The physical origin of super-competitive accretion during the formation of the first supermassive black holes

Numerical simulations have shown the occurence of a scenario termed ''super-competitive accretion'', a term that describes a situation where only the central few objects grow supermassive while a larger number of stars compete for the reservoir, with significant accretion flows of $\gtrsim0.1$ M$_\odot$ yr$^{-1}$. This scenario particularly implies that the presence of fragmentation will not necessarily impeed the formation of a central massive object. We here explore this phenomenon using analytical estimates for growth via collisions and accretion, considering accretion due to self-gravity as well as Bondi-Hoyle accretion. Particularly, we explore under what conditions the accretion onto the central massive object breaks down, and derive a criterion that depends on the mass of the most massive object and the mass in fragments. For compact clusters with sizes about $0.1$~pc, we further find that the mass growth by collisions is comparable to the growth via accretion. Our results are validated through the comparison with numerical simulations, and we overall conclude that super-competitive accretion is a valid mechanism for the formation of very massive objects in the early Universe.