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We set up cosmological perturbation theory and study the cosmological implications of the so-called "generalized Galileon" developed in 6, 7. This is the most general scalar field theory whose Lagrangian contains derivatives up to second order while keeping second order equations of motion, and contains as sub-cases k-inflation, G-inflation and many other models. We calculate the power spectrum of the primordial curvature perturbation, finding a modification of the usual consistency relation of the tensor-to-scalar ratio in k-inflation or perfect fluid models. Finally we also calculate the bispectrum, which contains no new shapes beyond those of k-inflation.
Gao et al. (Thu,) studied this question.