By Charles J. Woodford a Thesis Submitted in Conformity

Centre-of-mass motion and precession of the orbital plane in binary black hole simulations

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

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 . 62 3.20 Comparison of errors for 1SF and 2SF coefficients from a linear fit in symmetric mass ratio. . . . 63

viii List of Appendices

Note on SpEC 67

Spin-Weighted Spherical Harmonics 71

Post-Newtonian Correction to the c.m. 73

Linear Momentum Flux from hl.m modes 74

Fundamental Frequencies from Kerr Geodesics 75

ix Chapter 1

Introduction

Black holes are among the most exotic objects in our universe. They are often used to probe the limits of scientiﬁc understanding as they encapsulate the extremes of what our universe allows. Black holes belong to a wider class of compact objects, which includes neutron stars and white dwarfs, but are set apart as black holes do not emit light and cannot be observed in the usual astronomical fashion.

There is much we still do not understand about black holes; however all black holes share some essential characteristics. In particular, black holes are extremely compact, and represent extreme curvature in the fabric of space and time - being often referred to as a singularity. This is best showcased by the formation method we know: stellar mass black holes form from large dying stars [1–3]. In short, for stars whose cores are less massive than 1.4 solar masses at the end of their lives, the outer layers are blown oﬀ and without outward pressure from nuclear fusion, the core contracts to form a white dwarf star. The contraction stops due to electron degeneracy pressure, which is the result of the Pauli exclusion principle, stating that two or more of the same particle in the same state cannot exist at the same time in the same space, applied to electrons. For iron cores that are between 1.4 and three solar masses, the collapse due to gravity overcomes electron degeneracy pressure and halts due to neutron degeneracy, creating a neutron star. Neutron stars have the density of an atomic nucleus, but are tens of kilometres in diameter. Despite our understanding of the reason why and how neutron stars come into existence, there is still a considerable amount to learn about them. Lastly, if the core of a large dying star is suﬃciently compact, gravity overcomes neutron degeneracy pressure and fully collapses in on itself, creating a black hole. Typically, when a star at least 10 times the mass of our Sun reaches the end of its life, a black hole is formed.

It is expected that many stellar mass black holes, with masses up to a few dozen times the mass of our Sun, are formed this way [4]. However, we also know that there are supermassive black holes, which are hundreds of thousands of solar masses or more [5], and expect that there are black holes with masses in between stellar and supermassive [6].

Regardless of their mass, black holes have an event horizon that prevents light from escaping - which is why they are not typically detected using visible light. The event horizon is a surface at which the gravitational ﬁeld

1 Chapter 1. Introduction 2 surrounding the singularity exerts such force that the escape velocity is the speed of light. Since nothing in the known universe can travel faster than the speed of light in its own and local reference frames, nothing can escape if captured, and nothing can be emitted if created. Einstein’s theory of general relativity, states that space and time can be considered as four dimensions of the same geometric manifold and that the curvature of spacetime is directly related to the energy and momentum of any matter and radiation locally present. The consequences of general relativity include the bending of spacetime and causal structure, or light cones, and so black holes and their event horizons may be deﬁned in another way. Light cones expand from any one event in spacetime at the speed of light in all directions, such that information of the event is available to everything inside the light cone and unknowable outside the light cone. Spacetime curves and bends from the presence of mass, and this curvature of spacetime can cause light cones to tilt – essentially bending rays of light and changing the course of information travelling throughout the universe. In the case of black holes, the curvature becomes so intense that light cones in the region of the forming black hole fold onto each other and there is no path out of that region. In this way, the event horizon designates the boundary of the region where light cones tilt such that there is no path for information to travel out of the black hole.

Mathematically, general relativity is encapsulated by the Einstein ﬁeld equations [7–9],

1 8πG R − Rg = T , (1.1) µν 2 µν c4 µν where gµν is the metric, which contains information about the shape and curvature of spacetime, Rµν is the Ricci tensor which describes the curvature of spacetime, R is the trace of the Ricci tensor, G is the gravitational constant, c is the speed of light in a vacuum, and Tµν is the stress-energy tensor, which contains information about any matter and radiation. Here, µ, ν = 0, 1, 2, 3 where 0 represents time and 1, 2, 3 represent the three spatial µν 2 µ ν dimensions. In particular, R = g Rµν. For ﬂat Minkowski spacetime, a line element reads ds = ηµνdx dx = 2 2 2 2 2 −c dt + dx + dy + dz . In this case, ηµν = gµν. For more realistic and complicated spacetimes, gµν will become more general and need to be solved for using the equations stated in Eq. 1.1. In this work, the rescaling G = c = 1 is used unless stated otherwise.

