Spin-induced black hole spontaneous scalarization: Analytic treatment in the large-coupling regime

It has recently been demonstrated numerically that spinning black holes whose dimensionless rotation parameter is larger than the critical value (J/M²)crit≡ācrit=1/2 can support spatially regular hairy scalar configurations which are characterized by a nonminimal negative coupling to the Gauss-Bonnet curvature invariant. Motivated by this physically important observation, we explore, using analytical techniques, the physical and mathematical properties of the composed spinning-Kerr-black-hole-nonminimally-coupled-linearized-scalar-field configurations in the large-coupling -η1 regime (here η is the dimensionless coupling parameter of the composed black-hole-scalar-field theory). In particular, we derive a remarkably compact analytical formula for the spin-dependent critical existence line ηcrit=ηcrit(ā) of the composed black-hole-field system. The physical significance of this critical line stems from the fact that, for a given value of the black-hole dimensionless rotation parameter ā, it marks the boundary between bald (scalarless) Kerr black holes and nonlinearly coupled hairy spinning-black-hole-scalar-field configurations.