Black holes in Gauss–Bonnet and Chern–Simons-scalar theory

We carry out the stability analysis of the Schwarzschild black hole in Gauss–Bonnet and Chern–Simons-scalar theory. Here, we introduce two quadratic scalar couplings ([Formula: see text]) to Gauss–Bonnet and Chern–Simons terms, where the former term is parity-even, while the latter one is parity-odd. The perturbation equation for the scalar [Formula: see text] is the Klein–Gordon equation with an effective mass, while the perturbation equation for [Formula: see text] is coupled to the parity-odd metric perturbation, providing a system of two coupled equations. It turns out that the Schwarzschild black hole is unstable against [Formula: see text] perturbation, leading to scalarized black holes, while the black hole is stable against [Formula: see text] and metric perturbations, implying no scalarized black holes.

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