Repopulating the pair-instability mass gap without sustained growth to massive IMBHs: the case of 47\,Tuc

Abstract

We model the formation and retention of the most massive black hole (BH) in 47~Tuc using the semi-analytical code \texttt{cBHBd}, coupling cluster evolution with binary BH dynamics and computing merger-remnant masses, spins, and gravitational-wave recoil kicks via numerical-relativity surrogate prescriptions. We evolve 80\,000 cluster realisations spanning initial masses, densities, IMFs, and metallicities, in both a baseline scenario ($m_{\max }=130{\mathrm{M}}_{\odot }$) and an extended-IMF scenario with $\sim 50-110$ primordial BH seeds above the pair-instability gap ($M_{\mathrm{BH}}\sim 130-700{\mathrm{M}}_{\odot }$). Selecting models reproducing 47~Tuc's present-day mass and half-mass radius, we find hierarchical mergers alone yield a most massive retained BH of $M_{\mathrm{BH}}\sim 45-70{\mathrm{M}}_{\odot }$ with spin $\ddot{I}{‡}_{\mathrm{BH}}\sim 0.65$, limited to $\sim 1-3$ mergers, as second-generation remnants acquire spin $\ddot{I}‡\sim 0.7$ that amplifies recoil kicks in subsequent generations. When primordial seeds are included, the retained-mass distribution becomes bimodal -- in $\sim 90\%$ of realisations all seeds are ejected, but in $\sim 10\%$ a massive seed ($M_{\mathrm{BH}}\gtrsim 450{\mathrm{M}}_{\odot }$) survives -- while the joint mass-spin distribution is trimodal; seeds surviving via stellar-mass BH mergers retain low spin ($\ddot{I}‡\lesssim 0.3$), whereas seed-seed mergers produce high-mass, high-spin remnants ($\ddot{I}‡\sim 0.65-0.7$), yielding 90th-percentile retained masses of $\sim 500-1100{\mathrm{M}}_{\odot }$. Both scenarios are consistent with the $3\ddot{I}ƒ$ dynamical upper limit of $578{\mathrm{M}}_{\odot }$. Our results favour a dark-remnant subsystem over a single massive IMBH and provide a spin-mass diagnostic testable with LIGO-Virgo-KAGRA, the Einstein Telescope, Cosmic Explorer, and LISA.

10 pages