Nonlinear completions of the Ricci coupling in black-hole scalarization

Abstract

We study spontaneous scalarization of static, spherically symmetric black holes in scalar--Gauss--Bonnet gravity with a Ricci coupling $F=1 βϕ^{2}/4+γβ^{2}ϕ^{4}+O(ϕ^{6})$ that nonlinearly completes the quadratic one. For arbitrary couplings and potential, we derive the near-horizon regularity condition and horizon curvature invariants in closed form; scalarized branches terminate where this condition degenerates, at finite horizon curvature. A weakly nonlinear expansion fixes the bifurcation direction exactly: for a massless scalar, its governing coefficient is quadratic in $β$, changes sign at $β=1.1156$, and depends linearly on a quartic self-interaction. At equal horizon values of the scalar, completions differ in mass, to leading order, in proportion to $γ$. A radial-perturbation master equation for arbitrary couplings confirms the exchange of stability at the bifurcation and turning points. Consequently, the domain of radially stable black holes shrinks as $γ$ increases over the range studied; it is smaller for the exponential completion than for the quadratic truncation.