Slow-Roll Inflation with Exponential Potential in Scalar-Tensor Models

A study of the slow-roll inflation for an exponential potential in the frame of the scalar-tensor theory is performed, where non-minimal kinetic coupling to curvature and non-minimal coupling of the scalar field to the Gauss-Bonnet invariant are considered. Different models were considered with couplings given by exponential functions of the scalar field, that lead to graceful exit from inflation and give values of the scalar spectral index and the tensor-to-scalar ratio in the region bounded by the current observational data. Special cases were found, where the coupling functions are inverse of the potential, that lead to inflation with constant slow-roll parameters, and it was posible to reconstruct the model parameters for given $ns$ and $r$. In first-order approximation the standard consistency relation maintains its validity in the model with non-minimal coupling, but it modifies in presence of Gauss-Bonnet coupling. The obtained Hubble parameter during inflation, $H\sim 10^{-5} M_p$ and the energy scale of inflation $V^{1/4}\sim 10^{-3} M_p$, are consistent with the upper bounds set by latest observations.