Existence of nonlinearly scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with polynomial couplings

Abstract

Nonlinearly scalarized black holes are investigated in Einstein-scalar-Gauss-Bonnet (EsGB) theory with polynomial coupling functions $ζ(φ)$ satisfying $ζ''(0) = 0$, where $ζ'(φ) = 0$ features besides $φ=0$ solutions with constant $φ_{\rm s} \ne 0$. We determine the threshold amplitudes for Gaussian pulses, above which Schwarzschild black holes (SBHs) %become unstable and may transition to scalarized black holes for two coupling functions: $ζ(φ)=αφ^4-βφ^8$ and $ζ(φ)=αφ^4-βφ^6$. In contrast, for the quartic coupling function $ζ(φ)=αφ^4$ SBHs are stable. Treating $ζ(φ)R_{GB}^2$ as an effective potential $V_\text{eff}$ provides an explanation for the ``plateau" and the divergence observed in the time evolution. We then construct the branches of nonlinearly scalarized black holes in the probe limit and with backreaction. While the pattern of the solution branches in the probe limit exhibits universal features, the presence of backreaction reveals a distinct dependence on the coupling strength $β$.

Figures (17)

• Figure 1: Time evolution of the scalar field on the SBH background with $M/\lambda=0.025$ for $\alpha=\frac{1}{4}$ and $\beta=\frac{100}{8}$. The threshold amplitudes $A_{\rm th}$ are 0.05 (a) and 0.1 (b), indicating the formation of a scalarized phase for $A \ge A_{\rm th}$ for coupling functions $\zeta_1(\phi)$ and $\zeta_2(\phi)$. (c) For $\zeta_3(\phi)$ the scalar field diverges within a short timescale for large initial amplitudes or decays for small initial amplitudes of the Gaussian wave.
• Figure 2: Time evolution of the scalar field on the SBH background with $M/\lambda=0.025$ for $\zeta_1=\frac{1}{4}\phi^4-\beta\phi^8$. The horizontal red dashed lines denote the scalar field values $\phi_H$ at the horizon of the final configurations: (a) $0.4472$, (b) $0.3162$, (c) $0.1778$.
• Figure 3: The effective potential $V^3_\text{eff}$ for $\zeta_3(\phi)=\frac{1}{4}\phi^4$ for $M=1$ and $\lambda=1$. (a) Three-dimensional plot of $V^3_\text{eff}(r,\phi)$ as a function of the scalar $\phi$ and the radial coordinate $r$. (b) Two-dimensional plot of $V^3_\text{eff}(\phi)$ as a function of $\phi$ for a fixed $r$.
• Figure 4: Three-dimensional plot of the effective potential $V^1_\text{eff}$ for $\zeta_1(\phi)=\frac{1}{4}\phi^4-\frac{25}{8}\phi^8$ as a function of the scalar $\phi$ and the radial coordinate $r$ for $M=1$ and $\lambda=1$.
• Figure 5: Zoomed figure for the effective potential $V^1_\text{eff}$ for $M=1$ and $\lambda=1$. (a) Three-dimensional $W$-shape. (b) Two-dimensional $W$-profile of $V^1_\text{eff}(\phi)$ as a function of $\phi$ at a fixed $r$, with the minimum of the scalar field located at $\phi_{\rm s}=\pm 0.4472$.