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By Charles J. Woodford a Thesis Submitted in Conformity Centre-of-mass motion and precession of the orbital plane in binary black hole simulations by Charles J. Woodford A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto c Copyright 2020 by Charles J. Woodford Abstract Centre-of-mass motion and precession of the orbital plane in binary black hole simulations Charles J. Woodford Doctor of Philosophy Graduate Department of Physics University of Toronto 2020 This work focuses on the inherent gauge ambiguity in general relativity (GR), related gauge effects in numerical relativity (NR), and the extraction of gauge-invariant, observable measures from NR. Gauge ambiguity is an inescapable feature of GR, analogous to not knowing which reference frame a system is in. This can create unphysical effects in data from strong gravity regime simulations, which ultimately become a source of error if not accounted for. NR involves solving Einstein’s equations on a computer. Here, NR is used to obtain accurate solutions for compact binary systems, namely binary black holes (BBH). BBH are observed through the emission of gravitational radiation as the binary undergoes inspiral, merger, and ringdown of the remnant black hole. Gravitational wave (GW) information from NR is represented as waveforms, typically decomposed into spin-weighted spherical harmonics, as is the case for the Simulating Extreme Spacetimes (SXS) collaboration. The first project is an analysis of the centre-of-mass (c.m.) in simulations of BBH. The c.m. is considered the origin for the decomposed waveforms, and unphysical movement in the c.m. erroneously affects the reported GWs. We establish that there is an initial displacement from the origin and a velocity kick that causes an overall linear drift in the c.m. We develop techniques to characterize, analyze, and “remove” effects of c.m. motion on the GWs by transforming the frame of reference, otherwise known as a gauge-transformation. The resulting GWs are proven to be more accurate and reliable. The removal of the linear c.m. drift is now a permanent post-processing step for all simulations in the SXS catalog. The second project is an investigation into the precession of the orbital plane of BBH, which is resilient against a larger set of gauge transformations than completely gauge-dependent quantities and therefore more reliable for comparisons between simulations, codes, and theories. We compare information extracted from simulations of BBH to Self-force perturbation theory (SF). The rate of precession of the orbital plane is extracted from NR and compared with Kerr geodesics, which allowed for the first and second order SF coefficients to be extracted. ii Acknowledgements This work would not have been possible without the continuous guidance and support of my supervisor Dr. Norman Murray. This degree had its fair share of abrupt changes, including a change of supervisor, two changes of internal committee, and two changes of research focus from numerical relativity to exoplanets back to numerical relativity. Despite not working with Norm directly on any of the research presented here, I am grateful for his helpful discussions and support over the last 3 years. I also thank my committee members Dr. Amar Vutha and Dr. Harald Pfeiffer. In addition to sitting on my committee, Harald has been involved with my conference presentations and main research project, and is a co-author for said main research project. I would like to thank Dr. Michael Boyle and Dr. Aaron Zimmerman for being co-authors on the respective research projects and for enabling me to complete this work with their guidance. Although not directly involved in any one project, I also thank Dr. Saul Teukolsky for encouraging me to return to research in numerical relativity and for standing in as a research supervisor when Norm could not. In general, I thank the members of the Simulating Extreme Spacetimes Collaboration for helpful conversations, guidance, and support regarding simulations of binary black holes, and to the Canadian Institute of Theoretical Astrophysics for being my home for the past five years as I completed this work. This work was supported in part by NSERC of Canada Grant No. PGSD3-504366-2017, the University of Toronto Fellowship from the Faculty of Arts and Science, Conference Grants from the School of Graduate Studies at the University of Toronto, Cray Incorporated Fellowships in Physics, Faculty of Arts and Science Program-Level Fellowship, and the E.F. Burton Fellowships in Physics. The computations described in this work were performed on the Wheeler cluster at Caltech, which is supported by the Sherman Fairchild Foundation and by Caltech, and on the GPC and Gravity clusters at the SciNet HPC Consortium, funded by the Canada Foundation for Innovation under the auspices of Compute Canada, the Government of Ontario, Ontario Research Fund–Research Excellence, and the University of Toronto. iii Contents 1 Introduction 1 2 Centre of mass corrections 7 2.1 Motivation . .8 2.2 The need for c.m. corrections . 