Kerr black holes with self-interacting scalar hair: hairier but not heavier

The maximal ADM mass for (mini-)boson stars (BSs) – gravitating solitons of Einstein’s gravity minimally coupled to a free, complex, mass $\mu$, Klein-Gordon field – is $M_{\mathrm{A}\mathrm{D}\mathrm{M}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\sim M_{Pl}^2\mathrm{/}\mu$. Adding quartic self-interactions to the scalar field theory, described by the Lagrangian ${\mathcal{L}}_I=\lambda \mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^4$, the maximal ADM mass becomes $M_{\mathrm{A}\mathrm{D}\mathrm{M}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\sim \sqrt{\lambda }{M}_{Pl}^3\mathrm{/}{\mu }^2$. Thus, for mini-BSs, astrophysically interesting masses require ultra-light scalar fields, whereas self-interacting BSs can reach such values for bosonic particles with Standard Model range masses. We investigate how these same self-interactions affect Kerr black holes with scalar hair (KBHsSH) [1], which can be regarded as (spinning) BSs in stationary equilibrium with a central horizon. Remarkably, whereas the ADM mass scales in the same way as for BSs, the $\mathit{\text{horizon}}\mathit{\text{mass}}$ $M_H$ does not increases with the coupling $\lambda$, and, for fixed $\mu$, it is maximized at the “Hod point”, corresponding to the extremal Kerr black hole obtained in the vanishing hair limit. This mass is always $M_{\rm H}^{\rm max }\sim M_{\rm Pl}^2/\mu$. Thus, introducing these self-interactions, the black hole spacetimes may become considerably “hairier” but the trapped regions cannot become “heavier”. We present evidence this observation also holds in a model with ${\mathcal{L}}_I=\beta \mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^6-\lambda \mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^4$; if it extends to $\mathit{\text{general}}$ scalar field models, KBHsSH with astrophysically interesting horizon masses $\mathit{\text{require}}$ ultra-light scalar fields. Their existence, therefore, would be a smoking gun for such (beyond the Standard Model) particles.