University of Groningen The Final State Tsang, Ka Wa DOI: 10.33612/diss.160948839 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2021 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Tsang, K. W. (2021). The Final State: the fate of relativistic compact objects after merger. University of Groningen. https://doi.org/10.33612/diss.160948839 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 03-10-2021 The Final State The fate of relativistic compact objects after merger Ka Wa Tsang A thesis presented for the degree of Doctor of Philosophy Nikhef The Netherlands 2020 Summer © 2020 Ka Wa Tsang ISBN: 978-94-6419-125-7 Printed in the Netherlands by: Gildeprint Cover design by: Helena CYC This work is part of the research program supported by the Nederlandse organisatie voor Weten- schappelijk Onderzoek (NWO). It was carried out at the Nationaal Instituut voor Subatomaire Fysica (Nikhef) in Amsterdam, the Netherlands. The Final State The fate of relativistic compact objects after merger PhD thesis to obtain the degree of PhD at the University of Groningen on the authority of the Rector Magnificus Prof. T. N. Wijmenga and in accordance with the decision by the College of Deans. This thesis will be defended in public on Friday 05 Mar 2021 at 16:15 hours by Ka Wa Tsang born on 1 June 1992 in Hong Kong Supervisor Prof.dr. C.F.F. van den Broeck Co-supervisor Prof.dr. A. Mazumdar Assessment Committee Prof.dr. J. van den Brand Prof.dr. D. Roest Prof.dr. R. Snellings ø)述ª^的®耀；y¼³揚他的K段。 這日0£日|ú言語；這夜0£夜³ú知X。 !言!語，_!r音可}。 它的Ï6通遍)下，它的言語³00u。 (iÇ19：1-4上) The heavens are telling the glory of God, and the firmament proclaims the work of his hands. Every day they pour forth speech, and every night they tell knowledge. There is no speech and there are no words; their sound is inaudible. Yet in all the world their line goes out, and their words to the end of the world. (Psalm 19:1-4a) CONTENTS Page 1 Introduction 1 2 Gravitational waves 7 2.1 Conventions and notations . 7 2.1.1 Units and constants . 7 2.1.2 Einstein summation convention . 7 2.1.3 Metric tensor . 8 2.1.4 Covariant derivatives . 8 2.1.5 Riemann tensor . 9 2.2 General relativity . 10 2.2.1 Einstein field equations . 10 2.2.2 Geodesic equation . 11 2.2.3 Geodesic deviation equation . 12 2.3 Linearized theory . 12 2.3.1 Weak-field metric . 12 2.3.2 Gauge transformations . 13 2.3.3 Lorentz transformations . 14 2.3.4 Transverse-traceless gauge . 14 2.3.5 Plane-wave solution . 16 2.3.6 Interaction with matter . 16 2.3.7 Generation . 20 2.4 Compact binary coalescence . 25 2.4.1 General features of an inspiral . 25 2.4.2 Waveform approximants . 27 3 Data analysis 39 3.1 Beam pattern functions . 39 3.2 Signal extraction . 43 3.2.1 Noise sources . 43 3.2.2 Characterization of noise . 43 3.2.3 Matched filtering . 44 3.3 Bayesian inference . 46 3.3.1 Frequentist vs Bayesian . 47 3.3.2 Bayes's theorem . 47 i CONTENTS 3.3.3 Charactization of posterior . 48 3.3.4 Combining posteriors . 49 3.3.5 Model selection . 50 3.3.6 Nested sampling . 50 3.3.7 Markov chain Monte Carlo . 53 3.3.8 Reversible-jump Markov chain Monte Carlo . 55 4 Ringdown 57 4.1 Testing the no-hair conjecture . 58 4.2 Model . 59 4.3 Simulations . 60 4.4 Start time . 61 4.5 Results . 63 4.6 Conclusions . 64 5 Echoes 67 5.1 Morphology-independent search . 69 5.2 Basis functions . 70 5.3 Simulations . 70 5.3.1 Priors . 71 5.3.2 Parameter estimation . 73 5.3.3 Background distribution . 75 5.4 Search on GWTC-1 . 75 5.4.1 Setup . 75 5.4.2 Results and discussion . 76 6 Postmerger 81 6.1 Introduction . 81 6.2 Types of remnant . 83 6.3 Postmerger morphology . 84 6.3.1 Time domain . 84 6.3.2 Frequency domain . 88 6.4 Model functions and simulation . 91 6.4.1 Lorentzian approximants . 91 6.4.2 Validating the parameter estimation pipeline . 93 6.4.3 Inspiral and postmerger consistency . 96 6.5 Summary . 