In dark energy models where a scalar field \phi is coupled to the Ricci scalar R of the form e ^ { -2 Q ( \phi - \phi _ { 0 } ) / M _ { pl } } R , where Q is a coupling constant , \phi _ { 0 } is today ’ s value of \phi , and M _ { 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 \leq 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 _ { 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 .