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Constraining non-commutative space-time from GW150914

The gravitational wave signal GW150914, recently detected by LIGO and Virgo collaborations, is used to place a bound on the scale of quantum fuzziness of non-commutative space-time. We show that the leading non-commutative correction to the phase of the gravitational waves produced by a binary system appears at the 2nd order of the post- Newtonian expansion. This correction is proportional to $Λ^2 \equiv |θ^{0i}|^2/(l_P t_P)^2$, where $θ^{μν}$ is the antisymmetric tensor of non-commutativity. To comply with GW150914 data, we find that $\sqrtΛ \lesssim 6$, namely less than one order of magnitude above the Planck scale. This is the most stringent bound on non-commutative scale, exceeding the previous constraints from particle physics processes by $\sim 15$ orders of magnitude.