The Pennsylvania State University The Graduate School LOOP QUANTUM COSMOLOGY: ANISOTROPIES AND INHOMOGENEITIES A Dissertation in Physics by Edward Wilson-Ewing c 2011 Edward Wilson-Ewing Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2011 The dissertation of Edward Wilson-Ewing was reviewed and approved∗ by the following: Abhay Ashtekar Eberly Professor of Physics Dissertation Advisor, Chair of Committee Martin Bojowald Associate Professor of Physics Nigel Higson Evan Pugh Professor of Mathematics Paul Sommers Professor of Physics Jayanth Banavar Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. Abstract In this dissertation we extend the improved dynamics of loop quantum cosmol- ogy from the homogeneous and isotropic Friedmann-Lemaˆıtre-Robertson-Walker space-times to cosmological models which allow anisotropies and inhomogeneities. Specifically, we consider the cases of the homogeneous but anisotropic Bianchi type I, II and IX models with a massless scalar field as well as the vacuum, in- homogeneous, linearly polarized Gowdy T 3 model. For each case, we derive the Hamiltonian constraint operator and study its properties. In particular, we show how in all of these models the classical big bang and big crunch singularities are re- solved due to quantum gravity effects. Since the Bianchi models play a key role in the Belinskii, Khalatnikov and Lifshitz conjecture regarding the nature of generic space-like singularities in general relativity, the quantum dynamics of the Bianchi cosmologies are likely to provide considerable intuition about the fate of such sin- gularities in quantum gravity. In addition, the results obtained here provide an important step toward the full loop quantization of cosmological space-times that allow generic inhomogeneities; this would provide falsifiable predictions that could be compared to observations. iii Table of Contents List of Figures vii Acknowledgments viii Chapter 1 Introduction 1 1.1 Quantum Gravity . 1 1.2 Loop Quantum Cosmology . 3 1.3 Organization . 8 Chapter 2 Bianchi Type I Models 10 2.1 Introduction . 10 2.2 Hamiltonian Framework . 13 2.3 Quantum Theory . 19 2.3.1 LQC Kinematics . 20 ˆ k 2.3.2 The Curvature Operator Fab ................. 22 2.3.3 The Quantum Hamiltonian Constraint . 28 ˆ 2.3.4 Simplification of Cgrav ...................... 32 2.4 Properties of the LQC Quantum Dynamics . 37 2.4.1 Relation to the LQC Friedmann Dynamics . 38 2.4.2 Effective Equations . 41 2.4.3 Relation to the Wheeler-DeWitt Dynamics . 44 2.5 Discussion . 46 Chapter 3 Bianchi Type II Models 51 iv 3.1 Introduction . 51 3.2 Classical Theory . 54 3.2.1 Diagonal Bianchi Type II Space-times . 54 3.2.2 The Bianchi II Phase Space . 57 3.3 Quantum Theory . 61 3.3.1 LQC Kinematics . 61 ˆi 3.3.2 The Connection Operator Aa ................. 63 3.3.2.1 Application to the Open FLRW Model . 66 3.3.3 The Quantum Hamiltonian Constraint . 66 3.3.3.1 A More Convenient Representation . 67 ˆ 3.3.3.2 The Fourth Term in CH ............... 68 ˆ 3.3.3.3 The Fifth Term in CH ................ 71 3.3.3.4 Singularity Resolution . 72 3.3.3.5 The Explicit Form of the Hamiltonian Constraint . 74 3.4 Effective Equations . 76 3.5 Discussion . 79 Chapter 4 Bianchi Type IX Models 82 4.1 Introduction . 82 4.2 Classical Theory . 84 4.3 Quantum Theory . 90 4.3.1 LQC Kinematics . 90 4.3.2 The Quantum Hamiltonian Constraint . 92 4.4 Effective Equations . 100 4.5 Discussion . 103 Chapter 5 From Bianchi I to the Gowdy Model 107 5.1 Introduction . 107 5.2 Bianchi I T 3 Model in Vacuo: Kinematics . 109 5.2.1 Vacuum Bianchi I Hamiltonian Constraint . 110 5.2.2 Superselection Sectors . 112 5.3 Bianchi I T 3 Model in Vacuo: Physical Structure . 113 5.3.1 Solutions to the Hamiltonian Constraint . 114 ˆ 5.3.2 The Operator U6 ........................117 5.3.3 Physical Hilbert Space . 120 5.4 Hybrid Quantization of the Gowdy T 3 Cosmologies . 120 5.4.1 Kinematical Structure and Hamiltonian Constraint Operator 121 5.5 Discussion . 124 v Chapter 6 Summary and Outlook 127 Appendix A Parity Symmetries 130 Appendix B The Closed Friedmann-Lemaˆıtre-Robertson-Walker Model 134 B.1 The Hamiltonian Constraint Operator . 135 B.2 Comments on the Different Quantizations . 136 Bibliography 138 vi List of Figures 2.1 Depiction of the LQG quantum geometry state corresponding to the LQC state |p1, p2, p3i, with the edges of the spin network traversing through the fiducial cell V. The LQG spin-network has edges paral- lel to the three axes selected by the diagonal Bianchi I symmetries, each carrying a spin label j = 1/2. 23 2.2 Edges of the spin network traversing the 1-2 face of V and an ele- mentary plaquette associated with a single flux line. This plaquette 2 encloses the smallest quantum of area, ∆ `Pl. The curvature oper- ˆ k ator F12 is obtained from the holonomy around such a plaquette. 