There are several solutions for the Einstein field equations that describe black holes, such as Schwarzschild spacetime [10–16] and Kerr spacetime [17–25]. Schwarzschild spacetime describes an isolated, non-rotating black hole of mass MBH. The Kerr solution represents spinning black holes with spin parameter a = |S~|/MBH ∈ [0, MBH], where S~ is the spin vector. In this notation, a = 0 implies not spinning and a = MBH implies maximally spinning. There are many different metrics to describe different spacetimes, and although there are more complicated metrics to describe black holes, the work presented here is based on Kerr and so the relevant parameter space includes the mass and spin of each black hole.

It is essential to compare theoretical work with observations. However, the nature of black holes makes them difficult to observe. Very little to no observable light is reflected or emitted by a black hole, and thus the name “black hole”. One way to still observe a black hole is by its effect on the matter in its environment. This is the

2 Chapter 1. Introduction 3 rationale behind the observation of the supermassive black hole in galaxy M87 by the Event Horizon Telescope [26], which observed the hot gas accreting onto the supermassive black hole. This is also the mechanism operating in X-ray binaries, which are very luminous in X-rays and typically comprised of a relatively average star and compact object, such as a neutron star or black hole [27]. Binary black holes (BBHs), i.e. two black holes orbiting each other, are of particular interest as they are still invisible in light, but are detectable through gravitational waves. BBHs radiate energy and their orbits decay, until the black holes merge and become a large black hole that is slightly less massive than the sum of the two original black holes due to a large outﬂux of radiation during the merger. The radiated gravitational energy manifests as ripples in spacetime, hence the name gravitational waves. The metric in a region containing gravitational radiation can be expressed as

gµν = ηµν + hµν, (1.2) where hµν describes the effect of gravitational radiation on the spacetime, and is represented as a perturbation onto the flat Minkowski spacetime described by ηµν. In this way, hµν represents a gravitational wave. Gravitational waves were observed for the first time by LIGO (Laser Interferometer Gravitational Wave Observatory) [28, 29] on September 14, 2015 as the LIGO collaboration was preparing for their first observing run [30–32]. Since this initial detection using two interferometers on opposite sides of the United States, LIGO, in collaboration with the Italy-based detector VIRGO, have detected a confirmed 12 binary black holes [30, 33–36], and 2 binary neutron stars [37, 38] along with several dozen announced event candidates [39]. With the introduction of gravitational-wave detectors, the pursuit of BBH gravitational waveforms has intensified in an attempt to create and fill vast waveform template banks that are comprised of accurate waveforms incorporating all known physics for the best possible parameter estimation for compact binaries and tests of general relativity. Gravitational waveforms created through numerical relativity, or solving Einstein’s equations on a computer, are generally the most accurate waveforms available, and are used for systematic studies of gravitational wave analysis [40,41] and to validate and improve semi-analytic and analytic models of BBHs, which in turn are used for gravitational wave detection and parameter estimation [42–44]. Before numerical relativity became available to use consistently and on a large scale, much work was done to better characterize systems of compact objects analytically. As general relativistic equations are not solvable analytically, there are several perturbation theories that attempt to approximate the solutions for a given range of system parameters, such as Post-Newtonian (PN) [15, 45–48], Self-force (SF) [49–57], and re-combinations therein, such as the Effective One-Body (EOB) formalism [58–60]. v The PN formalism focuses on expanding the Newtonian approximations in powers of c , where v is the speed of the orbiting bodies. PN approximations do well for the inspiral, or early stage, of a BBH evolution, but tend to be inaccurate for the late inspiral and are not applicable to the merger and ringdown stages, where “ringdown” refers to the “settling” of the final black hole after merger. The SF formalism focuses on expanding in inverse mass ratio 1 , q = m1 ≥ 1, and so is best suited for systems q m2 with large mass ratio.

