Primordial non-gaussianities in general modified gravitational models of inflation

We compute the three-point correlation function of primordial scalar density perturbations in a general single-field inflationary scenario, where a scalar field phi has a direct coupling with the Ricci scalar R and the Gauss-Bonnet term G. Our analysis also covers the models in which the Lagrangian includes a function non-linear in the field kinetic energy X = -(partial derivative phi)(2)/2, and a Galileon-type field self-interaction G (phi, X)rectangle phi, where G is a function of phi and X. We provide a general analytic formula for the equilateral non-Gaussianity parameter f(NL)(equil) associated with the bispectrum of curvature perturbations. A quasi de Sitter approximation in terms of slow-variation parameters allows us to derive a simplified form of f(NL)(equil) convenient to constrain various inflation models observationally. If the propagation speed of the scalar perturbations is much smaller than the speed of light, the Gauss-Bonnet term as well as the Galileon-type field self-interaction can give rise to large non-Gaussianities testable in future observations. We also show that, in Brans-Dicke theory with a field potential (including f (R) gravity), f(NL)(equil) is of the order of slow-roll parameters as in standard inflation driven by a minimally coupled scalar field.