Super-Eddington growth of the first black holes
  The assembly of the first super massive black holes (SMBHs) at $z \gtrsim 6$
is still a subject of intense debate. If black holes (BHs) grow at their
Eddington rate, they must start from $\gtrsim 10^4 \, M_\odot$ seeds formed by
the direct collapse of gas. Here we explore the alternative scenario where
$\sim 100 \, M_\odot$ BH remnants of the first stars grow at super-Eddington
rate via radiatively inefficient slim accretion disks. We use an improved
version of the cosmological, data-constrained semi-analytic model
GAMETE/QSOdust, where we follow the evolution of nuclear BHs and gas cooling,
disk and bulge formation of their host galaxies. Adopting SDSS J1148+5251
(J1148) at $z = 6.4$ as a prototype of luminous $z \gtrsim 6$ quasars, we find
that $\sim$ 80% of its SMBH mass is grown by super-Eddington accretion, which
can be sustained down to $z \sim 10$ in dense, gas-rich environments. The
average BH mass at $z \sim 20$ is $M_{\rm BH} \gtrsim 10^4 \,M_\odot$,
comparable to that of direct collapse BHs. At $z = 6.4$ the AGN-driven mass
outflow rate is consistent with the observations and the BH-to-bulge mass ratio
is compatible with the local scaling relation. However, the stellar mass in the
central 2.5 kpc is closer to the value inferred from CO observations. Finally,
$\sim 20 %$ of J1148 progenitors at $z=7.1$ have BH luminosities and masses
comparable to ULAS J1120+0641, suggesting that this quasar may be one of the
progenitors of J1148.