24 vii Acknowledgments The work presented in this dissertation has been supported by Le Fonds qu´eb´ecois de la recherche sur la nature et les technologies, the Natural Sciences and Engi- neering Research Council of Canada, the National Science Foundation under grants PHY04-56913 and PHY0854743, the George A. and Margaret M. Downsborough Endowment, the Eberly research funds of the Pennsylvania State University and the Edward A. and Rosemary A. Mebus, David C. Duncan, and Home Braddock and Nellie and Oscar Roberts Graduate Fellowships of the Pennsylvania State University. I am very grateful for this support. I am most indebted to my advisor Abhay Ashtekar who has taught me so much about quantum gravity, who has always been available to answer my questions and who has, above all, shown me how to be rigorous in my research. I also thank my other collaborators for contributing in many ways to my under- standing of physics: Alejandro Corichi, Guillermo Mena Marug´an, Carlo Rovelli, Adam Henderson, Mercedes Mart´ın-Benito and Francesca Vidotto, as well as Mar- tin Bojowald who taught my graduate general relativity courses. I am also much obliged to Guillermo and Carlo for hosting me in Madrid and Marseille respectively, I am privileged to have had the opportunity to visit and work with them. I am thankful for having had the opportunity to interact with many people during my graduate studies here at Penn State; I would like to thank Miguel Campiglia, Dah-Wei Chiou, Jacobo Diaz-Polo, Jonathan Engle, Mikhail Kagan, Alok Laddha, Tomas Liko, Elena Magliaro, Simone Mercuri, William Nelson, Joseph Ochoa, Javier Olmedo, Tomasz Paw lowski, Claudio Perini, Andrew Ran- dono, Juan Reyes, Parampreet Singh, David Sloan, Victor Taveras, Manuel Tiglio, Artur Tsobanjan, Kevin Vandersloot and Nicolas Yunes. I also appreciate all that Randi Neshteruk and Kathleen Smith have done for me; their help has been invaluable. Finally, a big thank you to my family, Ron, Theresa, Tessa and Laura Rose, for their continued love, support and encouragement. It has meant a lot to me! viii For my family ix Chapter 1 Introduction 1.1 Quantum Gravity The two theories of quantum mechanics and general relativity revolutionized the field of physics in the twentieth century; both of these theories have been remark- ably successful and have greatly enhanced our understanding of the natural world. While quantum mechanics describes the small scale behaviour of elementary parti- cles, atoms and molecules, general relativity is the classical theory of gravity used to describe large systems such as the solar system, galaxies or even the universe as a whole. By and large, quantum effects are not relevant in the study of the macro- scopic systems described by general relativity, and since gravity is such a weak force, it can safely be ignored in the small-scale regimes where quantum effects are important. Due to this split, only one or the other of these theories is necessary to describe the vast majority of physical phenomena since the effects due to the other will be negligible. However, in some cases it is expected that quantum effects and gravity will simultaneously be important, especially when a large mass is confined in a small region as happens, for example, in the cases of black holes and the very early universe when the matter energy density was extremely high. In order to accurately describe the physics of these systems, one must develop a theory which combines the two: a theory of quantum gravity. To date, it has been extremely difficult to construct such a theory. There are currently several candidate theories including asymptotic safety, causal dynamical 2 triangulations, causal sets, noncommutative geometry, string theory, supergravity and the focus of this dissertation, loop quantum gravity. At this point, it is im- portant to note that all of these theories are currently incomplete and that none of them have yet offered any falsifiable predictions. Nonetheless, recent progress in many of these diverse approaches has been encouraging; in particular, there have recently been some interesting results in the field of loop quantum gravity. For an introduction to loop quantum gravity, see, e.g., [1, 2, 3]. Loop quantum gravity (LQG) is a background independent, nonperturbative approach to quantum gravity where the space-time geometry is treated quantum mechanically from the very beginning. One starts from a classical theory of gravity i where the elementary variables are an SU(2)-valued connection Aa and its con- a jugate momentum, the densitized triad Ei (i.e., triads with density weight one) [4, 5, 6, 7].