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

TL;DR

This paper provides an analytical explanation for the universal spacing observed in the discrete spectrum of scalarized Schwarzschild black holes in the linearized regime, complementing previous numerical findings on black hole scalarization.

Contribution

It analytically derives the universal behavior of the scalarization spectrum, advancing understanding of black hole scalar hair formation in Gauss-Bonnet gravity.

Findings

Universal asymptotic spacing $oxed{ ext{~}2.72}$ in the scalarization spectrum.

Analytical explanation for the discrete resonant spectrum of scalarized black holes.

Confirmation of numerical results through linearized analytical treatment.

Abstract

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.