Alive and well: mimetic gravity and a higher-order extension in light of GW170817

The near-simultaneous multi-messenger detection of the gravitational wave (GW) event GW170817 and its optical counterpart, the short $\gamma$-ray burst GRB170817A, implies that deviations of the GW speed from the speed of light are restricted to being of ${\cal O}(10^{-15})$. In this note, we study the implications of this bound for mimetic gravity and confirm that in the original setting of the theory, GWs propagate at the speed of light, hence ensuring agreement with the recent multi-messenger detection. A higher-order extension of the original mimetic theory, appearing in the low-energy limit of projectable Ho\v{r}ava-Lifshitz gravity, is then considered. Performing a Bayesian statistical analysis where we compare the predictions of the higher-order mimetic model for the speed of GWs against the observational bound from GW170817/GRB170817A, we derive constraints on the three free parameters of the theory. Imposing the absence of both ghost instabilities and superluminal propagation of scalar and tensor perturbations, we find very stringent 95\% confidence level upper limits of $\sim 7 \times 10^{-15}$ and $\sim 4 \times 10^{-15}$ on the coupling strengths of Lagrangian terms of the form $\nabla^{\mu}\nabla^{\nu}\phi\nabla_{\mu}\nabla_{\nu}\phi$ and $(\Box \phi)^2$ respectively, with $\phi$ the mimetic field. We discuss implications of the obtained bounds for mimetic theories. This work presents the first ever robust comparison of a mimetic theory to observational data.