Includes bibliographical references. / In this thesis we consider four different topics in the field of cosmology, namely, black hole topology, the averaging problem, the effect of surface terms on the dynamics of classical and quantum fields, and the generation of an open universe through inflation with random initial conditions. It should be mentioned that while the research for this thesis was being done, no large effort was made to pursue a single theme. One reason for the diversity of the topics in this thesis is that the results which came out of this research were not always the results which were expected to be found when the investigation was started. Another reason for looking at several topics is simply that once a problem has been solved, then it is natural to move on to another problem which has not yet been solved. For those readers who value that a thesis is centered around a single unifying theme, let me mention that each of the four topics in this thesis are indeed related. Namely, each topic which we discuss focuses on an aspect of the global dynamics of the universe, in a situation where this is non-trivially different from the local dynamics. The non-trivial relation between global and local dynamics is rarely addressed in cosmology. Partially this is because of the difficulties which arise when one considers a realistic universe with infinitely many coupled degrees of freedom. Hence, it is a common practice to rely on simplifications which reduce the number of degrees of freedom, or the couplings between them. Further, there are few direct observations which probe the large-scale dynamics of the universe, or none at all, depending on the length scale and the type of cosmological model which one considers. As a consequence, there is a considerable freedom in choosing a priori assumptions or simplifications in the field of cosmology, without being able to falsify the validity thereof. For instance, when we analyse the relation between field perturbations at spatial infinity and perturbations here and now, we assume that quantum field theory, as we know it, is valid everywhere between here and spatial infinity. Although one cannot avoid making certain fundamental assumptions, the type of simplifications which are adopted in a calculation plays a less fundamental role. It is the objective of this thesis to improve our understanding of the large scale dynamics of the universe by showing rigorously what one can and what one cannot derive from certain fundamental assumptions. Interestingly, our results are often quite different from the results which are based on the same assumptions, but which involve certain commonly made simplifications as well.

Includes bibliographical references. / The work presented in this thesis forms part of a programme undertaken in collaboration with G.F.R. Ellis and R. Maartens, the primary aim of which is to examine in detail how cosmological observations may be used to determine the large scale structure of space-time (see, e.g., refs. [1-5]). In order to place this work in context as part of the ongoing cosmological enterprise, it is necessary to review briefly the main objectives of cosmology, the basic assumptions underlying most modern attempts to achieve these objectives, and the observational as well as philosophical status of these assumptions.

Numerical simulations have become an indispensable tool for understanding the complex non-linear behavior of many physical systems. Here we present two numerical investigations in cosmology. The first is posed in the context of inhomogeneous solutions to General Relativity. We lay out formalism for calculating observables in an arbitrary spacetime, for an arbitrary placed observer. In particular, we calculate the area distance, redshift and transverse motion across the observers sky. We apply our method to the Szekeres metric, and develop code in MATLAB to implement it. We successfully demonstrate that the code works for the FLRW and LT special cases, and then investigate some Szekeres models with no spherical symmetry. The second project is posed in the context of chameleon gravity. Recently, it was argued that the conformal coupling of the chameleon to matter fields created an issue for early universe cosmology. As standard model degrees of freedom become non-relativistic in the early universe, the chameleon is attracted towards a "surfing" solution, so that it arrives at the potential minimum with too large a velocity. This leads to rapid variations in the chameleon's mass and excitation of high energy modes, casting doubts on the classical treatment at Big Bang Nucleosynthesis. We propose the DBI chameleon, a consistent high energy modification of the chameleon theory that dynamically renders it weakly coupled to matter during the early universe thereby avoiding the breakdown of calculability. We demonstrate this explicitly with numerical simulations.

The lightcone gauge is a set of what are called the observational coordinates adapted to our past lightcone. We develop this gauge by producing a perturbed spacetime metric that describes the geometry of our past lightcone where observations are usually obtained. We then connect the produced observational metric to the perturbed Friedmann-Lemaître-Robertson-Walker metric in the standard general gauge or what is the so-called 1+3 gauge. We derive the relations between these perturbations of spacetime in the observational coordinates and those perturbations in the standard metric approach, as well as the dynamical equations for the perturbations in observational coordinates. We also calculate the observables in the lightcone gauge and rederive them in terms of Bardeen potentials to first order. A verification is made of the observables in the perturbed lightcone gauge with those in the standard gauge. The advantage of the method developed is that the observable relations are simpler than in the standard formalism. We use the perturbed lightcone gauge in galaxy surveys and galaxy number density contrast. The significance of the new gauge is that by considering the null-like light propagations, the calculations are much simpler since angular deviations are not considered. Standard cosmology based on General Relativity is generally believed to have serious shortcomings, such as the unexplained issues of dark matter and dark energy. As a remedy, many alternative theories of gravitation have been proposed over the years, one of which is ƒ(R) gravity. We explore classes of irrotational-fluid cosmological models in the context of ƒ(R) gravity in an attempt to put some theoretical and mathematical restrictions on the form of the ƒ(R) gravitational Lagrangian. In particular, we investigate the consistency of the linearised dust models for shear-free cases as well as in the limiting cases when either the gravito-magnetic or gravito-electric components of the Weyl tensor vanish. We also discuss the existence and consistency of classes of non-expanding irrotational spacetimes in ƒ(R)-gravity. Furthermore, we explore exact ƒ(R) gravity solutions that mimic Chaplygin-gas inspired ΛCDM cosmology. Starting with the original, generalized and modified Chaplygin gas equations of state, we reconstruct the forms of ƒ(R) Lagrangians. The resulting solutions are generally quadratic in the Ricci scalar, but have appropriate ΛCDM solutions in limiting cases. These solutions, given appropriate initial conditions, can be potential candidates for scalar field-driven early universe expansion (in ation) and dark energy-driven late-time cosmic acceleration.

