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Compact Gravitational Wave Sources SCUOLA DI DOTTORATO “VITO VOLTERRA” DOTTORATO DI RICERCA IN FISICA – XXII CICLO Compact Gravitational Wave Sources THESIS SUBMITTED TO OBTAIN THE DEGREE OF DOCTOR OF PHILOSOPHY (“DOTTORE DI RICERCA”) IN PHYSICS OCTOBER 2009 BY Francesco Pannarale Greco Program Coordinator Thesis Advisor Prof. Enzo Marinari Prof. Valeria Ferrari A mia Mamma If you have an apple and I have an apple and we exchange these apples, then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas. G.B. Shaw Acknowledgments As a first thing, I would like to thank my supervisor, Valeria Ferrari, for all of the encour- agement and support over the past three years and for always pushing further ahead my understanding of the physics behind problems. I am also deeply indebted to Leonardo Gualtieri for the many countless and endless discussions, the many answers he gave me, the many right and important questions he posed. I want to thank Luciano Rezzolla, for being such an inspiring thesis referee. Ever since the very first time I spoke to him, I have been learning so much physics and asking myself many fascinating questions, which is one of the foundations of this job. I also want to thank him and everybody at the Albert Einstein Institute for the three exciting weeks of research in Golm during my last year of PhD and for the hospitality and friendliness I encountered when visiting the Numerical Relativity group. I would also like to acknowledge Omar Benhar, Kostas Kokkotas and Anna Watts, with whom I have had brief but valuable work-related discussions and very pleasant non-work- related chats. I thank Stefania Marassi and Riccardo Ciolfi, the two other youngsters of the Rome rela- tivity group along with myself, for the support, the numerous useful tips and the entertaining company they guaranteed all along these three years. I am grateful to Corrado Mascia for the help he gave me when I was trying to solve a problem during my second year of PhD and for being the excellent calculus professor I was lucky to have during my undergrad courses. I owe a lot to Emanuele Berti, for all his precious advice and for being the friend that every frightened young scientist would like to have, and to Andrea Passamonti, for the great time I had with him in Pisa and for being so kind to me every time I contacted him. I thank my Mother, of course, but unfortunately words cannot be enough to express how grateful I am for all her help and all she has done for me and given me in these twenty-six years. Dedicating this thesis to her is a way to say that if I accomplished what I did, it was mainly because of her. It would have never been possible without her support. Now I have to walk on my own, I know I will be alright given all the things she has taught me. I will never forget how nice it was to finally be able to host my cousin Maria Antonietta and to get to know her better. She is such a special and wise person, and I wish her all the best in her life. The wonderful family atmosphere and the immense affection I found when I was hosted by Angela, Enzo, Antonio and her for my last conference will never be forgotten either. I am very very grateful for those fantastic days with them! Gianvito Laterza and Barbara Reeckman both deserve special motioning because of two very important conversations I had with them. They were crucial for me to be where I am now, and I owe them a lot for listening to me and for giving me their advice and full support. I thank all the “saletta”-related people for being fantastic friends, for their support, for the dinners we shared, for the parties we held, for all the dumb things and the smart things we have done together so far: Valentina, Valery, Antonio, Antonino, Paolo, Tommaso, Maria, Giancarlo, Andrea, Alessio, Chiara, Sergio, Pierre, Leone, Alessandro, Lisa. The first time I walked into our... ehm... office-playground-house, I could have never imagined what a lucky day that day was for me: I was about to meet all you great human beings by simply opening a door. I will miss our good old days and I hope our roads will cross as often as possible. During my eight years at La Sapienza, I have met special people who are still very present in my life even though we do not go to class together every morning anymore: Claudio and Nicoletta, with whom I have a very important and tight bond and who were great hosts in Copenhaghen, Federica, who is always very sweet and will run to see you any time you ask her to, Raffaello, who had me laugh throughout these years and has now moved to a better work life in my hometown. Luca deserves a place all to himself since he is the only one who is both a high school friend and a “saletta”-member. This summer he has finally switched to Macs: I had to wait thirteen years for this historical moment. I am very happy for him and I am quite sure Bill didn’t even notice, but