3 Chapter 1. Introduction 4

The EOB formalism is a re-combination of the PN equations that empirically improves the convergence of the PN series. Various versions of EOB are favoured for waveform template generation over large spans of parameter space. Each of these formalisms lose accuracy at some limit in parameter space or within the BBH evolution. Matching the output of these methods to numerical relativity simulation data is essential to tweaking and ensuring the validity of these formalisms. As computing hardware, coding practices, and numerical methods advanced, the general relativistic equations, which characterize BBH systems, were better able to be solved numerically. Frans Pretorius utilized the general harmonic method in 2005 with his breakthrough simulations, and enabled a leap forward for numerical relativity and solving Einstein’s equations via computer [61–63]. In the decade and a half since, many improvements have been made to numerical relativity codes to make simulations of compact binaries more efficient and accurate, as well as compatible with improving hardware and computing techniques [64–68]. The code used in the work presented here is the Spectral Einstein Code (SpEC) [69, 70], created and maintained by members of the Simulating Extreme Spacetimes Collaboration (SXS). All of the data used and presented here is produced by SpEC and most is available on the SXS gravitational waveform catalog [71]. Any coding or analysis packages used that are not a part of SpEC are referenced accordingly in their respective chapters. While numerical relativity waveforms are the most accurate BBH waveforms, it is necessary to test and confirm their validity, accuracy, and reproducibility. There have been numerous discussions on how to measure the accuracy of a numerical relativity simulation and some sources of error in these simulations have been investigated, including numerical truncation errors, error due to extraction at finite radius or imperfect extrapolation to infinite radius, and errors between simulations of different lengths that otherwise have identical parameters [72–74] (see Appendix A). Though not strictly a source of error like those named above, there are also consequences due to the gauge freedom of general relativity that may be confused for errors if not properly understood, and will effectively become sources of error if ignored [75–77]. Gauge freedom in general relativity affects all numerical relativity simulations, and thus far no numerical relativity results in the literature have been in a completely specified gauge due to this inherent gauge freedom. Gauge effects on gravitational waves are essentially arbitrary coordinate transformations. By definition, they do not affect the physics of a system but can influence the interpretation of observable quantities if not taken into account. When comparing gravitational waves, common gauge ambiguities are time translations and phase rotations. This can be readily seen in the methods for detections of gravitational waves by optimizing matched-filtering over time and phase of the signal [77]. By extension, time and phase ambiguities are also present in numerical, analytic, and semi-analytic gravitational waveforms and associated extracted information, along with coordinate transformations that make comparing results across codes and techniques difficult. By making some standard approximations (outlined below), it is possible to avoid a complete diffeomorphism freedom, i.e. smooth mappings between spacetimes, and restrict to a smaller gauge group.

4 Chapter 1. Introduction 5

Since gravitational wave detectors are very large distances away from the sources of gravitational waves, only asymptotic features of the radiation need to be considered. If the source is isolated, the model spacetime approximating our Universe can be assumed to be asymptotically flat. This enables restrictions to be imposed on the gauge in the asymptotic regime, and the most common gauge used is the Bondi gauge [77–84]. The Bondi gauge describes a spacetime that approaches flat, or Minkowski, spacetime at large radii from the source of the gravitational radiation. The Bondi gauge is based on the Bondi coordinates xµ = (u, r, xA), where u = t − r is retarded time, r is the 2 a b a 1 2 3 distance from the source to the observer, typically expressed in this formalism as r = δaby y where y = (y , y , y ) are spatial Cartesian coordinates, and xA are the two angular coordinates θ, φ. Much of the work presented here is in Cartesian coordinates and retarded time, or in full Bondi coordinates. The allowed gauge transformations on the Bondi gauge, and any gauge describing gravitational radiation, are symmetry transformations of the Bondi gauge metric, and are called the Bondi-Metzner-Sachs (BMS) group [77–79, 81, 84–88]. The BMS group is infinitely large, and permits Lorentz boosts, rotations, and translations. BMS transformations may be represented on the u, θ, φ coordinates as:

