We perform the stability analysis of the Schwarzschild black hole in $f(R)$ gravity with the parity-violating Chern–Simons (CS) term coupled to a dynamical scalar field $\theta$. For this purpose, we transform the $f(R)$ gravity into the scalar-tensor theory by introducing a scalaron $\varphi$, providing the dynamical CS modified gravity with two scalars. The perturbation equation for the scalar $\theta$ is coupled to the odd-parity metric perturbation equation, providing a system of two coupled second-order equations, while the scalaron is coupled to the even-parity perturbation equation. This implies that the CS coupling affects the Regge–Wheeler equation, while $f(R)$ gravity does not affect the Zerilli equation. It turns out that the Schwarzschild black hole is stable against the external perturbations if the scalaron is free from the tachyon.

• Received 14 September 2011

DOI:https://doi.org/10.1103/PhysRevD.84.104029