Novel exact ultracompact and ultrasparse hairy black holes emanating from regular and phantom scalar fields

In the framework of a simple gravitational theory that contains a scalar field minimally coupled to gravity, we investigate the emergence of analytic black-hole solutions with nontrivial scalar hair of secondary type. Although it is possible for one to obtain asymptotically (anti)de Sitter solutions using our setup, in the context of the present work, we are solely interested in asymptotically flat solutions. At first, we study the properties of static and spherically symmetric black-hole solutions emanating from both regular and phantom scalar fields. We find that the regular-scalar-field-induced solutions are solutions describing ultracompact black holes, while the phantom scalar fields generate ultrasparse black-hole solutions. The latter are black holes that can be potentially of very low density since, contrary to ultracompact ones, their horizon radius is always greater than the horizon radius of the corresponding Schwarzschild black hole of the same mass. Then, we generalize the above static solutions to slowly rotating ones and compute their angular velocities explicitly. Finally, the study of the axial perturbations of the derived solutions takes place, in which we show that there is always a region in the parameter space of the free parameters of our theory that allows the existence of both ultracompact and ultrasparse black holes.