Abstract:
The asymptotic behaviour at late times of inhomogeneous axion-dilaton cosmologies is investigated. The space-times considered here admit two abelian space-like Killing vectors. These space-times evolve towards an anisotropic universe containing gravitational radiation. Furthermore, a peeling-off behaviour of the Weyl tensor and the antisymmetric tensor field strength is found. The relation to the pre-big-bang scenario is briefly discussed.

Abstract:
This is a review of cosmological models prepared for the Pont d'Oye workshop on the origin of structure in the universe. The classes of models are discussed in turn, and then some of their uses are considered.

Abstract:
We study a class of inhomogeneous and anisotropic $G_2$ string cosmological models. In the case of separable $G_2$ models we show that the governing equations reduce to a system of ordinary differential equations. We focus on a class of separable $G_2$ M-theory cosmological models, and study their qualitative behaviour (a class of models with time-reversed dynamics is also possible). We find that generically these inhomogeneous M-theory cosmologies evolve from a spatially inhomogeneous and negatively curved model with a non-trivial form field towards spatially flat and spatially homogeneous dilaton-moduli-vacuum solutions with trivial form--fields. The late time behaviour is the same as that of spatially homogeneous models previously studied. However, the inhomogeneities are not dynamically insignificant at early times in these models.

Abstract:
Some exact solutions for the Einstein field equations corresponding to inhomogeneous $G_2$ cosmologies with an exponential-potential scalar field which generalize solutions obtained previously are considered. Several particular cases are studied and the properties related to generalized inflation and asymptotic behaviour of the models are discussed.

Abstract:
In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods: LeVeque's multi-dimensional high-resolution integration scheme which allows accurate evolution of solutions containing discontinuities or steep gradients, and an adaptive mesh refinement (AMR) algorithm which enables the local resolution of a simulation to vary dynamically in response to the behaviour of the evolved solution. A family of hyperbolic formulations of the Einstein equations is derived by generalization of an evolution system proposed by Frittelli and Reula, and numerical solutions produced using these formulations are compared to solutions produced using alternative reductions of the evolutions equations. Properties of the harmonic time slicing condition are also investigated, and analytic and numerical results concerning the formation of coordinate singularities are presented. Numerical simulations are performed of planar cosmologies, described using Gowdy's reduction of the Einstein equations, and U(1)-symmetric cosmologies, described using Moncrief's reduction of the Einstein equations, with the spacetimes in both cases being vacuum. Numerical studies follow up on the work of Berger, Moncrief and co-workers, with attention being focused on the small-scale features that develop in the models and the behaviour of linear and nonlinear gravitational waves.

Abstract:
In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaitre models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i.e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.

Abstract:
We present exact inhomogeneous and anisotropic cosmological solutions of low-energy string theory containing dilaton and axion fields. The spacetime metric possesses cylindrical symmetry. The solutions describe ever-expanding universes with an initial curvature singularity and contain known homogeneous solutions as subcases. The asymptotic form of the solution near the initial singularity has a spatially-varying Kasner-like form. The inhomogeneous axion and dilaton fields are found to evolve quasi-homogeneously on scales larger than the particle horizon. When the inhomogeneities enter the horizon they oscillate as non-linear waves and the inhomogeneities attentuate. When the inhomogeneities are small they behave like small perturbations of homogeneous universes. The manifestation of duality and the asymptotic behaviour of the solutions are investigated.

Abstract:
The averaging problem for inhomogeneous cosmologies is discussed in the form of a disputation between two cosmologists, one of them (RED) advocating the standard model, the other (GREEN) advancing some arguments against it. Technical explanations of these arguments as well as the conclusions of this debate are given by BLUE.

Abstract:
We examine the effects of spatial inhomogeneities on irrotational anisotropic cosmologies by looking at the average properties of anisotropic pressure-free models. Adopting the Buchert scheme, we recast the averaged scalar equations in Bianchi-type form and close the standard system by introducing a propagation formula for the average shear magnitude. We then investigate the evolution of anisotropic average vacuum models and those filled with pressureless matter. In the latter case we show that the backreaction effects can modify the familiar Kasner-like singularity and potentially remove Mixmaster-type oscillations. The presence of nonzero average shear in our equations also allows us to examine the constraints that a phase of backreaction-driven accelerated expansion might put on the anisotropy of the averaged domain. We close by assessing the status of these and other attempts to define and calculate `average' spacetime behaviour in general relativity.

Abstract:
A combination of analytic and numerical methods has yielded a clear understanding of the approach to the singularity in spatially inhomogeneous cosmologies. Strong support is found for the longstanding claim by Belinskii, Khalatnikov, and Lifshitz that the collapse is dominated by local Kasner or Mixmaster behavior. The Method of Consistent Potentials is used to establish the consistency of asymptotic velocity term dominance (AVTD) (local Kasner behavior) in that no terms in Einstein's equations will grow exponentially when the VTD solution, obtained by neglecting all terms containing spatial derivatives, is substituted into the full equations. When the VTD solution is inconsistent, the exponential terms act dynamically as potentials either to drive the system into a consistent AVTD regime or to maintain an oscillatory approach to the singularity.