98 7 Conclusions 99 A Numerical relativity configurations 101 B Spin-weighted spherical harmonics 105 Publications 107 List of acronyms 109 Bibliography 111 Public summary 125 Acknowledgements 131 ii 1 CHAPTER 1 INTRODUCTION Gravity is a daily natural phenomenon by which all massive objects tend to attract towards each other. In 1687, Isaac Newton (1642-1727) proposed Newton's law of universal gravitation in his book, Philosophiæ Naturalis Principia Mathematica, to quantify gravity. It states that every point mass exhibits an attractive gravitational force on other point mass along the line intersecting both points with a magnitude directly proportional to their masses and inversely proportional to the square of the distance apart, m1m2 F~ 1 = ^r21; (1.1) − r2 where F~ 1 is the force acting on the first object due to the second object, m1 and m2 are the masses of the first and second object respectively, r is the distance separated between the two objects, ^r21 is the unit vector pointing from the second object to the first object, and the negative sign represents that gravity is attractive. Together with the Newton's law of motions, F~ 1 = m1~a1; (1.2) where ~a1 is the acceleration of the first object, it successfully explains a set of famous empirical laws, namely Kepler's laws of planetary motion, which are 1. `The orbit of a planet is an ellipse with one of whose foci being occupied by the Sun', 2. `The vector drawn from the Sun to the planet's position sweeps equal areas in equal times', and 3. `The squares of the periods are directly proportional to the cubes of the major axes of the ellipses'. Despite the success of Newtonian gravity, it has serious drawbacks both theoretically and experimentally. For example, theoretically it assumes infinite speed of gravity. In other words, one object can gravitationally influence another object instantaneously. It violates special relativity, in which no signals can travel faster than the speed of light to avoid violation of causality. Experimentally it fails to explain the advance of perihelion of the Mercury first observed in 1859. In 1915, Albert Einstein (1879-1955) proposed an alternative theory of gravity, that is the famous General Relativity (GR) in which gravity is no longer a result of force but spacetime curvature. GR satisfies the correspondence principle, meaning that it reduces to Newtonian 1 CHAPTER 1. INTRODUCTION 1 Figure 1.1: Orbital decay of the Hulse{Taylor binary consistent to the energy loss carried away by GWs [2]. gravity in the weak gravitational field and non-relativistic limits. John Wheeler, an American theoretical physicist, summarised GR in one sentence, i.e. \matter tells spacetime how to curve and curved spacetime tells matter how to move". In fact, compared to Newtonian gravity, not only matter can curve spacetime but also radiation and dark energy. GR not only explains what Newtonian gravity cannot explain but also makes new predic- tions. All classical tests are consistent with GR suggesting the adoption of GR over Newtonian gravity. They are 1. The advance of perihelion of the Mercury, 2. The bending of light around a massive object, and 3. The gravitational redshift of light. In GR, the orbit of the Mercury is indeed not an ellipse as predicted by Newtonian gravity but its perihelion precesses with an amount consistent with the observed shift. In addition, GR predicts that the starlight bends around a massive object twice as much as what Newtonian gravity predicts and it was confirmed later on by Arthur Eddington (1882-1944) during the total solar eclipse on 29 May 1919. Finally, the gravitational redshift of light was verified in the Pound{Rebka experiment in 1959 [1]. More importantly, GR predicts the existence of black holes and gravitational waves. A black hole (BH) is a highly compact astrophysical object that significantly deforms its surrounding 2 1 Figure 1.2: The GW event GW150914 observed by the LIGO Hanford (H1, Left) and Livingston (L1, Right) detectors [3]. spacetime to an extent that a boundary is created, namely the event horizon, inside which neither matter nor radiation can escape from it.