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.
