A class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior

Motivated by the conjecture that the cosmological constant problem could be solved by strong quantum effects in the infrared, we use the exact flow equation of quantum Einstein gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function ${\mathcal{F}}_k\mathrm{}\left(V\right)\mathrm{}$ of the Euclidean space-time volume V. A partial differential equation governing its dependence on the scale k is derived and its fixed point is analyzed. For the more restrictive truncation of theory space where ${\mathcal{F}}_k\mathrm{}\left(V\right)\mathrm{}$ is of the form $V+V\ln V,$ ${V+V}^2,$ and $V+\sqrt{V},$ respectively, the renormalization group equations for the running couplings are solved numerically. The results are used in order to determine the k-dependent curvature of the $S^4$-type Euclidean space-times which are solutions to the effective Einstein equations, i.e., stationary points of the scale dependent effective action. For the $V+V\ln V$-invariant (discussed earlier by Taylor and Veneziano) we find that the renormalization group running enormously suppresses the value of the renormalized curvature which results from Planck-size bare parameters specified at the Planck scale. Hence one can obtain very large, almost flat universes without fine-tuning the cosmological constant.