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Astrophysical Implications of the Bumblebee Model of Spontaneous Lorentz Symmetry Breaking

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Astrophysical Implications of the Bumblebee Model of Spontaneous Lorentz Symmetry Breaking Astrophysical implications of the Bumblebee model of Spontaneous Lorentz Symmetry Breaking Gonçalo Dias Pereira Guiomar Thesis to obtain the Master of Science Degree in Engineering Physics Supervisor(s): Prof. Dr. Vítor Manuel dos Santos Cardoso Prof. Dr. Jorge Tiago Almeida Páramos Examination Committee Chairperson: Prof. Dr. Ana Maria Vergueiro Monteiro Cidade Mourão Supervisor: Prof. Dr. Jorge Tiago Almeida Páramos Members of the Committee: Prof. Dr. Amaro José Rica da Silva November 2014 Acknowledgments This thesis could not have been possible without the help and guidance of my supervisor, Professor Jorge Páramos, with whom I rediscovered the joy of doing Physics. His immense patience when dealing with my incompetence, along with his immense knowledge of unorthodox working places made this work a fun and rewarding experience. Also, I would like to thank Professor Vítor Cardoso for his help and availability in the process of realizing this thesis. For my family, I am truly grateful for your continuous support and for providing me the opportunity of realizing my goals, no matter how uncertain they might have seem in the past. For my friends, who accompanied me throughout this journey, thank you for joining me in my culinary digressions. This last paragraph I dedicate to Geisa, for helping me collapse onto a better state of being. i Abstract In this work the Bumblebee model for spontaneous Lorentz symmetry breaking is considered in the context of spherically symmetric astrophysical bodies. A discussion of the modied equations of motion is presented and constraints on the parameters of the model are perturbatively obtained. Along with this, a detailed review of this model is given, ranging from the questioning of the basic assumptions of General Relativity, to the role of symmetries in Physics and the Dark Matter problem. Keywords General Relativity, Bumblebee Model, Lorentz Symmetry Breaking, Stellar Equilibrium (English) iii Resumo Neste trabalho, consideramos o modelo Bumblebee para a quebra espontânea da simetria de Lorentz no contexto de corpos celestes com simetria esférica. Uma discussão das equações de movimento mod- icadas é apresentada, juntamente com os constragimentos do modelo obtidos de modo perturbativo. De modo a contextualizar o modelo e o problema em questão, uma revisão é apresentada, onde se abordam temas tais como os fundamentos da Relatividade Geral, o papel das simetrias na física e o problema da matéria escura. Palavras Chave Relatividade Geral, Modelo Bumblebee, Quebra da simetria de Lorentz, Equilíbrio Estelar v Contents 1 Introduction 2 1.1 Thesis Outline.........................................3 1.2 Essential concepts.......................................3 1.3 General Relativity.......................................4 1.4 The limits of Einstein's Relativity..............................5 2 Astrophysics 9 2.1 Introduction........................................... 10 2.2 Tolman-Openheimer-Volkov equation............................. 10 2.3 Polytropes and the Lane-Emden Equation.......................... 12 3 Symmetry Breaking 15 3.1 Introduction........................................... 16 3.2 Explicit and Spontaneous Breaking of Symmetries..................... 17 3.3 Observer and Particle LSB.................................. 20 3.4 Standard Model Extension and LSB............................. 21 4 Vector Theories 25 4.1 Introduction........................................... 26 4.2 Aether Theories......................................... 26 4.3 The Bumblebee Model..................................... 34 5 Astrophysical constraints on the Bumblebee 37 5.1 Introduction........................................... 38 5.2 Static, spherically symmetric scenario............................ 38 5.3 Perturbative Eect of the Bumblebee Field......................... 39 5.4 Numerical analysis....................................... 41 6 Conclusions 45 Bibliography 47 vii List of Figures 2.1 Lane-Emden solution for a spherical body in hydrostatic equilibrium............ 14 3.1 Snowakes generated through a Linenmayer rule set for the rst, second and third iterations............................................. 16 3.2 The Mexican Hat potential.................................. 19 4.1 The Bullet Cluster....................................... 28 5.1 Prole of the relative perturbations pb=p0, pV =p0, ρb/ρ0 and ρV /ρ0 induced by the Bumblebee............................................ 42 5.2 Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb..... 43 5.3 Allowed region (in grey) for a relative perturbation of less than 1% for pV and ρV .... 