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 (eta) over bar of the theory belongs to a discrete set {[(eta) over bar (-)(n), (eta) over bar (+)(n)]}(n=0)(n=infinity) 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) equivalent to root(eta) over bar (+)(n+1) - root(eta) over bar (+)(n) similar or equal to 2.72. Motivated by this intriguing observation, in the present paper we study 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) similar or equal to 2.72 which characterizes the discrete resonant spectrum {(eta) over bar (+)(n)}(n=0)(n=infinity) of the composed hairy black-hole-linearized-scalar-field configurations.