Adiabatic evolution of the self-interacting axion field around rotating black holes

Ultra-light axion fields, motivated by string theory, form a large condensate (axion cloud) around rotating black holes through superradiant instability. Several effects due to the axion cloud, such as the spin-down of black holes and the emission of monochromatic gravitational waves, open a new window to search for axions by astrophysical observations. When the axion self-interaction is considered, the evolution of clouds is altered significantly, and an explosive phenomenon called a bosenova can happen. Thus, it is necessary to understand the precise evolution of self-interacting clouds for the detection of axions by astrophysical observations. In this paper, we propose a new method to track the whole process of the growth of self-interacting axion clouds employing the adiabatic approximation. We emphasize that our method relies neither on the non-relativistic approximation nor on perturbative treatment of the self-interaction, which is often used in the literature. Our main finding is that the evolution of clouds in the strongly self-interacting regime depends on the strength of the gravitational coupling between the axion and the black hole. For a large coupling, the cloud evolves into a quasi-stationary state where the superradiant energy gain is balanced with the energy dissipation to infinity by the self-interaction. On the other hand, when one decreases the size of the coupling, clouds become unstable at some energy, which would be interpreted as the onset of a bosenova.