44 ix List of Tables 5.1 Selected stars that used as models for the numerical analysis of the Bumblebee pertur- bation............................................... 43 5.2 Table of non-dimensional parameters............................. 44 xi Abbreviations GR - General Relativity TOV - Tolman-Oppenheimer-Volko LE - Lane-Emden EP - Equivalence Principle LLI - Local Lorentz Invariance LPI - Local Position Invariance EP - Equivalence Principle WEP - Weak Equivalence Principle SEP - Strong Equivalence Principle LSB - Lorentz Symmetry Breaking PPN - Parametrized Post Newtonian CPT - Charge Parity Time Symmetry GZK - Greisen-Zatsepin-Kuzmin CMBR - Cosmic Microwave Background Radiation CMBR - Cosmic Microwave Background Radiation HI-RES - High Resolution Fly's Eye SME - Standard Model Extension SM - Standard Model SSB - Spontaneous Symmetry Breaking QED - Quantum Electrodynamics MOND - Modied Newtonian Dynamics FLRW - Friedmann-Lemaitre-Roberston-Walker VEV - Vacuum Expectation Value 1 1 Introduction Contents 1.1 Thesis Outline...................................3 1.2 Essential concepts.................................3 1.3 General Relativity.................................4 1.4 The limits of Einstein's Relativity.......................5 2 1.1 Thesis Outline 1.1 Thesis Outline This work will begin by a small introduction of the relevant concepts needed from General Relativity in the subsection 1.2 and the current observed limitations of the theory in subsection 1.4. The Tolman- Oppenheimer-Volko equations are then introduced in Chapter2 along with the Lane-Emden (LE) model for spherical bodies, which will be needed in order to understand Chapter5. In Chapter3, the concept of Lorentz Symmetry Breaking (LSB), in both its explicit and spontaneous form, is introduced as a way to portray one of the fundamental properties of the model being used, the Bumblebee Model. This will later be introduced in4 as a particular case of the more general Aether Theories. The two nal chapters (5,6) contain the kernel of this work: The application of the models presented as a way of constraining their parameters using a spherical astrophysical body modeled by a polytrope. This thesis closely follows the work done in [1]. 1.2 Essential concepts Presently, Einstein's general theory of relativity serves as a tool to understand a wide range of phenomena. From the dynamics of compact astrophysical bodies, such as stars and black holes, to cosmology, its striking predictions have sustained this theoretical framework and made it one of the most relevant scientic achievements in the history of science [2]. The experimental conrmation of the existence of gravitational lensing, time dilation and gravi- tational redshift have solidied this fact but this by no means imply that the theory is completely correct; in fact, such consistence is a motivating factor for testing its limits even further. Questioning the basic assumptions of General Relativity is thus a valid way to achieve this goal, as they are the fundamental rule set from which it emerges. These basic assumptions can be expressed in the following manner [3]: Weak Equivalence Principle - Bodies in free fall have the same acceleration independently of • their compositions. Local position invariance - The rate at which a clock ticks is independent of its position. • Local Lorentz invariance - The rate at which a clock ticks is independent of its velocity. • This work shall focus on the second and third basic assumptions presented above. A more detailed discussion will be presented in3 in how this principle can be used to test the validity of physical theories. As a short introduction to the review that will be made in the following section, a small appetizer is given in the context of special relativity, showing how one can test the limits of such theory. The tests are based on testing the principles of relativity (physical laws are independent of the inertial frame of reference used to describe them) and the constancy of the speed of light. In the case of inertial frames of reference (which can always be found, at least in the vicinity of any given point in spacetime), this translates into invariance under Lorentz transformations, a tenet of Special Relativity: deviations from these transformations would also imply deviations from the underlying principles. 3 1. Introduction This can be approached through the Robertson-Sexl-Mansouri formalism, which consists on the known Lorentz transformations, vx t 2 x vt t0 = − c ; x0 = − ; y0 = y; z0 = z; (1.1) 1 v2 1 v2 − c2 − c2 q q altered in a way as to express a preferred frame of reference Σ(T; X~ ). The transformations now take the form, t ~.~x ~x 1 1 ~v~x T = − ; X~ = ~v + ~vT: (1.2) a d − d − b v2 With these transformations, it can be shown [3] that through an expansion of a; b; d and ~ around v=c2 one can obtain the following expression for the relative shift in the two way
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