Mass transport in galaxy discs limits black hole growth to sub-Eddington
  rates
  Super-massive black holes (SMBHs) observed to have masses of $M_\bullet \sim
10^9 \, \mathrm{M_\odot}$ at $z\gtrsim6$, $<1$ Gyr after the Big Bang, are
thought to have been seeded by massive black holes which formed before growing
concurrently with the formation of their host galaxies. We model analytically
the idealised 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 parametrised 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 $M_\bullet \sim 1.8 \times 10^7 \, \mathrm{M_{\odot}}$ at $z=6$.
Only in the extreme case of $\sim 10^{16}$ M$_{\odot}$ haloes at $z=6$ produces
SMBH masses of $\sim 10^9$ M$_{\odot}$. 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 $M_\bullet \sim 10^9 \, \mathrm{M_\odot}$ SMBHs by $z
\sim 6$. 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.
