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Numerical Relativity Beyond General Relativity

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Numerical Relativity Beyond General Relativity Numerical Relativity beyond General Relativity Thesis by Maria Okounkova In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2019 Defended May 1st, 2019 ii © 2019 Maria Okounkova ORCID: 0000-0001-7869-5496 All rights reserved iii Space may be the final frontier but it’s made in a Hollywood basement - Red Hot Chili Peppers, Californication iv ACKNOWLEDGEMENTS I have many wonderful colleagues to thank for the work presented in this thesis. Thank you to Saul Teukolsky, my advisor, for all of the physics discussions we have had over the years. Your suggestions and advice are always helpful. Thank you to Mark Scheel, for teaching me about numerical relativity, and patiently answering questions about code. Thank you to (Professor!) Leo Stein, for motivating me to delve deeper into theory. Thank you to Daniel Hemberger, for all of the work that you have put into developing a good, working code, and for your ever-inspiring coding practices. Thank you to Swetha Bhagwat, for being an amazing co-author. Thank you to the other members of the Simulating eXtreme Spacetimes (SXS) collaboration, for all of your help over the years. Thank you to my additional coauthors, Matthew Giesler, Duncan Brown, and Stefan Ballmer. Thank you to my committee members, Rana Adhikari, Yanbei Chen, Sergei Gukov, and Alan Weinstein, for your interest in this work. v ABSTRACT Einstein’s theory of general relativity has passed all precision tests to date. At some length scale, however, general relativity (GR) must break down and be reconciled with quantum mechanics in a quantum theory of gravity (a beyond-GR theory). Binary black hole mergers probe the non-linear, highly dynamical regime of gravity, and gravitational waves from these systems may contain signatures of such a theory. In this thesis, we seek to make gravitational wave predictions for binary black hole mergers in a beyond-GR theory. These predictions can then be used to perform model-dependent tests of GR with gravitational wave detections. We make predictions using numerical relativity, the practice of precisely numerically solving the equations governing spacetime. This allows us to probe the behavior of a binary black hole system through full inspiral, merger, and ringdown. We choose to work in dynamical Chern-Simons gravity (dCS), a higher-curvature beyond-GR effective field theory that couples spacetime curvature to a scalar field, and has motivations in string theory and loop quantum gravity. In order to obtain a well- posed initial value formalism, we perturb this theory around GR. We compute the leading-order behavior of the dCS scalar field in a binary black hole merger, as well as the leading-order dCS correction to the spacetime metric and hence gravitational radiation. We produce the first numerical relativity beyond-GR waveforms in a higher-curvature theory of gravity. This thesis contains additional results, all of which harness the power of numerical relativity to test GR. We compute black hole shadows in dCS gravity, numerically prove the leading-order stability of rotating black holes in dCS gravity, and lay out a formalism for determining the start time of binary black hole ringdown using information from the strong-field region of a binary black hole simulation. vi PUBLISHED CONTENT AND CONTRIBUTIONS [1] Maria Okounkova, Mark A. Scheel, and Saul A. Teukolsky. “Evolving Metric Perturbations in dynamical Chern-Simons Gravity”. In: Phys. Rev. D99.4 (2019), p. 044019. doi: 10.1103/PhysRevD.99.044019. arXiv: 1811.10713 [gr-qc]. M.O. conceived of the project, performed the analytical calculations, wrote the code, performed the simulations, analyzed the data, and wrote the manuscript. [2] Maria Okounkova, Mark A Scheel, and Saul A Teukolsky. “Numerical black hole initial data and shadows in dynamical Chern–Simons gravity”. In: Clas- sical and Quantum Gravity 36.5 (Feb. 2019), p. 054001. doi: 10.1088/ 1361-6382/aafcdf. M.O. conceived of the project, performed the analytical calculations, wrote the code, performed the simulations, analyzed the data, and wrote the manuscript. [3] Swetha Bhagwat et al. “On choosing the start time of binary black hole ringdowns”. In: Phys. Rev. D97.10 (2018), p. 104065. doi: 10 . 1103 / PhysRevD.97.104065. arXiv: 1711.00926 [gr-qc]. M.O. conceived of the project jointly with S.B., and the two wrote the code, performed the analytical calculations, performed the computations, analyzed the data, and wrote the manuscript. [4] Maria Okounkova et al. “Numerical binary black hole mergers in dynamical Chern-Simons gravity: Scalar field”. In: Phys. Rev. D96.4 (2017), p. 044020. doi: 10.1103/PhysRevD.96.044020. arXiv: 1705.07924 [gr-qc]. M.O. performed the analytical calculations, wrote the code, performed the simulations, analyzed the data, and wrote the manuscript. vii TABLE OF CONTENTS Acknowledgements............................... iv Abstract..................................... v Published Content and Contributions...................... vi Table of Contents................................ vii Chapter I: Introduction ............................. 1 1.1 A century of general relativity..................... 1 1.2 Gravity beyond general relativity ................... 1 1.3 Testing general relativity in the strong-field regime.......... 3 1.4 A brief introduction to numerical relativity.............. 7 1.5 Pushing numerical relativity beyond general relativity . 10 1.6 Looking forward............................ 16 Chapter II: Numerical binary black hole mergers in dynamical Chern-Simons gravity: Scalar field............................. 19 2.1 Introduction .............................. 19 2.2 Formalism ............................... 22 2.3 Results................................. 30 2.4 Discussion and future work ...................... 49 2.A Scalar field evolution formulation................... 52 2.B Pontryagin density in 3+1 split..................... 54 Chapter III: Numerical black hole initial data and shadows in dynamical Chern-Simons gravity............................ 57 3.1 Introduction .............................. 57 3.2 Solving for general metric perturbation initial data.......... 59 3.3 Solving for metric perturbations in dCS................ 70 3.4 Physics with dCS metric perturbations ................ 74 3.5 Conclusion............................... 86 3.A Perturbed extended conformal thin sandwich quantities . 86 3.B Reconstructing the perturbed spacetime metric............ 88 Chapter IV: Evolving Metric Perturbations in dynamical Chern-Simons Grav- ity and the stability of rotating black holes in dynamical Chern-Simons Gravity.................................... 91 4.1 Introduction .............................. 91 4.2 Dynamical Chern-Simons gravity................... 93 4.3 Evolving metric perturbations..................... 96 4.4 Evolving dCS metric perturbations ..................107 4.5 Results and discussion.........................113 4.A Perturbed 2-index constraint......................115 4.B Code tests ...............................120 Chapter V: Binary black hole collisions in dynamical Chern-Simons gravity . 125 viii 5.1 Introduction ..............................125 5.2 Methods ................................127 5.3 Perturbations to quasi-normal modes . 132 5.4 Results.................................136 5.5 Implications for testing general relativity . 147 5.6 Conclusion...............................153 5.A Choosing a perturbed gauge......................154 5.B Computing perturbed gravitational radiation . 156 Chapter VI: Numerical relativity simulation of GW150914 beyond general relativity...................................159 6.1 Introduction ..............................159 6.2 Results.................................161 6.3 Conclusion...............................168 Chapter VII: On choosing the start time of binary black hole ringdown . 170 7.1 Introduction ..............................170 7.2 Theory.................................173 7.3 Numerical implementation.......................189 7.4 Results.................................196 7.5 Conclusion...............................218 7.A Kerr-NUT parameters .........................221 Bibliography ..................................223 1 C h a p t e r 1 INTRODUCTION 1.1 A century of general relativity Over one hundred years ago, Albert Einstein put forth the theory of general relativity (GR), coupling spacetime to the matter and energy contained within [82]. In the century following this discovery, there was much progress in exploring the properties of this classical theory. The theory was found, for example, to contain black hole solutions [180]. Later, it was discovered that the theory contained spinning black hole solutions [112, 197]. Swiftly, scientists began to think not only about single black holes, but binary black hole systems. In binaries, two black holes orbit one another, inspiraling closer together through the emission of gravitational radiation, and ultimately merging in a violent, energetic process, to form one black hole. Theoretically computing the gravitational radiation (or gravitational waves) emitted by binary systems was of particular interest [135, 156]. The end of the century saw the first precise, numerical prediction of a full gravitational waveform from a binary black hole merger [161]. 1.2 Gravity beyond general relativity The same century, however, saw the development of quantum mechanics and quan- tum field theory as a description of nature. If the universe
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