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Open Tbode-Dissertation.Pdf The Pennsylvania State University The Graduate School THE ROBUSTNESS OF BINARY BLACK HOLE MERGERS AND WAVEFORMS A Dissertation in Physics by Tanja Bode © 2009 Tanja Bode Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2009 The dissertation of Tanja Bode was reviewed and approved∗ by the following: Lee Sam Finn Professor of Physics Chair of Committee Deirdre Shoemaker Assistant Professor of Physics (Adjunct) Dissertation Advisor Martin Bojowald Assistant Professor of Physics Michael Eracleous Associate Professor of Astronomy & Astrophysics Jayanth R. Banavar Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. Abstract In the past five years, the field of numerical relativity has changed dramatically. With many groups now able to simulate merging black holes and more breakthroughs coming almost monthly, there is much competition to pick off as many astrophysically relevant situations as possible. It is sensible, though, to step back from this competition and to look on the new techniques and results with more skeptical eyes than before. In this dissertation we look at two aspects of initial data from new perspectives and find the waveforms generated from merging binary black hole (BBH) systems to be robust to significant errors in the initial data. In the first study we find that, by adding tuneable auxiliary gravitational waves into a BBH spacetime, up to 1% extra Arnowitt-Deser-Misner (ADM) energy can be added to a standard BBH system before the waveforms are significantly altered. While this study is based on observations of spurious radiation found in all standard initial data sets to date, the second study takes a more general approach. With an eye towards setting up more complex black hole (BH) systems, we find that evolutions of skeleton approximate initial data based on solutions to the ADM Hamiltonian with point sources also yield robust gravitational waveforms that are accurate enough for use in matched template searches for gravitational wave signals in the Laser Interferometer Gravitational-wave Observatory (LIGO) band. We also consider the interpretation of a possible class of constraint violations as an unphysical negative energy field that is absorbed by the BHs. Both of these studies show that the change in apparent horizon (AH) masses during the evolution is a good way to gauge the robustness of the extracted waveforms. At the end of this dissertation we discuss ongoing work on evolving BBH mergers embedded in gaseous clouds. This is the first study with the new matter code Scotch which couples a hydrodynamic matter field to the fully-nonlinear spacetime evolution code. Evolving a wet BBH system will gauge how robust gravitational wave templates are given that true astrophysical sources are not in vacuum. This is a first step at considering the larger question of whether the presence of gas can overcome the “last parsec” problem, hastening the mergers and thereby increasing the expected merger rates. iii Table of Contents List of Figures vii List of Tables x List of Symbols xi Acknowledgments xiv Chapter 1 Introduction 1 1.1 Astrophysical Motivations for Numerical Binary Black Hole Studies . 1 1.1.1 Gravitational Wave Sources . ....... 2 1.1.2 Population Studies of Compact Binaries . ........... 3 1.2 Binary Black Hole Simulations: A Brief History . ............... 3 1.3 Robustness of Simulation Results . ........... 5 1.4 Conventions..................................... ..... 6 Chapter 2 Spacetime Formulations and MayaKranc 7 2.1 Decomposing the Einstein Equations: ‘3+1’ Decomposition . 7 2.1.1 ADMFormalism .................................. 8 2.1.2 Baumgarte-Shapiro-Shibata-Nakamura (BSSN) Formulation............ 11 2.1.3 SlicingConditions . ...... ..... ...... ...... ..... ..... 13 2.2 The MayaKranc Code..................................... 15 Chapter 3 Initial Data and Analysis 17 3.1 Binary Black Hole Initial Data . .......... 17 3.1.1 SpacetimeFullofHoles . ..... 19 3.1.2 Bowen-YorkApproach . 20 iv 3.1.3 Punctures..................................... 21 3.1.4 Solving the Constraints: A Few Comments . ......... 23 3.2 Analysis: Arnowitt-Deser-Misner (ADM) Mass and Momenta ............... 24 3.3 Analysis:Horizons ............................... ....... 25 3.3.1 ApparentHorizon............................... 26 3.3.2 Isolated and Dynamic Horizons . ....... 27 3.4 Analysis: Gravitational Waveforms . ............. 28 3.4.1 WeylScalars ................................... 28 3.4.2 TetradChoice.................................. 30 3.4.3 Physics from Ψ4 ................................... 32 3.4.4 Comments on Ψ4 Analyses.............................. 37 Chapter 4 The Effects of Spurious Radiation on Binary Black Hole Mergers 39 4.1 ANewtonianPerspective . ........ 