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 beta 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.