https://theses.gla.ac.uk/ Theses Digitisation: https://www.gla.ac.uk/myglasgow/research/enlighten/theses/digitisation/ This is a digitised version of the original print thesis. Copyright and moral rights for this work are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This work cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Enlighten: Theses https://theses.gla.ac.uk/ [email protected] SELF-SIMILAR COSMOLOGICAL MODELS by DAVID ALEXANDER B.Sc. Thesis submitted to the University of Glasgow for the degree of Ph.D. Department of Physics and Astronomy, The University, Glasgow G12 8QQ. November 1988 (c) David Alexander 1988 ProQuest Number: 10998215 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10998215 Published by ProQuest LLC(2018). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 For my Father A curious thing about tensors is tensors have traces and norms. Their tops are made out of vectors. Their bottoms are made out of forms. There's stress and pressure and one that measures the distance from P to Q. But the most wonderful thing about tensors, is the one called Gjjy. D. Alexander and A.G. Emslie, composed in *'The Aragon'', Byres Rd., Glasgow, May 1987. CONTENTS Page PREFACE (i) ACKNOWLEDGEMENTS (iii) SUMMARY (v) CHAPTER 1 REVIEW OF CURRENT COSMOLOGICAL IDEAS 1 1.1 Preamble 1 1.2 The Observable Universe 2 1.3 Theories for Large-Scale Structure in the Universe 6 1.4 The Early Universe 22 1.5 Exotic Cosmologies 31 CHAPTER 2 GLOBAL SYMMETRIES IN COSMOLOGY 41 2.1 Introduction 41 2.2 Symmetries of Spacetime 47 2.3 Self-Similar Symmetries 56 2.3.A Geometrical Approach 56 2.3.B Classical Hydrodynamical Approach 61 2.4 Applications of Self-Similarity 72 CHAPTER 3 SELF-SIMILAR INHOMOGENEOUS SPACETIMES WITH A NON-ZERO COSMOLOGICAL CONSTANT 80 3.1 Introduction 80 3.2 Formation of Self-Similar Solutions 85 3.3 Addition of a Matter Inhomogeneity Parameter A 97 3.4 Inhomogeneous (A*0) Solutions 100 3.4.A General Formulation 100 3.4.B A<0 103 3.4.C A>0 103 3.5 Conclusions 114 Page CHAPTER 4 SELF-SIMILAR IMPERFECT FLUID COSMOLOGIES 117 4.1 Introduction 117 4.2 Effect of Viscous Stresses on a Cosmological Fluid 118 4.3 General Formalism of a Viscous Cosmology 127 4.4 Self-Similar Representation of a Viscous Cosmology 133 4.5 Numerical Solutions to the "Viscous" Field Equations 139 4.5.A Solutions with Equation of State, P+T=0 142 4.5.B Solutions with Equation of State, P+T=€ 161 4.6 Conclusions 172 CHAPTER 5 FORMATION OF BLACK HOLES IN SELF-SIMILAR ANISOTROPIC UNIVERSES 175 5.1 Introduction 175 5.2 Black Hole Similarity Solutions 179 5.3 Effect of Anisotropy on the Formation of Primordial Black Holes 185 5.4 Conclusions 196 CHAPTER 6 FUTURE WORK 198 6.1 Review 198 6.2 Asymptotic Behaviour of Monotonic Self-Similar Solutions 199 6.3 Applications of Self-Similar Symmetry of the Second Kind 203 6.4 Geometric Interpretation of Self-Similar Symmetry 208 APPENDIX: Conformal Motions and Self-Similarity 209 REFERENCES 215 PREFACE In this thesis, cosmological models, which admit self-similar symmetries, are examined. Symmetries in cosmology have become increasingly important since the formulation of the Einstein field equations demonstrating the close correspondence between geometry and physics. Physical investigations of gravitational systems have benefited greatly from the applications of geometric techniques and, in particular, from the consideration of various symmetries which lead to a simplification of the relevant equations. One such symmetry is that of self-similarity, which was initially developed as a physical symmetry in the study of hydrodynamics and is now associated with the more global symmetries of conformal and homothetic motions. Self-similar symmetry is particularly useful in the study of cosmology, since the Universe can be treated as a hydrodynamic fluid (or a geometrical manifold) in which there are no characteristic scales and so may be expressible by the techniques of self-similarity. In Chapter 1, a general review of the current developments in the study of cosmology is given. In particular, the 'intrusion’ of particle physics into the realm of cosmology is discussed. The application of particle physics theories, which has resulted in a better understanding of the Universe at early epochs and has helped to explain the origin of large-scale structure in the Universe, is also outlined briefly. Finally, exotic