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 ($m_{\rm max} = 130\,\mathrm{M}_{\odot}$) and an extended-IMF scenario with ${\sim}\,50-110$ primordial BH seeds above the pair-instability gap ($M_{\rm BH} \sim 130-700\,\mathrm{M}_{\odot}$). 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 $M_{\rm BH} \sim 45-70\,\mathrm{M}_{\odot}$ with spin $\chi_{\rm BH} \sim 0.65$, limited to ${\sim}\,1-3$ mergers, as second-generation remnants acquire spin $\chi \sim 0.7$ 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 ($M_{\rm BH} \gtrsim 450\,\mathrm{M}_{\odot}$) survives -- while the joint mass-spin distribution is trimodal; seeds surviving via stellar-mass BH mergers retain low spin ($\chi \lesssim 0.3$), whereas seed-seed mergers produce high-mass, high-spin remnants ($\chi \sim 0.65-0.7$), yielding 90th-percentile retained masses of ${\sim}\,500-1100\,\mathrm{M}_{\odot}$. Both scenarios are consistent with the $3\sigma$ 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.