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Theories of dark energy that affect the speed of gravitational waves $c_{\rmGW}$ on cosmological scales naturally lead to a frequency-dependent transitionof that speed close to the LIGO/Virgo/KAGRA (LVK) band. While observations suchas GW170817 assure us that $c_{\rm GW}$ is extremely close to the speed oflight in the LVK band, a frequency-dependent transition below the LVK band is asmoking-gun signal for large classes of dynamical dark energy theories. Here wediscuss 1) how the remnants of such a transition can be constrained withobservations in the LVK band, 2) what signatures are associated with such atransition in the LISA band, and 3) how joint observations in the LVK and LISAbands allow us to place tight constraints on this transition and the underlyingtheories. We find that deviations of $c_{\rm GW}$ can be constrained down to alevel of $\sim 10^{-17}$ in the LVK ${\textit and}$ LISA bands even for mildfrequency-dependence, much stronger than existing bounds forfrequency-independent $c_{\rm GW} \neq c$. We use the strain data from GW170817to bound the deviation of $c_{\rm GW}$ to be less than $10^{-17}$ at 100 Hz andless than $10^{-18}$ at 500 Hz. We also identify a particularly interestingtype of transition in between the LVK and LISA bands and show how multi-bandobservations can constrain this further. Finally, we discuss what these currentand forecasted constraints imply for the underlying dark energy theories.

Probing the speed of gravity with LVK, LISA, and joint observations