cosmological theories which attempt to go beyond Einstein’s theory of general relativity are addressed. The application of differential geometry to cosmology is discussed in Chapter 2. The description of the Universe as a four-dimensional manifold is considered and the various symmetries which may be imposed on such a manifold are described. The symmetry of self-similarity is then introduced together with a discussion of its developmentin hydrodynamics and its applications to the study of the dynamics of the Universe. Spatially-inhomogeneous spacetimes with a non-zero cosmological constant are investigated in Chapter 3. The role of self-similarity is discussed and solutions of this spacetime, which admit a similarity symmetry, are considered. Integrals of the motion are determined and these are related to the degree of anisotropy and inhomogeneity of any given solution. The solutions found are then discussed in the context of the cosmological "no-hair" theorems which consider the effect of a large vacuum term on the expansion of the Universe. In Chapter 4, the effects of viscosity and shear on the evolution of a cosmological model are considered. A self-similar analysis is carried out in which the viscosity coefficients vary in a prespecified manner, and two different classes of solution are investigated. These solutions differ in the choice of the equation of state, which is chosen to represent the extreme cases of a 'viscous dust’ and a 'stiff’ Universe. The self-similar stiff solutions are then developed to consider the growth of primordial black holes in the early Universe in Chapter 5. Chapter 6 includes a brief review of the work of the thesis and suggests a few interesting lines for future research. The original work of this thesis is contained in Chapters 3-5 and also in the Appendix. The contents of Chapter 3 have been accepted for publication in Monthly Notices and different aspects of this work have also appeared in various conference proceedings. The work of Chapters 4 and 5 is currently being developed for publication. iii ACKNOWLEDGEMENTS The work in this thesis was carried out while the author was a research student in the Department of Physics and Astronomy, University of Glasgow and a research assistant in the Department of Physics, University of Alabama in Huntsville. I extend my thanks to the staff and students of both these departments, who have made my last three years both interesting and enjoyable. I would also like to thank them for tolerating my numerous inane prattlings and stupid questions. I would like to thank my joint supervisors, Robin Green in Glasgow and Gordon Emslie in Huntsville, for providing a calming influence or a kick, whenever one or the other was required. I am indebted to both Robin and Gordon for their friendship, encouragement and perserverance. I also thank Gordon for introducing me to the delights of Huntsville-Alabama, parachuting, Wendys and Margueritas (Long Live Casa Gallardo!!!), and for providing me with the occasional foray into the world of solar physics. Other members of the department deserve special mention. A standard, and well deserved acknowledgement, in Astronomy theses from Glasgow, is to Professor P. A. Sweet and, like my predecessors, I am grateful to 'The Prof.’ for his encouragement and for the many discussions of things astronomical and physical. Thanks: to Professor John Brown for helping me to procure funds for my trips to Alabama, Oxford and Cambridge and for providing me with the opportunity to continue my career in research; to Drs. John Simmons and Ken M cClements (now at Oxford) for allowing me to play squash under the pretence of 'collecting data’; to Drs. Alex MacKinnon and Bert van den iv Oord for straightening out my naive attempts at understanding solar physics; to Dr. Alan Thompson for sharing his knowledge of the multifarious computer systems which abound in the Astronomy Group; to Mr. Andrew Liddle for being another cosmologist, although on the other side of the fence; to Daphne, whose organisational and coffee-making abilities have helped smooth the way for many a student’s research; to Christine for typing the page numbers. I would like to thank Professor I. S. Hughes for allowing me to carry out my research in the Department of Physics and Astronomy and to the Science and Engineering Research Council for providing the necessary funding.