Gravitational-Wave Polarizations: aFundamentalTestofGravity Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Fisica Candidate Samuel Patrone ID number 1691855 Thesis Advisor Co-Advisor Dr. Francesco Pannarale Prof. Alan J. Weinstein Academic Year 2019/2020 Thesis defended on 21 July 2020 in front of a Board of Examiners composed by: Prof. Paolo Mataloni (chairman) Prof. Maria Chiara Angelini Prof. Gianluca Cavoto Prof. John Russo Prof. Naurang Saini Prof. Francesco Santanastasio Prof. Raffaella Schneider Gravitational-Wave Polarizations: a Fundamental Test of Gravity Master’s thesis. Sapienza – University of Rome © 2020 Samuel Patrone. All rights reserved This thesis has been typeset by LATEX and the Sapthesis class. Author’s email: [email protected] A Nonna Mena, Reginella del mio cuore «The last enemy that shall be destroyed is death.» J. K. Rowling iii Acknowledgments This thesis is based on a research project the author carried out in the context of the Summer Undergraduate Research Program (SURF) at the California Institute of Technology LIGOLab in 2019. The participation in this experience was made possible by the sponsorship of INFN Sezione di Roma and Scuola Superiore di Studi Avanzati Sapienza (SSAS). A heartfelt and sincere thanks to my mentor Alan J. Weinstein for all the precious insights and support provided during that wonderful research experience, and for having me introduced to the exciting world of gravitational waves. Furthermore, a special thank goes to all the people who made my overseas summer unforgettable, among which I want to personally acknowledge Aaron, Alvin, Gabriele, Luca, Rico, Shruthi, all the fellow SURF students and all the LIGOLab graduate students, post- docs and professors. Nothing of this would have been possible without the precious hospitality of the California Institute of Technology and the Closson’s family, who opened the door of their house and their heart to make me feel at home even miles away from it. Thank you, Clemmy, Tim, and Aubrey, for being my west coast American family! I would like to express my deepest gratitude to my advisor Francesco Pannarale for having worked with me tirelessly at the final version of this dissertation. His adjustments and suggestions were essential for the good result of this thesis. I would like to extend my enthusiastic thanks to all my professors, teachers, and mentors for having me introduced to the wonderful path of knowledge. One among the others, Prof. Mattiacci, who taught me that a journey of a thousand miles begins with a single step and who was next to me when I made the first one. This thesis is the last milestone of a five-year-long journey, that brought me closer to the understanding of the Universe. Although aware that many things are still obscure and my entire life would not be enough to even grasp a sense of them, I have concluded that no one of us is left alone in the mystery of our existence. Thanks to all my friends for having shared part of their unique being with me. In particular, Alessio, Leonardo, Pierluigi, and Tilde, for being there no matter what over the years. Gioacchino and Daniele, for having walked together through the academic jungle and managed to survive. What this thesis will not show is how hard it can be to follow your dreams despite the circumstances. Difficulties are part of our life. They are not against us, they make us. However, what I know for sure is that nothing of what I am now would have been possible without the wholehearted support of my family. To my parents, Silde and Amato, to Zia Rosy, Nonno Mimmo, Nonna Pea, Zio Tore, and Zia Sara’s east cost family, at risk of sounding sickeningly sweet and inappropriate, I want to say vi amo from the bottom of my heart. And that’s all it is. The last lines of these thoughts are dedicated to Nonna Mena and Viviana, who are just waiting for me around the corner. And - again - to You, because I am. iv Abstract This thesis presents the results of an original and independent research that aims at assessing with software simulations the possibility of inferring the polarization content of a transient gravitational wave (GW) within a Bayesian framework, given the extended network of five ground-based interferometric detectors available in the near future. A theoretical review on the phenomenon of gravitational radiation in Metric Theories of Gravity is provided and some characteristics of GWs in alternative theories of gravity are reproduced. In particular, it is shown that each theory makes different predictions on the GW polarization modes: this is one of the reasons for which GW polarization measurements are among the strongest fundamental tests we can perform on gravity. In the central part of the dissertation, we illustrate the working principles of GW interferometers as well as the main data analysis techniques involved in processing GW data. The response in amplitude of a single interferometer