Cosmic Bubble Collisions Observable Signature of A Classical Transition Wei Lin A Thesis Submitted to the Faculty of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Master of Science Graduate Program in Physics and Astronomy York University, Toronto, Ontario May, 2015 c Wei Lin, 2015 ii Abstract The theory of inflation was introduced to resolve many existing observational problems in cosmology. Inflation becomes eternal when a region of space continuously spawns non- inflation regions. This process arises from metastable vacua in a potential landscape. Our universe could be a realization of one of these many vacua predicted by the theory. In this thesis, we explored the idea that a universe can be born via a collision between two bubble universes. This process is known as a \Classical Transition". In this thesis, the potential observability of relics produced during the collision is studied. If a classical transition did happen in the past, its presence is imprinted on the CMB temperature anisotropy. In order to determine the pattern due to a primordial bubble collision, numerically simulating the entire history of a bubble collision is necessary. After running the simulation, we extracted the comoving curvature perturbation (which has a direct connection to the temperature fluctuations) from spatial slices. In our analysis, we 2 calculated the observed CMB quadrupole for two models, D2 ≈ 1100[µK ], as well as its correlation to the local curvature, Ωk. In both models we studied, we showed that the produced signatures resulting from a classical transition can vary a lot. In addition, we surveyed the boundary of a classical transition for polynomial potentials in the model where false vacua are above the inflationary plateau. We showed that p the position (in field space) of a classical transition boundary has a 1= relation to the slow-roll parameter, . In the second model where the false vacuum occupied by the nucleated bubbles are below the inflationary plateau, we confirmed that it is also possible to produce a classical transition regardless of the surrounding vacuum energy. However, the relative vacuum energy difference of the newly formed universe to its environment will potentially affect the post-collision behaviours of domain walls. In this study, we show there exist three different types of geometry in the domain wall (repulsive, marginal repulsive and oscillatory) when different surrounded vacuum energy is presented. We investigated a variety of vacua energy in the marginal limit in which the domain wall unveils a marginal repulsive geometry and showed that the new formed universe always has a vacuum energy slightly greater than its surroundings. Acknowledgements During this work, it is impossible for me to complete it without assistance from many people. First of all, I must thank Matthew C. Johnson for being a great supervisor and advisor. He leads me to this fascinating field of cosmology. In these couple years, I have a diverse experience in different research areas related to cosmology. The skills and broad knowledge required for studying the field of cosmology is more than I can imagine. Without his help, I would have not learnt this much. Moreover, I would like to thank my examine committee members, Michael Haslam and Roman Koniuk, for pro- viding advices in this thesis to make it more complete. Next, many thanks to a friend of mine, Yu-Wei Huang, for helping me to illustrate some beautiful pictures in this work. Doing research in the field of physics relies a strong background knowledge. For my undergraduate education in physics, I would like to thank those who had taught me and inspired me. Especially, I am grateful to Randy Lewis. He first introduced the subject of quantum mechanics to me and initiate my curiosity to physics. From time to time, when we have conversations about physics, his talk always enlighten me to think physics at a different angle. In the same year, I had my first research experience with Chorng-Yuan Huang from National Central University, Taiwan. Many thanks to him for supervising me, giving me this opportunity and sharing his research experience as his career. Last but not the least, I would like to thank my colleagues, Alex Terrana, Ryan Deneault, Jesse Rogerson and George Conidis for offering hospitality and comfortable research en- vironment, and of course, my family in Taiwan for supporting and encouraging me to continue my interest in physics. 11th May, 2015 Wei Lin iii Contents Abstract ii Acknowledgements iii Contents iv List of Figures vi Abbreviations viii 1 Introduction1 1.1 What's Inflation?................................2 1.1.1 Motivation...............................3 1.1.2 Potential Landscape..........................4 1.1.3 Different Inflationary Models.....................4 1.1.4 The Eternal Nature of Inflation....................6 1.2 Bubble Creation Mechanisms.........................8 1.2.1 Quantum Transition..........................8 1.2.2 Classical Transition.......................... 