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Diffeomorphism-invariant averaging in quantum gravity and cosmology

This thesis concerns research undertaken in two related topics concerning high-energy gravitational physics. The first is the construction of a manifestly diffeomorphism-invariant Exact Renormalization Group (ERG). This is a procedure that constructs effective theories of gravity by integrating out high-energy modes down to an ultraviolet cutoff scale without gauge-fixing. The manifest diffeomorphism invariance enables us to construct a fully background-independent formulation. This thesis will explore both the fixed-background and background-independent forms of the manifestly diffeomorphism-invariant ERG. The second topic is cosmological backreaction, which concerns the effect of averaging over high-frequency metric perturbations to the gravitational field equations describing the universe at large scales. This has been much studied the context of the unmodified form of General Relativity, but has been much less studied in the context of higher-derivative effective theories obtained by integrating out the high-energy modes of some more fundamental (quantum) theory of gravity. The effective stress-energy tensor for backreaction can be used directly as a diffeomorphism-invariant effective stress-energy tensor for gravitational waves without specifying the background metric. This thesis will construct the manifestly diffeomorphism-invariant ERG and compute the effective action at the classical level in two different schemes. We will then turn to cosmological backreaction in higher-derivative gravity, deriving the general form of the effective stress-energy tensor due to inhomogeneity for local diffeomorphism-invariant effective theories of gravity. This an exciting research direction, as it begins the construction of a quantum theory of gravity as well as investigating possible implications for cosmology.