Correlation functions on a curved background

We investigate gravitational correlation functions in a curved background with the help of nonperturbative renormalization group methods. Beta functions for eleven couplings are derived, two of which correspond to running gauge parameters. A unique ultraviolet fixed point is found, suitable for a UV completion in the sense of Asymptotic Safety. To arrive at a well-behaved flow in a curved background, the regularization must be chosen carefully. We provide two admissible choices to solve this issue in the present approximation. We further demonstrate by an explicit calculation that the Landau limit is a fixed point also for quantum gravity, and additionally show that in this limit, the gauge parameter $\beta$ does not flow.