40 4.2 Injecting Radiation into a BBH Evolution . ............. 41 4.2.1 The Teukolsky-Nakamura Wave . ...... 41 4.2.2 BBH+TNWInitialData ............................... 42 4.2.3 Configurations ................................. 43 4.3 Evolutions of the BBH+TNW ................................ 45 4.3.1 MergerTime .................................... 45 4.3.2 Dynamical and Radiated Quantities . ......... 50 4.3.3 FinalSpacetime................................ 52 4.3.4 Alignment of Amplitude and Phase . ....... 53 4.4 Conclusions..................................... ..... 54 Chapter 5 The Effects of Approximate Initial Data on Binary Black Hole Mergers 57 5.1 SkeletonInitialData ............................. ........ 58 5.2 Quasi-circular Initial Data . ........... 63 5.3 Hamiltonian Constraint Violations . ............. 65 5.4 SkeletonEvolutions..... ...... ..... ...... ...... .. ........ 66 5.5 Single Puncture Analysis . ......... 70 5.6 ImpactonDataAnalysis .. ...... ..... ...... ...... ... ....... 74 5.7 Conclusions..................................... ..... 75 Chapter 6 Theoretical Framework: Coupling Hydrodynamics to a Relativistic Code 77 6.1 Matter Evolution Equations . .......... 78 6.1.1 Valencia Formulation . ...... 79 6.2 Important Aspects of a (Conservative) Hydrodynamics Code ................ 80 6.2.1 Finite Differencing Scheme: Godunov Method and HRSC . 80 6.2.2 Reconstruction: Piecewise Parabolic Method (PPM) . .............. 83 v 6.2.3 RecoveryofPrimitives . ...... 83 6.2.4 AtmosphereHandling . 84 6.2.5 Physicality ................................... 84 Chapter 7 The Effects of Surrounding Gas on Binary Black Hole Mergers 85 7.1 InitialData ..................................... ..... 86 7.1.1 Constraint Solving with Matter . ......... 86 7.1.2 Choosing a Matter Field and Binary . ........ 87 7.2 CurrentProgress ................................. ...... 88 Chapter 8 Summary and Open Questions 90 Appendix A Weyl Scalars 93 A.1 Weyl Scalars on a Hypersurface . ......... 93 Appendix B Multipole Expansions 95 B.1 Spin-weighted Spherical Harmonics . ............ 95 B.2 Vector Spherical Harmonics . ......... 96 B.3 Tensor Spherical Harmonics . ......... 97 B.3.1 Mathews Tensor Spherical Harmonics . ......... 98 B.3.2 Zerilli Tensor Spherical Harmonics . .......... 99 B.3.3 Tensor Spherical Harmonics Decomposition . ............ 100 Appendix C Teukolsky-Nakamura Waves 103 C.1 MetricPerturbation ..... ...... ..... ...... ...... .. ........ 103 C.2 Expansion of Ki j ....................................... 105 Appendix D Tools for Coupled Hydrodynamic Simulations 107 D.1 Reconstruction: Piecewise Parabolic Method (PPM) . ................. 107 D.2 Riemann Solver: Modified Marquina . ......... 109 D.3 RecoveryofPrimitives .. ...... ..... ...... ...... ... ....... 112 D.3.1 Polytropic Equation of State . ........ 112 D.3.2 GeneralEquationofState . 113 Bibliography 114 vi List of Figures 2.1 Schematic of the ADM foliation with the break up of the metric into spatial and temporal variables. ......................................... 9 4.1 Comparison of the initial A˜xx between the Eppley packet widths, σ, for a wave amplitude of 1M3 added to the R1 BBH system. Being the conformal extrinsic curvature, this includes a 2 modulation by 1/ψ so A˜xx vanishes at the punctures. 43 4.2 We plot the effect of the TNW on the binding energy per unit reduced mass in the initial data. The potentials were calculated by solving the initial data using Ansorg’s code for various separations while keeping the individual AH masses and total ADM angular momentum fixed. ............................................. 44 4.3 Changes in merger times compared to the dry R1 run as a function of packet width with estimatederrorbars...... ...... ..... ...... ...... .... ...... 47 4.4 Changes in merger times compared to the dry R1 run as a function of wave amplitude with estimatederrorbars...... ...... ..... ...... ...... .... ...... 47 4.5 Changes in merger times compared to the fractional change in initial ADM energy of the spacetime with estimated error bars. .......... 48 4.6 Sample comparison of waveforms extracted at different radii. Plotted are the waveforms for the dry R1 run and A = 1M3, σ = 5M run extracted at 30M and 75M. ........... 48 4.7 Energy radiated across a sphere of radius r = 40M as calculated from the Weyl scalar Ψ4. 50 4.8 Angular momentum radiated across a sphere of radius r = 40M for σ of 3M as calculated from the Weyl scalar Ψ4. ................................... 51 4.9 Flux of angular momentum across a sphere of radius r = 40M for σ of 4M as calculated from the Weyl scalar Ψ4. ................................... 52 4.10 Closeup of angular momentum flux across a sphere of radius
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