Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations

We study static spherically symmetric black hole solutions with a linearly time-dependent scalar field and discuss their linear stability in the shift- and reflection-symmetric subclass of quadratic degenerate higher-order scalar-tensor (DHOST) theories. We present the explicit forms of the reduced system of background field equations for a generic theory within this subclass. Using the reduced equations of motion, we show that in several cases the solution is forced to be of the Schwarzschild or Schwarzschild--(anti-)de Sitter form. We consider odd-parity perturbations around general static spherically symmetric black hole solutions and derive the concise criteria for the black holes to be stable. Our analysis also covers the case with a static or constant profile of the scalar field.