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.