Second-order matter density perturbations and skewness in scalar-tensor modified gravity models

We study second-order cosmological perturbations in scalar-tensor models of dark energy that satisfy local gravity constraints, including f(R) gravity. We derive equations for matter fluctuations under a sub-horizon approximation and clarify conditions under which first-order perturbations in the scalar field can be neglected relative to second-order matter and velocity perturbations. We also compute the skewness of the matter density distribution and find that the difference from the Lambda CDM (CDM: cold dark matter) model is less than a few per cent even if the growth rate of first-order perturbations is significantly different from that in the Lambda CDM model. This shows that the skewness provides a model-independent test for the picture of gravitational instability from Gaussian initial perturbations including scalar-tensor modified gravity models.