$\Lambda$CDM star clusters at cosmic dawn: stellar densities, environment, and equilibrium

The James Webb Space Telescope (JWST) has opened a window on many new puzzles in the early Universe, including a population of high-redshift star clusters with extremely high stellar surface density, suggesting unique star formation conditions in the Universe's early evolution. We study the formation and evolution of these first star clusters and galaxies using an AREPO cosmological simulation box designed to resolve the intricate environments of the smallest halos hosting Population III star clusters at $z \geq 12$. Our approach, which prioritizes baryonic structure identification through a friends-of-friends algorithm, provides new insights into early star cluster formation and delivers predictions directly relevant to observations. We investigate the dynamical properties of these first star clusters and use numerical and analytical methods to understand the populations of virialized and non-virialized systems. Our findings indicate that high-$z$ star clusters in a feedback-free regime can achieve extreme surface densities, consistent with the systems detected by JWST. These results imply that JWST may have the opportunity to uncover stellar systems at high redshift whose dynamical state preserves evidence of the hierarchical structure formation process.