Globular cluster formation from inertial inflows: accreting extremely massive stars as the origin of abundance anomalies

We use the inertial-inflow model of massive star formation to describe the formation of globular clusters (GCs) in turbulent molecular clouds. A key aspect of this model is that the maximum stellar mass scales linearly with cloud mass, such that extremely massive stars (EMSs, ${10}^{3-4}\text{msun}$) form in massive GCs ($\gtrsim {10}^5\text{msun}$). The total wind mass loss is dominated by accreting EMSs (aEMSs), whose wind mass-loss rates have become comparable to their accretion rates ($\gtrsim {10}^{-2}\text{msun}{\text{yr}}^{-1}$). These winds pollute the intra-cluster medium with hot-hydrogen burning yields during GC formation. We propose a parameterised model for the evolution of the stellar mass function during GC formation ($\sim 1-2\text{myr}$), accounting for gas inflow, wind mass loss and mixing of aEMS yields with pristine gas that has initial proto-GC abundances. Low-mass stars ($\lesssim 1\text{msun}$) form continuously from this mixed gas and their abundances resemble observed abundance trends with GC mass and metallicity, specifically: (i) the helium spread in a typical GC is small ($\mathrm{\Delta }Y\simeq 0.01$) and increases with GC mass; (ii) the fraction of polluted stars increases with GC mass and metallicity; (iii) the extent of the Mg-Al anticorrelations is more pronounced in metal-poor and massive GCs. We conclude that GCs formed with a population of EMSs from gas with surface densities $\gtrsim {10}^3\text{msun}{\text{pc}}^{-2}$ and that nitrogen-rich galaxies discovered by the James Webb Space Telescope ({\it JWST}) are dominated by EMS-rich GCs that formed in the earliest phases of galaxy formation. These EMSs may have left behind intermediate-mass black holes with masses above the pair-instability gap ($\gtrsim 120\text{msun}$) that could be found with ongoing gravitational wave experiments.