The sustainable growth of the first black holes

Super-Eddington accretion has been suggested as a possible formation pathway of $10^9 \, M_\odot$ supermassive black holes (SMBHs) 800 Myr after the Big Bang. However, stellar feedback from BH seed progenitors and winds from BH accretion disks may decrease BH accretion rates. In this work, we study the impact of these physical processes on the formation of $z \sim 6$ quasar, including new physical prescriptions in the cosmological, data-constrained semi-analytic model GAMETE/QSOdust. We find that the feedback produced by the first stellar progenitors on the surrounding does not play a relevant role in preventing SMBHs formation. In order to grow the $z \gtrsim 6$ SMBHs, the accreted gas must efficiently lose angular momentum. Moreover disk winds, easily originated in super-Eddington accretion regime, can strongly reduce duty cycles. This produces a decrease in the active fraction among the progenitors of $z\sim6$ bright quasars, reducing the probability to observe them.