On the Gravitational Wave Spectrum of Compact Relativistic Objects Dissertation der Mathematisch-Naturwissenschaftlichen Fakultät der Eberhard Karls Universität Tübingen zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt von Sebastian H. Völkel aus Öhringen Tübingen 2019 Gedruckt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Eberhard Karls Universität Tübingen. Tag der mündlichen Qualifikation: 08.05.2020 Dekan: Prof. Dr. Wolfgang Rosenstiel 1. Berichterstatter: Prof. Dr. Kostas D. Kokkotas 2. Berichterstatter: Prof. Dr. Wilhelm Kley 3. Berichterstatter: Prof. Dr. Emanuele Berti on the gravitational wave spectrum of compact relativistic objects doctoral thesis in physics by sebastian h. völkel university of tübingen faculty of science institute for astronomy and astrophysics theoretical astrophysics first supervisor: prof. dr. kostas d. kokkotas second supervisor: prof. dr. wilhelm kley external reviewer: prof. dr. emanuele berti october 2019 i Preface This doctoral thesis has been carried out at the University of Tübingen in the Institute for Astronomy andAstro- physics under supervision of my first supervisor Prof. Dr. Kostas D. Kokkotas and my second supervisor Prof. Dr. Wilhelm Kley. As the title suggests, the present work is on the gravitational wave spectrum of compact rela- tivistic objects. In a series of projects, more specific questions in this context have been proposed and addressed. These have been reported in nine regular journal publications1 [ , 2, 3, 4, 5, 6, 7, 8, 9] and one proceedings [10]. For two of the publications, related blog articles in the accompanying blog of the journal have also been prepared [11, 12]. After acknowledgements, abstract, and table of contents, the thesis is organized into three main parts.Part I provides the theoretical foundations. Chapter 1 outlines the exciting field of general relativity and particularly gravitational wave physics suitable for any interested reader with a background in physics. Chapter 2 then provides a more detailed review of compact objects, while chapter 3 studies the perturbation equations describing gravitational waves. Chapter 4 explains the WKB methods for the direct and inverse spectrum problem. Part II is on applications and results. It contains the research output of this thesis and is based on the journal publications. Part III provides an overall discussion in chapter 13 and final conclusions in chapter 14. ii iii Acknowledgments Personal As all good journeys come to an end, so too must my undergraduate and doctoral studies in Tübingen. After these great years of freedom and learning, I want to thank the many people that accompanied me. First and foremost I express my deepest gratitude to my family, my brother Oliver and parents Heinz and Dorothea Völkel, for the continuous support and help on which I could always rely on. I am also indebted to Lisa-Marie Huber for the wonderful time we have spent together. Clearly this thesis would not have been possible without my first supervisor Kostas Kokkotas. My Gratitude is not limited to his scientific experience that guided me towards the research projects, but also includes the possibilities for valuable and honest discussion. His liberal way of supervising provided me with various oppor- tunities to follow my scientific interests and facilitated the many conferences and summer schools I visited. The scientific idea proposed by him to work on the inverse spectrum problem of compact objects ignited mycreative urge and turned out to be a fruitful endeavour. At this point, I also have to include my second supervisor Wil- helm Kley, whose door has always been open for me. I want to thank both for their support in me, especially for the PhD scholarship Landesgraduiertenförderung and the 69th Lindau Nobel Laureate Meeting. From my times as undergraduate student, I also want to thank Andrea Santangelo, for the opportunity to work on a research project in his high energy astrophysics group. Furthermore, I thank him and Josef Jochum, for their trust in me in assisting them in their observational cosmology lecture. I will also not forget the (not quite) isospectral brass drums that David Wharam arranged for both of our curiosity. During my years in the theoretical astrophysics group, many people came and left, but I always felt at home. The list of people that made this possible is (un)fortunately longer than I can acknowledge here. Nevertheless, I want to explicitly mention those that accompanied me for the longest time and helped me whenever I needed them. These are Andreas Boden, Gela Hämmerling, Severin Frank, Pantelis Pnigouras, János Schmidt, Christian Krüger, Andrew Coates, Dimitra Tseneklidou, Salvatore Vultaggio, Lorenzo Küchler, Verena Krall, Panagiotis