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

¹ Dipartimento di Fisica “G. Occhialini”, Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
² Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy

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Received: 5 December 2025
Accepted: 19 March 2026

Abstract

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, m_BH,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, m_BH ≲ 25 M_⊙, in contrast to high-spin BHs, a_* ≥ 0.7, which are predominantly high-mass, m_BH ≳ 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 m_BH ≥ 100 M_⊙ the median spin is a^-_* ∼ 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, m_BH ~ 10³ M_⊙.