Perturbation theory and averages in inhomogeneous cosmologies.

This thesis will discuss the standard model of cosmology, ΛCDM, along with standard perturbation theory. We will thereafter, explore alternatives to ΛCDM. In Chapter 2 we shall discuss a brief history of the development of GR and introduce the standard model of cosmology. We will also discuss various pieces of observational evidence that support ΛCDM and those that ΛCDM struggles to explain. Finally, we will introduce inhomogeneous cosmologies and the basis of backreaction. In Chapter 3 we introduce the 3+1 and 1+3 formalisms. The 3+1 formalism in particular is the basis for numerical relativity (NR) and averaging procedures that Buchert uses to develop his theory of backreaction. In particular, we will introduce the notion of 3–dimensional hypersurfaces in a 4–dimensional space- time, different types of curvature, and the various projections of the Einstein equations from a 4–dimension manifold to the 3–dimensional hypersurfaces. In Chapter 4 we introduce how the standard model of cosmology accounts for inhomogeneities in the CMB and models the evolution of structure — standard perturbation theory. We will discuss the mathematical basis of perturbation theory, the solutions to the perturbed Einstein equations, and the problem of gauges, with examples of various gauges. In Chapter 5 we present the latest form of the Buchert formalism [3] with additional derivations and details. We will discuss the 3+1 formalism in the context of the Buchert formalism with three timelike congruences, introduce the fluid-extrinsic averaging formalism, and the fluid-intrinsic averaging formalism. We shall make contact with various parts of the literature and discuss challenges that the Buchert formalism overcomes/improves on. In Chapter 6 we present the post-Newtonian formalism which goes beyond standard linear perturbation theory with the aim of modelling the formation of non-linear structures. We will also investigate the various gauges used in simulations to determine their viability in the post-Newtonian expansion. We also conduct and original investigation extending the Buchert averaging scheme to the post- Newtonian expansion with an example of how the averaged evolution equations of Buchert may be used in simulations. In Chapter 7 we perform an original investigation of backreaction effects in a particular Szekeres model. To do so we will introduce the cosmic quartet and the pseudo cosmic quartet. A similar investigation was undertaken by Bolejko, however, we use different parameters, a different density profile, and the aforementioned pseudo cosmic quartet. We will discuss our results and then compare them with Bolejko’s results