General-relativistic instability in rapidly accreting supermassive stars: The impact of rotation

Context. Supermassive stars (SMSs) collapsing via the general-relativistic (GR) instability are invoked as the possible progenitors of supermassive black holes. Their mass and angular momentum at the onset of the instability are key in many respects, in particular regarding the possibility for observational signatures of direct collapse. Accretion dominates the evolution of SMSs and, similar to rotation, it has been shown to impact their final properties significantly. However, the combined effect of accretion and rotation on the stability of these objects is not known. Aims. Here, we study the stability of rotating, rapidly accreting SMSs against GR perturbations and derive the properties of these stars at death. Methods. On the basis of hylotropic structures, which are relevant for rapidly accreting SMSs, we define rotation profiles under the assumption of local angular momentum conservation in radiative regions, which allows for differential rotation. We account for rotation in the stability of the structure by adding a Newtonian rotation term in the relativistic equation of stellar pulsation, which is justified by the slow rotations imposed by the ΩΓ-limit. Results. We find that rotation favours the stability of rapidly accreting SMSs as soon as the accreted angular momentum represents a fraction of f ≳ 0.1% of the Keplerian angular momentum. For f ∼ 0.3−0.5%, the maximum masses consistent with GR stability are increased by an order of magnitude compared to the non-rotating case. For f ∼ 1%, the GR instability cannot be reached if the stellar mass does not exceed 10 7 − 10 8 M ⊙ . Conclusions. These results imply that, as in the non-rotating case, the final masses of the progenitors of direct collapse black holes range in distinct intervals depending on the scenario considered: 10 5 M ⊙ ≲ M ≲ 10 6 M ⊙ for primordial atomically cooled haloes and 10 6 M ⊙ ≲ M ≲ 10 9 M ⊙ for metal-rich galaxy mergers. The models suggest that the centrifugal barrier is inefficient to prevent the direct formation of a supermassive black hole at the collapse of a SMS. Moreover, the conditions of galaxy mergers appear to be more favourable than those of atomically cooled haloes for detectable gravitational wave emission and ultra-long gamma-ray bursts at black hole formation.