Mass transport in galaxy discs limits black hole growth to sub-Eddington rates

Supermassive black holes (SMBHs) observed to have masses of M∙∼109M⊙ at z ≳ 6, <1 Gyr after the big bang, are thought to have been seeded by massive black holes that formed before growing concurrently with the formation of their host galaxies. We model analytically the idealized growth of seed black holes, fed through gas inflow from growing proto-galaxy discs. The inflow depends on the disc gravitational stability and thus varies with black hole and disc mass. We find that for a typical host halo, the efficiency of angular momentum transport, as parametrized by the disc viscosity, is the limiting factor in determining the inflow rate and the black hole accretion rate. For our fiducial case, we find an upper black hole mass estimate of M∙∼1.8×107M⊙ at z = 6. Only in the extreme case of ∼1016 M⊙ haloes at z = 6 produces SMBH masses of ∼109 M⊙. However, the number density of such haloes is many orders of magnitude below the estimated 1 Gpc−3 of SMBHs at z = 6, indicating that viscosity driven accretion is too inefficient to feed the growth of seeds into M∙∼109M⊙ SMBHs by z ∼ 6. We demonstrate that major mergers are capable of resolving the apparent discrepancy in black hole mass at z = 6, with some dependence on the exact choice of orbital parameters of the merger.