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The Three Ss of Gravitational Wave Astronomy: Sources, Signals, Searches

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The Three Ss of Gravitational Wave Astronomy: Sources, Signals, Searches The Three Ss of Gravitational Wave Astronomy: Sources, Signals, Searches Thesis by Ilya Mandel In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2008 (Submitted March 11, 2008) ii c 2008 Ilya Mandel All Rights Reserved iii Acknowledgements Any credit for this thesis is shared with my colleagues and collaborators. All short- comings are entirely my own. Kip Thorne has been a generous mentor, encouraging of my successes, patient with my failures, and unwavering in his dedication to the pursuit and teaching of science. Although some of my recent chapter drafts were as full of his trademark red and blue ink as my first paper written under his supervision, I hope that I have learned something from his wisdom and experience about how to do research and how to write about it. Jon Gair is both a senior colleague and one of my closest friends. While fondly recalling discussing bumpy black holes along the way to the top of Mount Kilimanjaro or serenading bears in Yosemite, I look forward to our future joint adventures both in and out of physics. I am lucky to have met Cole Miller several years ago, to become infected with his passion for astrophysics, and to have the opportunity of learning from him. Alexander Silbergleit, Michael Heifetz, Francis Everitt, and the rest of the Gravity Probe B team got me interested in gravitational physics while I was an undergraduate, and continued to care for me long after I left Stanford. Of course, I am grateful to all of my co-authors whose work is included in this thesis: Pau Amaro-Seoane, Stanislav Babak, Jeandrew Brink, Duncan A. Brown, Jeff Crowder, Curt J. Cutler, Hua Fang, Marc Freitag, Jonathan R. Gair, Chao Li, Geoffrey Lovelace, M. Coleman Miller, Kip S. Thorne, and Michele Vallisneri. A number of people have given useful advice on various aspects of the research presented here. These include Ronald J. Adler, Luc Bouten, Yanbei Chen, Teviet iv Creighton, Steve Drasco, Clovis Hopman, Yuri Levin, Lee Lindblom, Robert Owen, Yi Pan, E. Sterl Phinney, Frans Pretorius, Richard Price, Mark A. Scheel, and Alexander S. Silbergleit. I thank Curt Cutler, Sterl Phinney, Kip Thorne, and Alan Weinstein for serving on my candidacy and thesis committees and for their advice and encouragement. I am grateful for the support I received over the years from the Caltech Physics, Mathematics, and Astronomy division in the form of a Teaching Assistant Stipend and reading grants, and from the following sources: NSF Grant PHY-0099568, NASA Grant NAG5-12834, and the Brinson Foundation. Last, but not least, a big thank you to Alejandro, Craig, Cynthia, Dennis, Donal, JoAnn, Lenya, Paul, Pavlin, Prasun, Tanya, and all of the other flatmates and friends whose company I have enjoyed. BB, you are the best! Mom, thank you for everything. v Abstract As gravitational wave astronomy prepares for the first detections of gravitational waves from compact-object binary inspirals, theoretical work is required on the study of (i) gravitational-wave sources, (ii) the signals emitted by those sources, and (iii) the searches for those signals in detector data. This thesis describes work on all three fronts. (i) We discuss intermediate-mass-ratio inspirals (IMRIs) of black holes or neu- tron stars into intermediate-mass black holes (IMBHs) that could be detected with Advanced LIGO. We analyze different mechanisms of IMRI formation and compute IMRI event rates of up to tens of events per year for Advanced LIGO. We study the spin evolution of IMBHs that grow through a series of minor mergers. We ex- plore how a deviation of an IMRI’s central body from a Kerr black hole influences geodesics, including the possibility of chaotic orbital dynamics. We also address the scientific consequences of extreme-mass-ratio inspiral (EMRI) detections by LISA for astrophysics and general relativity, and the difficulties associated with detecting and analyzing EMRI signals. (ii) We study the periodic standing-wave approximation (PSWA), which can potentially provide accurate waveforms in the last inspiral cycles of a comparable-mass black-hole binary. Using a simple model, we find that the so- lution to Einstein’s equations for inspiraling black holes can be recovered to a high accuracy by the addition a perturbative radiation-reaction field to the standing-wave, noninspiraling solution. (iii) We demonstrate the utility of searching for and analyz- ing tracks in time-frequency spectrograms of a gravitational-wave signal as a means of estimating the parameters of a massive black-hole binary inspiral, as observed by LISA. vi Contents Acknowledgements iii Abstract v 1 Introduction 1 1.1 Sources: Extreme- and Intermediate- Mass-Ratio Inspirals ...... 2 1.1.1 Intermediate-Mass-Ratio Inspirals . .. 2 1.1.1.1 IMRI Overview—Chapter 2 . 2 1.1.1.2 IMRIRates—Chapter3 . 4 1.1.1.3 IMRIs and IMBH Spin—Chapter 4 . 5 1.1.2 Geodesics in Bumpy Spacetimes—Chapter 5 . 6 1.1.3 LISAEMRIs—Chapter6. 