Stability of the de-Sitter universe: one-loop nonlocal f (R) gravity

With the method of the background field expansion, we investigate the one-loop quantization of the Euclidean nonlocal f (R) model in the de-Sitter universe. We obtain the ghost-free condition (GFC) based on the transformation from the Jordan frame to the Einstein frame and the classical stability condition (CSC) satisfied \({f}_{RR}^{\left(0\right)}-{\phi }_{0}{F}_{RR}^{\left(0\right)}<0\) . We present the on-shell and off-shell one-loop effective action and quantum stability condition (QSC) by utilizing the generalized zeta-function. We find that under the fulfillment of GFC, CSC and QSC are inconsistent.