Feedback Limits to Maximum Seed Masses of Black Holes
  The most massive black holes observed in the Universe weigh up to $\sim
10^{10} \, \mathrm{M_{\odot}}$, nearly independent of redshift. Reaching these
final masses likely required copious accretion and several major mergers.
Employing a dynamical approach, that rests on the role played by a new,
relevant physical scale - the transition radius - we provide a theoretical
calculation of the maximum mass achievable by a black hole seed that forms in
an isolated halo, one that scarcely merged. Incorporating effects at the
transition radius and their impact on the evolution of accretion in isolated
haloes we are able to obtain new limits for permitted growth. We find that
large black hole seeds ($M_{\bullet} \gtrsim 10^4 \, \mathrm{M_{\odot}}$)
hosted in small isolated halos ($M_h \lesssim 10^9 \, \mathrm{M_{\odot}}$)
accreting with relatively small radiative efficiencies ($\epsilon \lesssim
0.1$) grow optimally in these circumstances. Moreover, we show that the
standard $M_{\bullet}-\sigma$ relation observed at $z \sim 0$ cannot be
established in isolated halos at high-$z$, but requires the occurrence of
mergers. Since the average limiting mass of black holes formed at $z \gtrsim
10$ is in the range $10^{4-6} \, \mathrm{M_{\odot}}$, we expect to observe them
in local galaxies as intermediate-mass black holes, when hosted in the rare
haloes that experienced only minor or no merging events. Such ancient black
holes, formed in isolation with subsequent scant growth, could survive, almost
unchanged, until present.