7 1.2 Signals: the Periodic Standing-Wave Approximation—Chapter7 .. 8 1.3 Searches: Mock LISA Data Challenge—Chapter 8 . .. 9 2 Prospects for Detection of Gravitational Waves from Intermediate- Mass-Ratio Inspirals 14 3 Rates and Characteristics of Intermediate-Mass-Ratio Inspirals De- tectable by Advanced LIGO 26 3.1 Introduction................................ 27 3.2 Astrophysical Setting, Capture Mechanisms, and Typical Eccentricities 29 3.2.1 Hardening of a CO–IMBH Binary via Three-Body Interactions 31 3.2.2 KozaiResonance ......................... 35 vii 3.2.3 DirectCaptures.......................... 38 3.2.4 Tidal Capture of a Main-Sequence Star . 40 3.2.5 TidalEffects............................ 41 3.2.5.1 TidalDisruption . 41 3.2.5.2 TidalCapture. .. .. 44 3.3 EventRates................................ 50 3.3.1 Advanced LIGO IMRI Sensitivity . 50 3.3.2 Number Density of Globulars with a Suitable IMBH . 54 3.3.3 IMRI Rate per Globular Cluster and Event Rate . 56 3.4 Effect of Eccentricity on Matched Filter Searches . ..... 58 3.5 Summary ................................. 61 3.6 Appendix A. Waveforms and Signal-to-Noise Ratio Calculation. 64 3.7 AppendixB.Ringdowns ......................... 69 4 Spin Distribution Following Minor Mergers and the Effect of Spin on the Detection Range for Low-Mass-Ratio Inspirals 79 4.1 Introduction................................ 80 4.2 SpinEvolution .............................. 82 4.3 Fokker-Planck Equation for Spin Evolution . .... 84 4.4 Spin Evolution via Monte Carlo Simulations . ... 88 4.5 Effect of Black-Hole Spin on Detection Ranges for Low-Mass-Ratio Inspirals .................................. 92 4.6 Appendix A. Fokker-Planck equation . 98 5 Observable Properties of Orbits in Exact Bumpy Spacetimes 102 5.1 Introduction................................ 103 5.2 Bumpy Black Hole Spacetimes . 108 5.2.1 SpacetimeProperties . 110 5.2.2 GeodesicMotion ......................... 111 5.3 IsolatingIntegrals............................. 117 5.3.0.1 Poincar´eMaps for the Manko-Novikov spacetimes . 119 viii 5.3.0.2 Frequency Component Analysis . 121 5.3.0.3 ComparisontoOtherResults . 121 5.3.0.4 Accessibility of the Ergodic Domain . 123 5.4 LastStableOrbit ............................. 127 5.4.1 CircularEquatorialOrbits . 127 5.4.2 Innermost Stable Circular Orbit . 128 5.5 Periapsis and Orbital-Plane Precessions . ..... 130 5.5.1 Epicyclic Frequencies . 132 5.5.2 Precessions ............................ 132 5.5.3 EffectofEccentricity . 136 5.6 Summary ................................. 140 5.7 Appendix A. Chaotic Motion in Newtonian Gravity . ... 143 5.8 Appendix B. Weak Field Precessions . 144 5.8.1 RelativisticPrecession . 144 5.8.2 Precession due to a Quadrupole Moment . 145 6 Detection and Science Applications of Intermediate- and Extreme Mass-Ratio Inspirals into Massive Black Holes 162 6.1 Background ................................ 163 6.2 EMRIdetection.............................. 166 6.2.1 Dataanalysisalgorithms . 166 6.2.1.1 Currentstatus . 167 6.2.1.2 Outstanding challenges . 171 6.2.2 Sourcemodelling . .. .. 174 6.2.2.1 Currentstatus . 174 6.2.2.2 Outstanding challenges . 179 6.3 Testingrelativitytheory . 185 6.3.1 Currentstatus........................... 186 6.3.1.1 Comparisonsofrivaltheories . 186 6.3.1.2 Tests of consistency within General Relativity . 187 ix 6.3.2 Outstandingchallenges . 192 6.4 EMRIscience ............................... 196 6.4.1 What can we learn from the characterisations of EMRI/IMRI dynamics, i.e., the observed eccentricities etc. of the orbits? . 200 6.4.2 What can we learn about the inspiralling compact objects from EMRIs/IMRIs? .......................... 203 6.4.3 What can we learn about the MBHs from EMRIs/IMRIs? . 203 6.4.4 What can we learn about cosmology and early structure forma- tionfromEMRIs/IMRIs? . 204 6.4.5 How can EMRIs/IMRIs be used to test GR, or (assuming GR is correct) that the central massive object is a Kerr BH? . 205 6.5 Conclusions ................................ 206 7 The Geometry of a Naked Singularity Created by Standing Waves Near a Schwarzschild Horizon, and its Application to the Binary Black Hole Problem 217 7.1 IntroductionandSummary. 218 7.2 The Mapping Between the BBH Problem and our Model Scalar-Field Problem .................................. 221 7.3 Standing-WaveScalarField . 224 7.3.1 Perturbative standing-wave solution . 224 7.3.1.1 Perturbative formalism for the standing-wave spacetime224 7.3.1.2 First-order metric perturbations due to the standing- wavescalarfield .................... 227 7.3.2 Time-averaged fully nonlinear standing-wave solution ..... 231 7.3.2.1 Formalism for nonlinear solution with back reaction. 232 7.3.2.2 Singular standing-wave spacetime . 233 7.3.2.3 Comparison of standing-wave and Schwarzschild space- times .......................... 238 7.4 DowngoingScalarField. 240 x 7.5 Reconstruction of Downgoing Scalar Field from Standing-Wave Scalar Field.................................... 242 8 A Three-Stage Search for Supermassive Black Hole Binaries in LISA Data 248 8.1 Introduction................................ 249 8.2 Stage 1: Search for Tracks in the Time–Frequency Plane .
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