Stability and quasinormal modes for black holes with time-dependent scalar hair

We investigate black hole solutions with time-dependent (scalar) hair in scalar-tensor theories. Known exact solutions exist for such theories at the background level, where the metric takes on a standard general relativity (GR) form (e.g., Schwarzschild–de Sitter), but these solutions are generically plagued by instabilities. Recently, a new such solution was identified by Bakopoulos et al. [Phys. Rev. D 109, 024032 (2024)], in which the time-dependent scalar background profile is qualitatively different from previous known exact solutions—specifically, the canonical kinetic term for the background scalar 𝑋 is not constant in this solution. We investigate the stability of this new solution by analyzing odd parity perturbations, identifying a bound placed by stability and the resulting surviving parameter space. We extract the quasinormal mode spectrum predicted by the theory, finding a generic positive shift of quasinormal mode frequencies and damping times compared to GR. We forecast constraints on these shifts (and the single effective parameter ^𝛽 controlling them) from current and future gravitational wave experiments, finding constraints at up to the 𝒪⁡(10−2) and 𝒪⁡(10−4) level for LIGO-Virgo-KAGRA and Laser Interferometer Space Antenna/TianQin, respectively. All calculations performed in this paper are reproducible via a companion Mathematica notebook.