We construct a manifestly diffeomorphism invariant Wilsonian (Exact) Renormalization Group for classical gravity, and begin the construction for quantum gravity. We demonstrate that the effective action can be computed without gauge fixing the diffeomorphism invariance, and also without introducing a background space-time. We compute classical contributions both within a background-independent framework and by perturbing around a fixed background, and verify that the results are equivalent. We derive the exact Ward identities for actions and kernels and verify consistency. We formulate two forms of the flow equation corresponding to the two choices of classical fixed-point: the Gaussian fixed point, and the scale invariant interacting fixed point using curvature-squared terms. We suggest how this programme may completed to a fully quantum construction.

We study the effective stress-energy tensor induced by cosmological inhomogeneity in f (R) = R + cR 2 and equivalent scalar-tensor theories, motivated both by models of early universe inflation and by phenomenological alternative cosmologies to the standard Λ-CDM. We use Green and Wald's framework for averaging over classical fluctuations of short-wavelength λ. By ensuring that the leading non-linear terms from the fluctuations of the Einstein terms and the corrections both contribute in the formal limit as λ → 0, we derive a diffeomorphism invariant effective stress-energy tensor whose trace is nonvanishing and of the right sign to potentially account for the current acceleration of the universe. However a more phenomenologically acceptable dark energy model would be required if this effect were to fully account for the current acceleration.

We calculate a general effective stress-energy tensor induced by cosmological inhomogeneity in effective theories of gravity where the action is Taylor-expandable in the Riemann tensor and covariant derivatives of the Riemann tensor. This is of interest as an effective fluid that might provide an alternative to the cosmological constant, but it also applies to gravitational waves. We use an adaptation of Green and Wald's weak-averaging framework, which averages over perturbations in the field equation where the perturbation length scales are small compared to the averaging scale. In this adaptation, the length scale of the effective theory, 1/M , is also taken to be small compared with the averaging scale. This ensures that the perturbation length scales remain in fixed proportion to the length scale of the effective theory as the cosmological averaging scale is taken to be large. We find that backreaction from higher-derivative terms in the effective action can continue to be important in the late universe, given a source of sufficiently high-frequency metric perturbations. This backreaction might also provide a window on exotic particle physics in the far ultraviolet.