to each polarization mode is shown to be determined by the geometry of the source-detector system (up to an overall normalization). In the last part of the thesis, the response of a network of GW interferometric detectors is studied in order to discriminate among different polarizations. The methodology developed — on simulated data — provides a powerful check for one of the fundamental predictions of General Relativity, and is thus of major significance in both the fundamental physics of gravity and the astrophysics of GW sources. We find that Bayesian inference can indeed accurately extract the polarization content of transient GW signals, with precision limited by the signal-to-noise ratio (SNR) in the response. As expected, the precision mostly improves with the total network SNR, with some exceptions due either to an SNR distributed very unevenly among the five detectors or to statistical fluctuations in the Bayesian sampler. Most of the code was written from scratch by the author. It is very flexible, as it allows for new ground-based interferometers with two or three arms to be easily added to (or removed from) the network. Such studies are of crucial importance since they can provide strong evidence in support of some particular configurations for future detectors. A technical note on this phenomenological study is publicly available on the LIGO Document Control Center [54]. v Contents Introduction viii 1 Metric Theories of Gravity and the Parametrized Post-Newtonian Formalism 1 1.1 Early Concepts and Newton’s Theory of Gravity . 2 1.2 The Einstein Equivalence Principle . 4 1.3 MetricTheoriesofGravity. 5 1.3.1 Definition ............................. 6 1.3.2 Classification . 6 1.3.3 The Strong Equivalence Principle . 7 1.4 The Parametrized Post-Newtonian Formalism . 8 1.4.1 The Geodesic Equation and The Newtonian Limit . 8 1.4.2 Post-Newtonian Variables and Expansion . 10 1.4.3 Post-Newtonian Potentials and Metric . 12 1.4.4 Post-Galilean Transformations and Preferred Frame Effects . 15 1.5 Metric Theories of Gravity: an Overview . 17 1.5.1 General Relativity . 17 1.5.2 Scalar-Tensor Theories . 18 1.5.3 Vector-Tensor Theories . 19 1.5.4 Other Kinds of Theories . 20 2 Gravitational-Wave Radiation in Metric Theories of Gravity 23 2.1 Wave Solutions in General Relativity . 24 2.1.1 A Century Long Debate . 24 2.1.2 Einstein’s Equations in Relaxed Form . 25 2.1.3 The DIRE Approach for Solving the Relaxed Field Equations 27 2.1.4 The TT-Gauge and the Quadrupole Formula . 29 Contents vi 2.2 Wave Solutions in Metric Theories . 31 2.2.1 Waveform............................. 32 2.2.2 Speed ............................... 32 2.3 Gravitational-Wave Polarizations . 36 2.3.1 Gravitational Potentials in the ppN Framework . 36 2.3.2 The E(2) Framework and Classification . 39 3 Gravitational-Wave Detection 45 3.1 Gravitational-Wave Interferometers . 46 3.1.1 The Michelson Interferometer . 46 3.1.2 Interaction of a Gravitational Wave and an Interferometer . 48 3.1.3 Fabry-Pérot Cavities . 48 3.2 Full Analysis of the Interferometer Response to a Gravitational Wave 50 3.2.1 Test Mass Displacement . 50 3.2.2 Polarizations . 51 3.2.3 Differential Arm Response Function . 53 3.3 Network of Interferometers and Overlaps . 58 3.4 Model-Independent Polarization Reconstruction . 63 3.4.1 Null Streams . 63 3.4.2 Sine-Gaussian Expansions . 63 4 Data Analysis Techniques 65 4.1 Noise.................................... 66 4.1.1 Random Processes . 66 4.1.2 Power Spectral Density and Noise Classification . 67 4.1.3 Typical Noise Sources for Gravitational-Wave interferometers 70 4.2 Matched Filter and Signal-to-Noise Ratio . 72 4.2.1 Filters, Kernels and Signal-to-Noise Ratio . 72 4.2.2 MatchedFilter .......................... 74 4.3 Bayesian Framework . 76 4.3.1 Statistical Inference Methods . 76 4.3.2 Bayes’Theorem ......................... 77 4.3.3 Bayesian Hypotheses on the Gravitational-Wave Polarization Content .............................. 78 4.3.4 Thelikelihood .......................... 79 4.3.5 Exploring the Parameter Space: the Nested-sampling Algorithm 80 Contents vii 5 A Phenomenological Study 83 5.1 Motivation, Methods and Goals . 84 5.1.1 Research Question and Methodology . 84 5.1.2 Preliminary Considerations on the Waveform . 86 5.2 Fully Polarized Gravitational Wave: the Network’s Response . 88 5.2.1 TheWaveform .......................... 88 5.2.2 Simulated Network Response . 90 5.3 Bayesian Inference of the Gravitational-Wave Polarization Content . 94 5.3.1 Tools ............................... 94 5.3.2 Results .............................. 96 Conclusions 102 A Visual Study of the Likelihood 104 B Python Code 107 Bibliography 138 viii Introduction «Und wir, die an steigendes Glück denken, empfänden die Rührung, die uns beinah bestürzt, wenn ein Glückliches fällt.» Rainer Maria Rilke, Duineser Elegien X Even if we do not realize it, we are constantly surrounded by waves.