One-loop modified gravity in a de Sitter universe, quantum-corrected inflation, and its confrontation with the Planck result

Motivated by issues on inflation, a generalized modified gravity model is investigated, where the model Lagrangian is described by a smooth function $f(R,K,\varphi )$ of the Ricci scalar $R$, the kinetic term $K$ of a scalar field $\varphi$. In particular, the one-loop effective action in the de Sitter background is examined on shell as well as off shell in the Landau gauge. In addition, the on-shell quantum equivalence of $f(R)$ gravity in the Jordan and Einstein frames is explicitly demonstrated. Furthermore, we present applications related to the stability of the de Sitter solutions and the one-loop quantum correction to inflation in quantum-corrected $R^2$ gravity. It is shown that, for a certain range of parameters, the spectral index of the curvature perturbations can be consistent with the Planck analysis, but the tensor-to-scalar ratio is smaller than the minimum value within the $1\sigma$ error range of the BICEP2 result.