10 1.3 Previous Work................................. 11 1.4 Observing A Bubble Collision Signature................... 13 1.4.1 Comoving Curvature Perturbation.................. 14 1.4.2 Simulation Procedure......................... 16 1.4.3 Simulation Inputs and Outputs.................... 21 1.4.4 Signatures Resulting from Different Mechanisms.......... 23 2 Modelling 25 2.1 Constructing Model and Goals........................ 25 2.2 Gaussian-Quadratic Potential Model..................... 26 2.2.1 Preliminary for AIP.......................... 27 2.2.2 Preliminary for BIP.......................... 30 3 Simulation Results and Analysis 33 3.1 CMB Quadrupole Moment.......................... 34 3.2 Simulation Predictions for D2 and Ωk .................... 36 3.3 Trends in Different Models........................... 43 3.4 Near The Classical Transition Boundary................... 52 3.4.1 Different Polynomial Potentials.................... 53 iv Contents v 4 Conclusion 59 A Preliminary Instanton Profile 63 00 B R and a0 Profiles 66 B.1 AIP....................................... 66 B.2 BIP....................................... 74 Bibliography 78 List of Figures 1.1 Old Inflation Model..............................5 1.2 New Inflation Model..............................6 1.3 Bubble Nucleations at Different Time Slices.................6 1.4 Potential Landscape..............................8 1.5 From Instanton to Bubble Collision...................... 10 1.6 Conformal Diagram of A Classical Transition................ 12 1.7 ξ verses x .................................... 17 2.1 Potential Landscape and Bubble Collision (BIP).............. 26 2.2 Gaussian Barrier for AIP........................... 28 2.3 Instanton Profiles for AIP........................... 29 2.4 Different Geometrical Behaviours of A Domain Wall for BIP....... 32 3.1 D2 for the AIP case with kinematics factor................. 37 3.2 D2 for the AIP case with potential shape factor............... 38 3.3 D2 for the BIP case.............................. 40 3.4 Ωk for the AIP case with kinematics factor................. 41 3.5 Ωk for the AIP case with potential shape factor............... 42 3.6 Ωk for the BIP case.............................. 43 3.7 Correlation between D2 and Ωk for the AIP case with kinematics factor. 44 3.8 Correlation between D2 and Ωk for the AIP case with potential shape factor 45 3.9 Correlation between D2 and Ωk for the BIP case.............. 46 3.10 Trend in R00 with Fixed Kinematics..................... 48 3.11 Trend in a with Fixed Kinematics...................... 49 3.12 Potential for the BIP case........................... 51 3.13 Contour Plot For a Marginal Repulsive Geometry.............. 52 3.14 Constant Spatial Surface at ξ0 = 0...................... 53 3.15 Perturbation Frozen Point (σ − β2)...................... 54 3.16 Perturbation Frozen Point (∆φ2 − β2).................... 55 3.17 Growth of Lumpiness............................. 56 3.18 Comparison Btween Potentials of Higher Powers.............. 57 3.19 Phase Diagram for a Classical Transition................... 58 3.20 Slow-roll Parameter verses Instanton Jump Size............... 58 A.1 Instanton Profiles for Constant R ....................... 63 A.2 Instanton Profiles for Constant φamp ..................... 64 A.3 Potential and Instanton Profile for various β2 ................ 65 B.1 R00 for the kinematics factor.......................... 66 vi List of Figures vii B.2 R00 for the kinematics factor (log scaled)................... 67 B.3 a0 for the kinematics factor.......................... 68 B.4 a0 for the kinematics factor (log scaled)................... 69 B.5 R00 for the potential shape factor....................... 70 B.6 R00 for the potential shape factor (log scaled)................ 71 B.7 a0 for the potential shape factor....................... 72 B.8 a0 for the potential shape factor (log scaled)................. 73 B.9 R00 for the BIP case.............................. 74 B.10 R00 for the BIP case (log scaled)....................... 75 B.11 a0 for the BIP case............................... 76 B.12 a0 for the BIP case (log scaled)........................ 77 Abbreviations GR General Relativity dS de Sitter space AdS Anti de Sitter space CDL Coleman De Luccia HM Hawking Moss CMB Cosmic Microwave Background LSS Large Scale Structure SZe Sunyaev Zel'dovich effect AIP Above Inflationary Plateau BIP Below Inflationary Plateau FRW Freidmann Robertson Walker viii Chapter 1 Introduction Cosmology is the study of the Universe. This includes