In dark energy models where a scalar field $phi$ is coupled to the Ricci scalar $R$ of the form $e^{-2Q (phi-phi_0)/M_{rm pl}}R$, where $Q$ is a coupling constant, $phi_0$ is todays value of $phi$, and $M_{rm pl}$ is the reduced Planck mass, we study how the recent Lunar Laser Ranging (LLR) experiment places constraints on the nonminimal coupling from the time variation of gravitational coupling. Besides a potential of the light scalar responsible for cosmic acceleration, we take a cubic Galileon term into account to suppress fifth forces in over-density regions of the Universe. Even if the scalar-matter interaction is screened by the Vainshtein mechanism, the time variation of gravitational coupling induced by the cosmological background field $phi$ survives in the solar system. For a small Galileon coupling constant $beta_3$, there exists a kinetically driven $phi$-matter-dominated-epoch ($phi$MDE) prior to cosmic acceleration. In this case, we obtain the stringent upper limit $Q le 3.4 times 10^{-3}$ from the LLR constraint. For a large $beta_3$ without the $phi$MDE, the coupling $Q$ is not particularly bounded from above, but the cosmological Vainshtein screening strongly suppresses the time variation of $phi$ such that the dark energy equation of state $w_{rm DE}$ reaches the value close to $-1$ at high redshifts. We study the modified gravitational wave propagation induced by the nonminimal coupling to gravity and show that, under the LLR bound, the difference between the gravitational wave and luminosity distances does not exceed the order $10^{-5}$ over the redshift range $0<z<100$. In dark energy models where the Vainshtein mechanism is at work through scalar derivative self-interactions, it is difficult to probe the signature of nonminimal couplings from the observations of standard sirens.
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