Gravitational wave from cosmic inflation in a gravity with two small four-derivative corrections

We investigate a model of inflationary cosmology where the minimally coupled scalar field theory is modified by additional correction terms. Among the most general ten correction terms remarked by Weinberg in context of effective field theory, we consider only two terms, $f_1(\phi) R^2$ and $f_2(\phi) R^{ab}R_{ab}$, following the work by Noh and Hwang where $f_1$ and $f_2$ are constant. The fourth order differential equations for the background universe and the tensor-type perturbation are derived out of this model. We show that these equations can be reduced to second order equations, supposing that $f_n$ are small. From these approximated equations, we find that the propagation speed of gravitational wave is slightly less than the speed of light due to $f_2$ term, and that the evolution of the tensor-type perturbation is conserved in the large scale limit.