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 compact with typical mass $\sim {10}^6{\mathrm{M}}_{\odot }$ and size $\sim 1\mathrm{p}\mathrm{c}$, as revealed recently by JWST observations at $z\sim 10$. Sufficiently high compactness can provide a time window for early-formed stellar black holes (BHs) to accrete primordial gas. We develop a model to determine the final spin distribution of stellar BHs which grow in mass via gas accretion within compact gaseous proto-stellar clusters. The velocity shear within a BH’s sphere of influence induces the formation of an accretion disk which is repeatedly disrupted by stochastic perturbations to the BH motion. We assume low initial BH spins $a_{\ast ,\mathrm{i}\mathrm{n}\mathrm{i}}=0.01$, and restrict initial BH masses below the upper BH mass gap, $m_{\mathrm{B}\mathrm{H},\mathrm{i}\mathrm{n}\mathrm{i}}<55{\mathrm{M}}_{\odot }$. Our analysis shows a strong BH spin-mass correlation, obtained within $\sim 10\mathrm{M}\mathrm{y}\mathrm{r}$ when gas is depleted. Low-spin BHs, $a_{\ast }\le 0.3$, are predominantly low-mass, $m_{\mathrm{B}\mathrm{H}}\lesssim 25{\mathrm{M}}_{\odot }$, in contrast to high-spin black holes, $a_{\ast }\ge 0.7$, which are predominantly high-mass, $m_{\mathrm{B}\mathrm{H}}\gtrsim 65{\mathrm{M}}_{\odot }$. Notably, there exist also low-spin, high-mass outliers with $\sim 1$ mass-gap BH per cluster expected to have $a_{\ast }\sim 0.1$. 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 transition mass $\sim 50{\mathrm{M}}_{\odot }$. For $m_{\mathrm{B}\mathrm{H}}\ge 100{\mathrm{M}}_{\odot }$ the median spin is ${\bar{a}}_{\ast }\sim 0.90$ with the central $68\mathrm{\%}$ of the distribution spanning $a_{\ast }\sim 0.70-0.96$, in striking agreement with the estimated spins of the gravitational-wave signal GW231123. These spin values persist up to the highest masses generated by our mechanism, $m_{\mathrm{B}\mathrm{H}}\sim {10}^3{\mathrm{M}}_{\odot }$.