high-redshift dark matter halos

It is well known that an initial population of seed black holes (BHs), formed in the nuclei of low-mass galaxies at high redshift, can simultaneously explain, through their subsequent growth by mergers and accretion, both the observed evolution of the quasar luminosity function (LF) and the distribution of remnant supermassive black hole (SMBH) masses measured in local galactic nuclei. Here we consider three very different initial conditions for this scenario: models in which initial seed BHs form in either all, or only a small fraction (fbh = 0.1 or 0.01) of high-redshift dark matter halos (with Mhalo = 5 × 109 M☉ at z = 6–10). We show that with a suitable and relatively minor adjustment of two global physical parameters (the radiative efficiency and mass accretion time-scale of quasar episodes), models with fbh ≈ 0.1 and 1 can accurately reproduce the observed quasar LF at redshifts 0 < z ≲ 6, as well as the remnant SMBH mass function at z = 0. However, SMBHs remain rare, and the normalization of the high-z quasar LF and the local SMBH mass function are both significantly underpredicted, if fbh ≲ 0.01. We also show that the merger history of SMBHs, in the mass range detectable by the future Laser Interferometer Space Antenna (LISA) instrument, generically looks different as fbh is varied; this should allow LISA to deliver useful constraints on otherwise degenerate models.