Instability of Schwarzschild Black Holes in Einstein-scalar-Gauss-Bonnet Gravity: Perturbative Approach and Time-Domain Analysis

We study the instability of Schwarzschild black holes and the appearance of scalarized solutions in Einstein-scalar-Gauss-Bonnet gravity performing a time-domain analysis in a perturbative scheme. First we consider a quadratic coupling function and we perform an expansion for a small perturbation of the scalar field around the Schwarzschild solution up to the second order; we do not observe any stable scalarized configuration, in agreement with previous studies. We then consider the cases of quartic and exponential coupling, using an expansion for small values of the Newton's constant, in order to include the nonlinear terms introduced by the coupling in the field equations; in this case we observe the appearance of stable scalarized solutions different from those found in literature. The discrepancy can be an artifact of the perturbative approach.