Massive Black Hole formation in proto-stellar clusters via early gas accretion

We review our semi-analytic model of stellar black hole (BH) mass growth by gas accretion in gas-rich stellar clusters during their birthstage within the first $\sim 10\,{\rm Myr}$ after the first stellar formation event. Such proto-stellar clusters are massive and compact, with typical masses $\sim 10^6\,{\rm M}_\odot$ and sizes $\sim 1\,{\rm pc}$, suggested by recent James Webb Space Telescope (JWST) observations. We find that the BH masses are shifted by the end of gas depletion to values within and above the BH mass gap, well within the range of components of the recent gravitational-wave (GW) signal GW231123, and up to masses $\sim 10^3\,{\rm M}_\odot$.