10 2.3 Centre-of-mass correction method . 12 2.3.1 Choosing the translation and boost . 13 2.3.2 Choosing the integration region . 15 2.3.3 Correlations between c.m. correction values and physical parameters . 17 2.4 Quantifying c.m. Correction Using Waveforms Alone . 18 2.4.1 Defining the method . 19 2.4.2 Results for the standard c.m.-correction method . 21 2.5 Improving the c.m. correction . 22 2.5.1 Post-Newtonian c.m. definition . 22 2.5.2 Linear-momentum recoil . 25 2.5.3 Causes of unphysical c.m. motion . 28 2.5.4 Epicycle quantification . 29 2.5.5 Position of the c.m. 33 2.6 Conclusions . 35 3 Fundamental frequency analysis for precessing systems 36 3.1 Motivation . 36 3.2 Previous work . 39 3.3 Methods and Meaning . 40 3.3.1 Testing the numerical relativity methods . 42 3.3.2 Ratios from Analytic and semi-Analytic methods . 44 3.4 Data and Analysis . 44 3.4.1 Results from numerical relativity . 46 iv 3.4.2 Comparisons with Kerr geodesics . 54 3.4.3 SF fitting and extraction . 55 3.5 Discussion . 62 4 Conclusion 64 Appendices 65 A Note on SpEC 67 B Spin-Weighted Spherical Harmonics 71 C Post-Newtonian Correction to the c.m. 73 D Linear Momentum Flux from hl;m modes 74 E Fundamental Frequencies from Kerr Geodesics 75 Bibliography 76 v List of Tables 3.1 Simulation parameters for precession of the orbital plane analysis. 45 3.2 Mass scaled frequency ranges for extracting 1SF and 2SF coefficients. 58 vi List of Figures 2.1 C.m. trajectories and their effect on waveform amplitude. .9 2.2 Magnitudes of c.m. offsets and drifts for all simulations in the SXS catalog . 14 2.3 Effect of differing beginning and ending times for COM correction values. 15 2.4 C.m. correction values compared with relevant simulation parameters. 17 2.5 Simplicity of the waveform for raw and c.m. corrected data. 22 2.6 Simplicity of the waveform for regular and optimized c.m. correction values. 23 2.7 Change in c.m. correction values for Newtonian, 1PN, and 2PN c.m. trajectories. 23 2.8 Differences between the Newtonian, 1PN, and 2PN c.m. correction values versus eccentricity. 24 2.9 Simplicity of the waveform for PN contributions to the c.m. 24 2.10 Comparison of motion caused by linear momentum flux to measured c.m. motion. 26 2.11 Illustration of epicycle correction for the two simulations shown in Figure 2.1........... 30 2.12 Contribution to c.m. motion which cannot be fitted by a linear drift. 31 2.13 Change in the c.m. correction when removing epicycles before fitting for the c.m. correction. 32 2.14 Difference in simplicity of the waveform with and without epicycle removal in the c.m. correction. 33 2.15 Comparison between the position of the c.m. and relevant simulation parameters. 34 3.1 Coordinate frames in BBH simulations. 37 3.2 Illustration of periastron advance in a binary system. 38 3.3 Angle between the primary spin vector and the orbital plane in an inclined BBH. 40 3.4 Stress-test model for frequency extraction method. 43 3.5 Rate of precession of the orbital plane from NR. 47 3.6 Rate of precession of the orbital plane evaluated at varying frequency windows. 48 3.7 Errors in extrapolation to the instantaneous rate of precession of the orbital plane from windowing. 49 3.8 Instantaneous rate of the precession of the orbital plane Kθφ for highest available resolution. 50 3.9 Kθφ for all available resolutions versus mass scaled frequency. 51 3.10 Kθφ for all available resolutions versus mass ratio. 52 3.11 Comparison of Kθφ in systems with varying inclination angles. 53 3.12 Comparison of Kθφ between NR and Kerr geodesics scaled with mass ratio. 55 vii 3.13 Comparison of Kθφ between NR and Kerr geodesics scaled with symmetric mass ratio. 56 3.14 Linear models for comparisons in Kθφ from NR and Kerr geodesics scaled with mass ratio. 57 3.15 1SF, 2SF coefficients from a linear model in mass ratio versus mass scaled frequency. 59 3.16 1SF, 2SF coefficients from a linear model in mass ratio. 59 3.17 Errors for 1SF and 2SF coefficients from a linear fit in mass ratio. 61 3.18 1SF and 2SF coefficients from a linear model in symmetric mass ratio. 61 3.19 1SF, 2SF coefficients from a linear model in symmetric mass ratio versus mass scaled frequency .
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