Onset of superradiant instabilities in rotating spacetimes of exotic compact objects

Exotic compact objects, horizonless spacetimes with reflective properties, have intriguingly been suggested by some quantum-gravity models as alternatives to classical black-hole spacetimes. A remarkable feature of spinning horizonless compact objects with reflective boundary conditions is the existence of a {\it discrete} set of critical surface radii, $\{r_{\text{c}}({\bar a};n)\}^{n=\infty}_{n=1}$, which can support spatially regular static ({\it marginally-stable}) scalar field configurations (here ${\bar a}\equiv J/M^2$ is the dimensionless angular momentum of the exotic compact object). Interestingly, the outermost critical radius $r^{\text{max}}_{\text{c}}\equiv \text{max}_n\{r_{\text{c}}({\bar a};n)\}$ marks the boundary between stable and unstable exotic compact objects: spinning objects whose reflecting surfaces are situated in the region $r_{\text{c}}>r^{\text{max}}_{\text{c}}({\bar a})$ are stable, whereas spinning objects whose reflecting surfaces are situated in the region $r_{\text{c}}<r^{\text{max}}_{\text{c}}({\bar a})$ are superradiantly unstable to scalar perturbation modes. In the present paper we use analytical techniques in order to explore the physical properties of the critical (marginally-stable) spinning exotic compact objects. In particular, we derive a remarkably compact {\it analytical} formula for the discrete spectrum $\{r^{\text{max}}_{\text{c}}({\bar a})\}$ of critical radii which characterize the marginally-stable exotic compact objects. We explicitly demonstrate that the analytically derived resonance spectrum agrees remarkably well with numerical results that recently appeared in the physics literature.