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Constraining the cosmological evolution of scalar-tensor theories with local measurements of the time variation of G

We determine the conditions for which the constraints from lunar laser ranging on the time evolution of the local gravitational constant can be extrapolated to impose constraints on the time evolution of the cosmological gravitational constant in scalar-tensor theories of modified gravity. We allow for the possibility that the scalar-tensor theories are non-linear and contain a screening mechanism. This results in strong constraints on the running of the cosmological Planck mass described by the Horndeski function $|α_M|\lesssim 0.002$. We find that our assumptions are valid for most Vainshtein and kinetic screening models, where the shift symmetry $ϕ\rightarrowϕ+ c$ holds, but are violated by some Chameleon and Symmetron screening models, where the macroscopic equivalence principle is broken.