What gravity waves are telling about quantum spacetime

We discuss various modified dispersion relations motivated by quantum gravity which might affect the propagation of the recently observed gravitational-wave signal of the event GW150914. We find that the bounds set by the data on the characteristic quantum-gravity mass scale M are too weak to constrain these scenarios and, in general, much weaker than the expected M>10⁴ eV for a correction to the dispersion relation linear in $1/M$. We illustrate this issue by giving lower bounds on M, plus an upper bound coming from constraints on the size of a quantum ergosphere. We also show that a phenomenological dispersion relation ω²=k²(1+αkⁿ/Mⁿ) is compatible with observations and, at the same time, has a phenomenologically viable mass M>10 TeV only in the quite restrictive range $0<n<0.68$. Remarkably, this is the domain of multiscale spacetimes but not of known quantum-gravity models.