No description available.

Includes bibliographical references. / The main work of this thesis can be summarised as: ■ An implementation of canonical quantisation to the covariant and gauge-invariant approach to cosmological perturbations. Standard results are reproduced. We discuss the advantages of this formalism over non-covariant and non gauge-invariant formalisms. ■ A characterisation of linear gravitational waves in a covariant way is achieved. The evolution equations for the electric and magnetic parts of the Weyl tensor are shown to be of different order. In particular, the electric part appears to have a third order evolution equation, while the magnetic part has a second order evolution equation. It is shown that the "silent" nature of the evolution equations for irrotational dust can be extended to the case of vortical dust. This may be relevant for the endpoints of gravitational collapse since the vorticity begins to grow as soon as density contrast becomes non-linear, as is the case in galaxies, showing that the irrotational silent universes are unstable. The main problem in accepting such vortical silent universes lies in proving integrability of the equations which has not been achieved so far, even in the irrotational case. ■ A review of issues in the Cosmic Microwave Background (CMB) is given, focussing particularly on points such as ergodicity, decaying modes, foreground contamination, recombination, spectral distortions and polarisation of the CMB. ■ A review of methods in gravitational lensing is presented, together with a hierarchy of distance measures in cosmology, forming an introduction to the following two chapters. ■ A common belief that photon conservation implies that the all-sky averaged area distance in inhomogeneous universes must be that of the background, matter-averaged Robertson-Walker area distance is dis proven. This means that there will in general be gravitational lensing effects even on large angular scales. ■ The realistic situation in which gravitational lensing leads to caustic formation is discussed. It is claimed that this invalidates many accepted beliefs concerning high-redshift observations in inhomogeneous universes. One application of importance is the CMB. Possible implications are discussed. ■ Random Gaussian fields are ubiquitous in modern statistical physics, and particularly important in CMB studies. Here we give accurate analytical functions approximating ∫e⁻ˣ²dx, the simplest of which is just the kink soliton.

Bibliography: pages 84-88. / The central aim of this thesis is to consider aspects of the spherically symmetric Lemaitre-Tolman-Bondi (LTB) solution as a model of the post-decoupling universe. To do this comprehensively is a massive task and is not our aim here. Indeed, far from it, we will concentrate on select instances of this programme and attempt in some places to indicate possibilities for further study. There are many solutions of the EFE which satisfy what we consider to be 'reasonable criteria' for a cosmology and others that do not. The LTB solution may be accepted as a reasonable cosmological model because ■ It allows non-empty solutions. ■ It allows expanding solutions. ■ It has a homogeneous and isotropic limit. ■ It allows for inhomogeneity.

Includes bibliographical references. / Recently the universe has been modeled in the covariant sense, in terms of fluid models and perturbations thereof, leading to Gauge Invariant Covariant (GIC) perturbations of these fluid models. It is well known that kinetic theory provides a physically sound and consistent description of the matter and radiation in the universe, so a perturbative theory of gas models using kinetic theory would be most helpful. This has been done to a large degree in the Gauge Invariant (GI) Bardeen approach to perturbation theory by studies of gases based on the relativistic Boltzmann equation. These treatments, however, were not fully covariant. The GI Bardeen approach is dependent on a co-ordinate choice, while in the full GIC perturbation theory full covariance is maintained along with gauge invariance by describing the theory in a particular set of perturbation variables that differ from the Bardeen choice but can be related to the Bardeen variables. The covariant formulation of the relativistic Boltzmann equation in terms of variables that are of use in the GIC theory for gases has been well described. In this thesis, I provide both a good introduction to the full GIC perturbation theory of a photon gas and matter fluid system in the linear theory as well as a solid grounding with respect to the exact FLRW fluid model upon which most of the original ideas and concepts of modern cosmology are based. The introduction to the exact FLRW model is done in the sense of the dynamical systems approach to cosmology which provides the easiest access to understanding the evolution of single and multi-fluid FLRW models.

