Global dynamics of the universe

Boersma, Jelle Pieter

Global dynamics of the universe

Doctoral Thesis

2000

Abstract

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 centred 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. This thesis is structured as follows. In the first chapter it is shown how different sections of the Kruskal geometry can be identified in a way which preserves time-orient ability of the spacetime. The existence of topologically different but locally identical solutions of Einstein's equations is well known, and not surprising considering the differential structure of these equations. also discuss the occurrence of Hawking radiation in topologically different black-hole geometries. Furthermore, we study the relation between black-hole solutions and circular cosmic strings. Assuming the existence of circular 1 cosmic string with deficit angle ranging between 0 and 211", we are able to construct a class of non-trivial vacuum solutions with properties similar to black-hole solutions but with a more complicated topology. In the second chapter of this thesis, we focus on the averaging problem in cosmology. The averaging problem occurs when one attempts to model a realistic inhomogeneous universe by a more symmetric model. Although averaging is often implied when studying realistic cosmological models, a rigorous treatment of averaging in cosmology appears to be surprisingly difficult. One difficulty which occurs when one tries to specify an averaging procedure is related to the large number of unphysical degrees of freedom which are present in the problem, namely, the coordinate freedom and the gauge freedom. The coordinate freedom manifests itself when one tries to evaluate the average of tensorial quantities, since the components of a tensor depend on the local choice of a frame. One may attempt to avoid this problem by specifying a local frame and evaluating some kind of average for each component separately. However, since there is no choice of frame which is preferred for physical reasons, this gives rise to a considerable amount of ambiguity. When one follows a perturbative approach, there is an additional freedom of choosing a gauge, which makes it ambiguous what one means by a perturbation of a physical quantity, even when this quantity does not depend on the local choice of frame. By specifying a choice of gauge, it becomes well defined what one means by a perturbation, but once again no choice of gauge seems to be preferred for physical reasons. In addition to these problems, there is an inherent ambiguity which is related to the freedom in choosing an averaging operation. Since there is generally more than one choice of averaging operation which is mathematically consistent, one needs to impose additional constraints which restrict the freedom of choosing an averaging operation. However, one would like to do so on the basis of a minimal set of assumptions. It is shown that each of these problems can be resolved in the case where perturbations theory can be applied. We use our results to calculate the lowest order non-trivial correction to the expansion of the observable universe, which is due to the fact that averaging does not commute with evaluating the (nonlinear) Einstein equations. In the third chapter of this thesis, we investigate the relation between surface terms which are evaluated at spatial infinity, and the local dynamics of a scalar field. Starting from the path-integral approach to quantum field theory, it is shown that the contribution of surface terms to the variation of the action functional cannot in general be neglected. The classical field equations can be derived by requiring that the variation of the action vanishes for all field perturbations, and it is shown that a surface term generally contributes a non-trivial source term to the classical field equations. This source term appears to vanish in spatially flat geometries, but it diverges in a spatially open geometries with super curvature perturbations. Rather surprisingly, it appears that the degrees of freedom of the scalar field which generate surface terms must have zero norm in the space of square integrable field 2 perturbations. Without restricting these zero-norm degrees of freedom, it follows that the local dynamics of the field are sensitive to details of the spacetime at spatial infinity. The main difficulty which we are confronted with consists of quantifying the zero-norm degrees of freedom. We briefly discuss a strategy for resolving this problem. In the fourth chapter we discuss different types of inflation. As is well known, the standard idea of inflation provides a simple explanation for the homogeneity of the observed universe. However, it appears to be much less straightforward to reconcile a period of inflation with the observed negative spatial curvature in the universe. Bubble inflation combines these two aspects, but it requires a rather restricted type of potential. After introducing the established ideas of standard inflation and bubble inflation, we focus on the dynamics of bubble spacetimes. It is shown that the often used thin-wall approach is not consistent with the assumption that the stress-energy is generated by a scalar field, although this assumption plays a crucial role in the theory of bubble-dynamics. In order to resolve this problem, we derive a simplified set of equations which describe the exact dynamics of a general spherically symmetric bubble spacetime. We then focus on the question of whether the restrictions on the shape of the potential, which are essential in the bubble inflation scenario, are necessary in order to explain the generation of negative spatial curvature during inflation. By studying the most generic situation where constant-scalar field hypersurfaces make a transition from being spacelike to being time like , it is shown that negative spatial curvature is generated under conditions which are more generic than the conditions which are generally assumed. The results which are presented in this thesis have been obtained through independent research, which was conducted by the author on an individual basis. The contents of the first three chapters have been published, [1] - [3], excluding the third section of the first chapter, which was added recently. The contents of the last chapter are currently being prepared for submission. None of the results which are obtained in this thesis have, to the best of my knowledge, been published elsewhere, or the original work has been cited.

Keywords

Mathematics and Applied Mathematics

Cosmology

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PhD / Doctoral

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