The supermassive black holes (SMBHs) massive enough (gsim few ×109 M sun) to power the bright redshift z ≈ 6 quasars observed in the Sloan Digital Sky Survey (SDSS) are thought to have assembled by mergers and/or gas accretion from less massive "seed" BHs.
If the seeds are the ~102 M sun remnant BHs from the first generation of stars, they must be in place well before redshift z = 6, and must avoid being ejected from their parent protogalaxies by the large (several ×102 km s-1) kicks they suffer from gravitational-radiation-induced recoil during mergers with other BHs. We simulate the SMBH mass function at redshift z > 6 using dark matter halo merger trees, coupled with a prescription for the halo occupation fraction, accretion histories, and radial recoil trajectories of the growing BHs. Our purpose is (1) to map out plausible scenarios for successful assembly of the z ≈ 6 quasar BHs by exploring a wide region of parameter space, and (2) to predict the rate of low-frequency gravitational wave events detectable by the Laser Interferometer Space Antenna (LISA) for each such scenario. Our main findings are as follows: (1) ~100 M sun seed BHs can grow into the SDSS quasar BHs without super-Eddington accretion, but only if they form in minihalos at z gsim 30 and subsequently accrete gsim60% of the time; (2) the scenarios with optimistic assumptions required to explain the SDSS quasar BHs overproduce the mass density in lower mass (few ×105 M sun lsim M bhlsim few × 107 M sun) BHs by a factor of 102-103, unless seeds stop forming, or accrete at a severely diminished rates or duty cycles (e.g., due to feedback), at z lsim 20-30. We also present several successful assembly models and their LISA detection rates, including a "maximal" model that gives the highest rate (~30 yr-1 at z = 6) without overproducing the total SMBH density.