The central problem studied in this thesis is, broadly speaking, the issue of coarsegraining
in GR approximations, and the effect of averaging on the field equations. The
important observation made is that there are some smoothing procedures implicit in the
standard, homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker cosmological
models. The point is that if such effects are not allowed for, we may actually be using
the wrong field equations in cosmology. There has been recently an increased effort in
this direction with some interesting results, as for example that the coarse-graining effects
could be non-negligible in the context of affecting the age of the universe.
The central idea explored at length in this thesis (see chapter three) is the possibility
of applying the Renormalization Group (RG) concepts in gravitation to tackle the averaging
problem. Research presented in this thesis produced results as follows: In section 3.6
an explicit smoothing-out procedure for inhomogeneous cosmologies is introduced. This
approach is implemented in a "3 + 1" formalism and invokes the coarse-graining arguments,
provided and supported by the real-space RG methods in an analogy with lattice
models of Statistical Mechanics. One of the results obtained is a re-interpretation of the
Ricci-Hamilton flow in terms of a RG flow, thereby providing the Ricci-Hamilton flow
with a physical meaning and showing how the averaging problem is rooted in the geometry.
Moreover, block variables are introduced and the recursion relations written down
explicitly, making thus possible a study of the system's critical behaviour. The criticality
is discussed and it is argued that it may be related to the formation of sheet-like structures
in the universe. Moreover, the explicit expression for the renormalized Hubble constant
is proposed. A discussion follows of the consequences of this approach in cosmology and
astrophysics.
Finally in chapter four the evolution of perturbations is studied in the dust-radiation
FLRW universe model. This is done in the framework of dynamical systems theory which
seems well suited to this purpose. The evolution of density perturbations is presented in
the form of phase diagrams. Some scales are discovered that are over-damped.
A number of ideas of relevance for future research are summarized in chapter five.
