Testing the Speed of Gravity with Black Hole Ringdown

We investigate how the speed of gravitational waves, $c_{GW}$, can be tested by upcoming black hole ringdown observations. We do so in the context of hairy black hole solutions, where the hair is associated with a new scalar degree of freedom, forecasting that LISA and TianQin will be able to constrain deviations of $c_{GW}$ from the speed of light at the ${\cal O}(10^{-4})$ level from a single supermassive black hole merger. We discuss how these constraints depend on the nature of the scalar hair, what different aspects of the underlying physics they are sensitive to in comparison with constraints derived from gravitational wave propagation effects, which observable systems will place the most stringent bounds, and that constraints are expected to improve by up to two orders of magnitude with multiple observations. This is especially interesting for dark energy-related theories, where existing bounds from GW170817 need not apply at lower frequencies and where upcoming bounds from lower-frequency missions will therefore be especially powerful. As such, we also forecast analogous bounds for the intermediate-frequency AEDGE and DECIGO missions. Finally, we discuss and forecast analogous black hole ringdown constraints at higher frequencies (so from LVK, the Einstein Telescope and Cosmic Explorer) and in what circumstances they can yield new information on top of existing constraints on $c_{GW}$. All calculations performed in this paper are reproducible via a companion Mathematica notebook.