Herbe, Sourabh Nampalliwar, Andrea Maselli, Kostas Glampedakis, Daniela Doneva, Stoytcho Yazadjiev, Roman Konoplya, Jacopo Soldateschi, Arthur Suvorov, Zeljko Grljusic and more. Furthermore I want to thank Andrew Coates, Gela Hämmerling, and Andreas Boden for their valuable time to proofread parts of the manuscript of this thesis. Of course I am also indebted to Heike Fricke, especially for all the help and management of the bureaucratic issues in academia. Honest thanks also go to many members of the neighboring computational physics group of Wilhelm Kley. Here I want to particularly thank Christoph Schäfer for all the positivity, ski touring and hospitality. I wish him and his family all the best. From all the fellow students that I had the privilege to meet during my undergraduate studies, I explicitly want to thank Kevin Marolt, for all our coffee breaks and outdoor activities, as well as Lucas Jordan, Pit Burgbacher, Philip Wolf, Viktor Bach, Martin Bohnert, Timo Schössler, Patrick Müller, Timo Storzer, Manuel Nonnenmacher, Daniel Mieg, and Jonas Leister, for the countless hours inside and outside the university. Finally, there are two friends from my early school days that strongly influenced my early academic interests and decisions. I thank Artur Lobanov for the many discussions and unique visits to his workplaces at CERN and at DESY. Speaking with him about physics, while he was already studying, motivated me a lot to pursue science. I also thank Patricia Wowra for coming up with the idea to visit the Institute for Astronomy and Astrophysics in Tübingen during a school internship (BOGY). It was a truly inspirational week and a key reason to study physics in Tübingen. iv v Funding Besides acknowledging individual people, I also want to give credit to the different organizations and funding bodies that made this work and some of my trips possible. First, I acknowledge support of the PhD scholar- ship Landesgraduiertenförderung by the State Ministry of Baden-Württemberg for Sciences, Research and Arts, awarded through the University of Tübingen. Furthermore, I am indebted to the Baden-Württemberg Stiftung for the financial support of some research projects by the Eliteprogramme for postdocs of Daniela Doneva.I acknowledge support from the COST Action CA16104 “GWverse” and Cost Action MP1304 “NewCompStar”. I thank the Wilhelm and Else Heraeus Foundation for visiting seminars in the Physikzentrum Bad Honnef and partial support for the 69th Lindau Nobel Laureate Meeting. Hereby, I also acknowledge the support of the Nobel Laureate Foundation for attending this unique meeting. I received travel support from the Deutscher Akademis- cher Austauschdienst e.V. (DAAD) trough the project on “Neutron Star Physics in the Era of Gravitational Wave Astronomy” for visits to Thessaloniki. Finally, the Kepler Center for Astro and Particle Physics at the University of Tübingen for attending repeatedly their PhD workshops. vi vii Zusammenfassung Kompakte Objekte sind für die Gravitationswellen-Astronomie, die in den letzten Jahren bedeutende Erfolge verbuchen konnte, von großem Interesse. Darunter befinden sich der erste direkte Nachweis von Gravitations- wellen, welche von verschmelzenden Schwarzen Löchern und Neutronensternen stammen, sowie der dazuge- hörige Nobelpreis in Physik im Jahre 2017. Das erste mit elektromagnetischer Strahlung aufgenommene Bild eines supermassiven Schwarzen Lochs wurde 2019 der Weltöffentlichkeit vorgestellt. Die vorliegende Disser- tation behandelt das Gravitationswellen-Spektrum von kompakten Objekten und basiert hierbei überwiegend auf Einsteins allgemeiner Relativitätstheorie. Die Dissertation umfasst mehrere Studien über unterschiedliche Aspekte axialer gravitativer Störungen sowie skalarer Testfeldern, welche in der Hintergrund-Raumzeit unter- sucht werden. Diese beschreiben Neutronensterne, Schwarze Löcher und verschiedene andere Modelle von ex- trem kompakten Objekten. Der überwiegende Teil dieser Arbeit widmet sich dem inversen Spektrum Prob- lem, bei dem Eigenschaften einer Quelle anhand seines Spektrums rekonstruiert werden sollen. Darüber hin- aus werden auch Studien über semi-klassische Eigenschaften Schwarzer Löcher behandelt. Bei diesen han- delt es sich sowohl um Hawking Strahlung, als auch um die Theorie der Oberflächenquantisierung Schwarzer Löcher. Um semi-analytische Berechnungen durchführen zu können, liegt der Fokus hierbei auf sphärisch symmetrischen und statischen Raumzeiten. Für diese entkoppeln die zugrundelegenden Störungsgleichungen voneinander und ergeben eine Wellengleichung, ähnlich zur zeitunabhängigen Schrödingergleichung mit einem Potentialterm. Wegweisende Abhandlungen gehen hier auf Regge und Wheeler 1957, sowie Zerilli 1970 zurück. Beide