Formalizing Slow-roll Inflation in Scalar-Tensor Theories of Gravitation

The viability of slow-roll approximation is examined by considering the structure of phase spaces in scalar-tensor theories of gravitation and the analysis is exemplified with a nonminimally coupled scalar field to the spacetime curvature. The slow-roll field equations are obtained in the Jordan frame in two ways: first using the direct generalization of the slow-roll conditions in the minimal coupling case to nonminimal one, and second, conformal transforming the slow-roll field equations in the Einstein frame to the Jordan frame and then applying the generalized slow-roll conditions. Two inflationary models governed by the potentials $V(\phi) \propto \phi^2$ and $V(\phi) \propto \phi^4$ are considered to compare the outcomes of two methods based on the analysis of $n_s$ and $r$ values in the light of recent observational data.