In Quantum Cosmology, universe states are treated as wave function solutions to a zero-energy Schroedinger equation that is hyperbolic in its second derivatives of spatial geometries and matter-fields. In order to select one wave function (that would in principle correspond to our Universe) out of infinitely many, requires an appropriate boundary condition. The Hartle-Hawking No Boundary and the Vilenkin Tunneling proposals are examples of such boundary conditions. We review their applications and shortcomings in the context of the Inflationary Scenario. Another boundary condition is that of S.W. Hawing and D.N. Page (1990) in the context of wormholes. Wormholes are generally considered to play a major role in setting the cosmological constant to zero and to provide a mechanism for black hole evaporation. It is significant that we are able to show that even the class of bulk matter wormhole instantons found by Carlini and Mijic (1990) are predicted in the quantum theory. However, unresolved issues and newfound problems seem to threaten the wormhole theory. Furthermore, since there are no a priori notions of time (and space) present in the quantum theory, it is important to show exactly how the notion of time is recovered over distances much larger than the Planck scale. A good notion of time is also essential for any quantum theory to predict the correct classical behaviour for the Universe today. The issue of time inevitably re-emerges throughout our work.

L'histoire séculaire des systèmes dynamiques puise ses originesdansle développement du cadre mathématique en astronomie.L'objet de cette thèse est l'étude de propriétés de la gravitation de ce point de vue de la dynamiqueà différentes échelles cosmologiques.Dans la théorie du potentiel, l'isochronie définit généralement le mouvement d'oscillation harmonique de pendules.En 1959, le mathématicien et astronome Michel Hénon étend cette définition afin de caractériser les oscillationsorbitales d'étoiles, autour du centre du système à symétrie sphérique auquel elles appartiennent.Dans ce cas, la période d'oscillation peut dépendre de l'énergie de l'étoile.Aujourd'hui, son potentiel isochrone est majoritairement utiliséen simulation numérique pour ses propriétés analytiques d'intégrabilité, mais demeure par ailleurs souvent méconnu.Dans cette thèse, nous revisitons la caractérisation géométrique de l'isochronie comme initiée par Michel Hénon etcomplétons ainsi la famille des potentiels isochrones en physique. La classification de cet ensemblesous l'action de divers groupes mathématiques met en évidence une relation privilégiéeentre les isochrones.Nous montrons alors la nature keplérienne intrinsèque aux isochrones, laquelle est au coeur dela nouvelle relativité isochrone que nous présentons.Les conséquences de cette relativité en mécanique céleste, à savoirla généralisation de la troisième loi de Kepler, celle de la transformation de Bohlin ou Levi-Civita,et le théorème de Bertrand, conduisent à l'analyse du résultat d'un effondrement gravitationnel.Une analyse isochrone est développée pour caractériser un état de quasi-équilibredesystèmes auto-gravitants isolés, comme certains amas stellaires ou galaxies dynamiquement jeunes,à partir de propriétés orbitales de leurs étoiles ou contenu physique.A l'échelle cosmologique, la dynamique de l'univers dépendde sa composition énergétique. Elle peut s'exprimer sous forme d'unsystème dynamique conservatif, bien connu en écologie pour décrire la dynamiquede populations variées. Ce modèle dit de Lotka-Volterra est exploité pour décrireun espacetemps globalement homogène et isotrope, dont les composantes peuventêtre en interaction non uniquement gravitationnelle.Dans cet univers jungle, des comportements dynamiques effectifs à grande échelle pourraient conduire à une expansionaccélérée de l'univers sans nécessité d'énergie noire. / Dynamical systems have a centuries-long history with roots going back to the mathematical development for astronomy. In the modern formalism, the present thesis investigates dynamical properties of gravitation at different astrophysical or cosmological scales.In potential theory, isochrony often refers to harmonic oscillations of pendulums. In 1959, the mathematician and astronomer Michel Hénon introduced an extended definition of isochrony to characterize orbital oscillations of stars around the center of the system to which they belong. In that case, the period of oscillations can depend on the energy of the star. Today, Michel Hénon’s isochrone potential is mainly used for its integrable property in numerical simulations, but is not widely known. In this thesis, we revisit his geometrical characterization of isochrony and complete the family of isochrone potentials in physics. The classification of this family under different mathematical group actions highlights a particular relation between the isochrones. The actual Keplerian nature of isochrones is pointed out and stands at the heart of the new isochronerelativity, which are presented together.The consequences of this relativity in celestial mechanics — a generalization of Kepler’sThird law, Bohlin or Levi-Civita transformation, Bertrand’s theorem — are applied to analyze the result of a gravitational collapse. By considering dynamical orbital properties, an isochrone analysis is developed to possibly characterize a quasi-stationary state of isolated self-gravitating systems, such as dynamically young stellar clusters or galaxies.At a cosmological scale, the dynamics of the universe depends on its energy content. Its evolution can be expressed as an ecological dynamical system, namely a conservative generalized Lotka-Volterra model. In this framework of a spatially homogeneous and isotropic spacetime, named Jungle Universe, the dynamical impact of a non-gravitational interaction between the energy components is analyzed. As a result, effective dynamical behaviors could account for an accelerated expansion of the universe without dark energy.