Spin-up and spin distribution of stellar black holes grown by gas accretion in proto-stellar clusters

Proto-stellar clusters, likely progenitors of globular clusters, are extremely compact with typical masses of ~106 M⊙ and sizes of ~1 pc, as has recently been revealed by James Webb Space Telescope observations at z ~ 10. Sufficiently high compactness can provide a time window for early-formed stellar black holes (BHs) to accrete primordial gas. We developed a semi-analytic model to follow BH spin-up and determine the final spin distribution of stellar BHs that grow in mass via gas accretion within compact gaseous protostellar clusters. The velocity shear within a BH’s sphere of influence induces the formation of an accretion disk that is repeatedly disrupted by stochastic perturbations to the BH motion. We assumed low initial BH spins of a*,ini = 0.01, consistent with stellar-evolution models with efficient angular-momentum transport, and we restricted initial BH masses to values below the upper BH mass gap, mBH,ini < 55 M⊙. Our analysis shows a strong BH spin-mass correlation, obtained within ~10 Myr when gas is depleted. Low-spin BHs, a* ≤ 0.3, are predominantly low-mass, mBH ≲ 25 M⊙, in contrast to high-spin BHs, a* ≥ 0.7, which are predominantly high-mass, mBH ≳ 65 M⊙. Notably, there exist also low-spin, high-mass outliers with ~1 mass-gap BH per cluster expected to have a* ~ 0.1. The general trend, however, expressed by the median spin as a function of final BH mass, is well fit by a high-spin saturating exponential with a transition mass of ~50 M⊙. For mBH ≥ 100 M⊙ the median spin is $\[\bar{a}_{*} \sim 0.90\]$, with the central 68% of the distribution spanning a* ~ 0.70–0.96, in striking agreement with the estimated spins of the BH components of the gravitational-wave signal GW231123. These spin values persist up to the highest masses generated by our mechanism, mBH ~ 103 M⊙.