Spontaneous scalarization of Gauss-Bonnet black holes: Analytic treatment in the linearized regime

It has recently been proved that nontrivial couplings between scalar fields and the Gauss-Bonnet invariant of a curved spacetime may allow a central black hole to support spatially regular scalar hairy configurations. Interestingly, former numerical studies of the intriguing black-hole spontaneous scalarization phenomenon have demonstrated that the composed hairy black-hole-scalar-field configurations exist if and only if the dimensionless coupling parameter $\bar\eta$ of the theory belongs to a discrete set $\{[\bar\eta^{-}_{n},\bar\eta^{+}_{n}]\}_{n=0}^{n=\infty}$ of scalarization bands. We have examined the numerical data that are available in the physics literature and found that the newly discovered hairy black-hole-linearized-massless-scalar-field configurations are characterized by the asymptotic universal behavior $\Delta_n\equiv \sqrt{\bar\eta^{+}_{n+1}}-\sqrt{\bar\eta^{+}_{n}}\simeq 2.72$. Motivated by this intriguing observation, in the present paper we study {\it analytically} the physical and mathematical properties of the spontaneously scalarized Schwarzschild black holes in the linearized (weak-field) regime. In particular, we provide a remarkably compact analytical explanation for the numerically observed universal behavior $\Delta_n\simeq 2.72$ which characterizes the discrete resonant spectrum $\{\bar\eta^{+}_{n}\}_{n=0}^{n=\infty}$ of the composed hairy black-hole-linearized-scalar-field configurations.