Self-Interacting Gravitational Atoms in the Strong-Gravity Regime

We numerically investigate free and self-interacting ultralight scalar fields around black holes in General Relativity. We focus on complex scalar fields $\Phi$ whose self-interactions are described by the quartic potential $V \propto \lambda |\Phi|^4$, and ignore the black hole spin in order to disentangle the effects of self interactions on the boson cloud. Using the spectral solver Kadath, we compute quasi-equilibrium configurations of the dominant eigenstates, including their backreaction on the spacetime metric. For scenarios with $- 10^{-2} \lesssim \lambda \lesssim 10^{-2}$ we find the mass of the self-interacting scalar cloud to be up to $\sim 70\%$ larger than that of a free scalar cloud, though the additional backreaction effect on the spacetime metric is only up to $\sim 1\%$ due to the low-density nature of the bosonic configurations. In this region of parameter space we observe approximate quadratic scalings between the mass of the cloud with $\lambda$, the scalar field amplitude, and the couplings between these two parameters. For systems with $\lambda$ beyond this range, the eigenfrequencies differ sufficiently from the known free-test-field values used as inputs in our numerical setup to make the results, though convergent, physically unreliable. This bounds the range of $\lambda$ in which the free scalar field solution remains a good approximation to self-interacting scalar field configurations. Our work is among the first nonperturbative explorations of self-interacting bosonic clouds around black holes, yielding detailed new insights into such systems in the nonlinear regime, while also overcoming technical challenges and quantifying limitations. Additionally, our results provide useful inputs for fully dynamical numerical relativity simulations and for future explorations of spinning black holes and real scalar fields.