Third order solutions of the cosmological density perturbations in the
  Horndeski's most general scalar-tensor theory with the Vainshtein mechanism
  We study the third order solutions of the cosmological density perturbations
in the Horndeski's most general scalar-tensor theory under the condition that
the Vainshtein mechanism is at work. In this work, we thoroughly investigate
the independence property of the functions describing the nonlinear
mode-couplings, which is also useful for models within the general relativity.
Then, we find that the solutions of the density contrast and the velocity
divergence up to the third order ones are characterized by 6 parameters.
Furthermore, the 1-loop order power spectra obtained with the third order
solutions are described by 4 parameters. We exemplify the behavior of the
1-loop order power spectra assuming the kinetic gravity braiding model, which
demonstrates that the effect of the modified gravity appears more significantly
in the power spectrum of the velocity divergence than the density contrast.
