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

Nonlinearly scalarized black holes are investigated in Einstein-scalar-Gauss-Bonnet (EsGB) theory with polynomial coupling functions $\zeta (\phi )$ satisfying ${\zeta }^{\mathrm{\prime }\mathrm{\prime }}(0)=0$, where ${\zeta }^{\mathrm{\prime }}(\phi )=0$ features besides $\phi =0$ solutions with constant ${\phi }_{\mathrm{s}}\mathrm{\ne }0$. 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: $\zeta (\phi )=\alpha {\phi }^4-\beta {\phi }^8$ and $\zeta (\phi )=\alpha {\phi }^4-\beta {\phi }^6$. In contrast, for the quartic coupling function $\zeta (\phi )=\alpha {\phi }^4$ SBHs are stable. Treating $\zeta (\phi )R_{GB}^2$ 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 $\beta$.