Lunar Laser Ranging constraints on nonminimally coupled dark energy and
  standard sirens
  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 today's 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.
