Instability of scalarized compact objects in Einstein-scalar-Gauss-Bonnet theories

We investigate the linear stability of scalarized black holes (BHs) and neutron stars (NSs) in Einstein-scalar-Gauss-Bonnet (GB) theories against odd- even-parity perturbations including higher multipole modes. show that angular propagation speeds $\ensuremath{\ell}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, with $\ensuremath{\ell}$ being moments, become imaginary hence BH solutions suffer from gradient instability. such an instability appears irrespective structure higher-order terms GB coupling function is caused purely due to existence leading quadratic term boundary condition value scalar field vanishes at spatial infinity. This indicates point mass-charge diagram where branches bifurcate Schwarzschild branch. also realized a nonlinear scalarization model perturbations. Our result suggests exterior static spherically-symmetric NS induced by same functions.