then again, we Mac-people our better aren’t we? What about listening to a bit of Oasis now? I wanted to thank more friends that gave me all there support in several moments along these three years: Alessia, Angelo, Dario and Orsetta — who I was very lucky and pleased to get back in touch with — Dario, Gabriele, Eleonora and Samrin — my lifelong friends who I know I will never lose — Donatella and Viviana — who were so nice to host me after my first conference, which I will never forget — Domenico — with whom the whole PhD adventure started — and Manuela — with whom I shared many of the topical moments of our PhD. Special thanks go to John Draskovic and Kyle Howe, who I was very lucky to meet at a school and to get to know more during their visit in Rome. I had a great time with them and I hope our roads will cross again soon. viii Contents Introduction1 1 An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs7 1.1 Milestones in Gravitational Wave History..................7 1.2 Gravitational Wave Emission.........................8 1.3 Gravitational Wave Detection......................... 11 1.4 Compact Objects............................... 12 1.5 Neutron Stars................................. 13 1.5.1 Neutron Star Observed Properties.................. 14 1.5.2 Neutron Star Anatomy........................ 18 1.5.3 Neutron Stars as Continuous Sources................ 21 1.6 Compact Binaries............................... 22 1.6.1 Compact Binary Coalescences.................... 25 1.7 Gamma-Ray Bursts.............................. 28 I Analytic Models of Mixed Binaries 33 2 BH-NS and NS-NS Binaries 35 2.1 Quasi-Equilibrium and Pre-Merger Simulations............... 35 2.2 Inspiral and Tidal Disruption......................... 36 2.3 Dynamical Calculations of Mergers..................... 37 2.3.1 NS-NS Mergers............................ 38 2.3.2 BH-NS Mergers........................... 39 3 The Affine Model 41 3.1 Kerr Geometry: Parallel Transport and Tidal Tensor............. 42 3.1.1 Newtonian Tidal Tensor....................... 42 3.1.2 Relativistic Tidal Tensor....................... 43 3.1.3 The Kerr Metric and the Symmetric Tetrad............. 44 3.1.4 The Parallel Propagated Tetrad.................... 46 3.1.5 The Tidal Tensor Field for Equatorial Geodesics.......... 47 3.2 The Affine Model Equations......................... 49 3.2.1 The Affine Constraint........................ 49 3.2.2 The Principal Frame......................... 51 3.2.3 The Neutron Star Internal Dynamics................. 53 ix 3.2.4 Polytropic EOS and Newtonian Self-Gravity............ 56 3.3 Orbit Descriptions within the Affine Model................. 57 4 Two Important Improvements of the Affine Model 61 4.1 Pseudo-Relativistic Self-Gravity....................... 62 4.2 Using any Barotropic EOS.......................... 66 4.3 A Note About Solving the EOS-Dependent Integrals Numerically..... 66 5 BH-NS Coalescing Binaries: Quasi-Equilibrium Approach 69 5.1 Formulation.................................. 69 5.2 Code Calibration and Tests.......................... 71 5.3 Comparison with Recent Relativistic Results................. 72 5.4 New Applications of the Quasi-Equilibrium Approach........... 77 5.4.1 Effects of Varying the NS EOS and the BH Spin.......... 77 5.4.2 Varying the NS Mass and Using Other Equations of State...... 80 5.4.3 Measuring the NS Radius with GWs................. 93 6 BH-NS Coalescing Binaries: Dynamical Approach 103 6.1 Formulation.................................. 104 6.1.1 The Orbit Hamiltonian........................ 105 6.1.2 The Hamiltonian for the NS Fluid.................. 106 6.1.3 GW Dissipation............................ 107 6.1.4 Equations of Motion......................... 108 6.1.5 Initial Conditions........................... 108 6.2 Results..................................... 109 6.3 Future Developments............................. 111 II Towards QNMs of Rapidly Rotating Neutron Stars 113 7 Perturbation Theory, Quasi-Normal Modes and Instabilities 115 7.1 Eulerian and Lagrangian Approaches..................... 116 7.1.1 Gauge Freedom and Gauge Transformations............ 118 7.2 Black Hole Oscillations............................ 118 7.3 Oscillations of Relativistic Stars....................... 121 7.3.1 Perturbing a Non-Rotating Star................... 122 7.3.2 Effects of Rotation.......................... 124 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods... 126 7.4.1 The Boundary Conditions...................... 127 7.4.2 Spectral Methods for Stellar Oscillations.............. 131 8 The Perturbed Einstein Equations for a Rapidly Rotating Star 135 8.1 Geometry of a Rotating Star......................... 135 8.2 Perfect Fluids................................. 138 8.3 Boundary Conditions............................. 140 8.4 Determining the Perturbation Equations..................
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