θ0 = θ(θ, φ),

φ0 = φ(θ, φ), (1.3)

u0 = K(θ, φ)[u − A(θ, φ)], where (θ, φ) → (θ0, φ0) is a conformal transformation of the sphere onto itself, K is the corresponding conformal factor given by dθ02 + sin2θ0dφ02 = K2(dθ2 + sin2θdφ2), and A is an arbitrary, suitably smooth real function on the sphere. The r coordinate is not considered here, as a transformation in r does not impact the structure of the BMS group and is arbitrary [86]. In the case of no rotation (ie. θ0 = θ, φ0 = φ), the BMS group further limits allowed transformations to supertranslations. Supertranslations include space and time translations that transform a set of null hypersurfaces into a different set of null hypersurfaces [86], and are discussed in detail in Chapter2. Since we do not know which gauge the results of numerical relativity simulations and general relativity analytic calculations are in, it is essential that comparisons between these methods have minimized gauge effects. Along the same line, comparing gauge-independent quantities both between numerical relativity and semi-analytic methods as well as between waveforms and observed data removes gauge effects as a potential source of error. This work focuses on minimizing gauge effects and the extraction of gauge-independent quantities from numerical simulations of BBHs, and is organized as follows:

• Chapter 2 discusses the published work regarding centre of mass corrections and the post-processing pipeline resulting from those ﬁndings for the SXS collaboration. The focus for minimizing gauge eﬀects using the centre of mass corrections is on post-processing rather than during a simulation as the gauge is not known at the beginning of a simulation and the minimizing techniques can estimate the majority of the needed

5 Chapter 1. Introduction 6

gauge transformations once all the simulation data is available. This also allows any and all simulations to be corrected and side-steps the need to rerun lengthy and expensive simulations.

• Chapter 3 discusses the extraction of observable gauge independent quantities and comparison of those quantities with known perturbation theory and analytic equivalents, as well as extractions of the ﬁrst and second order self-force corrections for these quantities. The quantities considered are potentially observable eﬀects derived from the frequencies that characterize the system. This technique is applied to precessing simulations to extract the observable precession of the orbital plane.

• Global results, conclusions, and future research proposals are presented in Chapter4.

6 Chapter 2

Centre of mass corrections

This chapter presents a detailed study of the centre-of-mass (c.m.) motion seen in simulations produced by the Simulating eXtreme Spacetimes (SXS) collaboration. This work has been published in Physical Review D., volume 100, issue 12, doi:10.1103/PhysRevD/100/124010, with co-authors Michael Boyle and Harald Pfeiﬀer [89], and as Section 2.4.2 in the SXS Collaboration 2019 catalog publication [90]. This chapter, in essence, is an edited, full edition of the associated ﬁrst author publication and results presented in both Refs. [89, 90].

In addition to the c.m. section in Ref. [90], I wrote the background portion of the publication and was heavily involved with the editing and rewriting process, which was ongoing from 2017 until its submission in 2019. I presented the results found by the SXS Collaboration in two American Physical Society April Meeting talks (2018, 2019), and gave a poster presentation on preliminary c.m. correction results at the Canadian Astronomy Society (CASCA) Conference in 2017. I am ninth author in the primary author list, ahead of the alphabetical author list.

In this work, the potential physical sources for the large c.m. motion in binary black hole simulations were investigated, and it was found that a signiﬁcant fraction of the c.m. motion cannot be explained physically, thus concluding that it is largely a gauge eﬀect. These large c.m. displacements cause mode mixing in the gravitational waveform, most easily recognized as amplitude oscillations caused by the dominant (2, ±2) modes mixing into subdominant modes. This mixing does not diminish with increasing distance from the source; it is present even in asymptotic waveforms, regardless of the method of data extraction. Herein, the current c.m.-correction method used by the SXS collaboration is described, and is based on counteracting the motion of the c.m. as measured by the trajectories of the apparent horizons in the simulations (see Appendix A). The potential methods to improve that correction to the waveform are investigated, and a complementary method for computing an optimal c.m. correction or evaluating any other c.m. transformation based solely on the asymptotic waveform data is provided.

7 Chapter 2. Centre of mass corrections 8

2.1 Motivation

This work addresses the translation and boost degrees of freedom in the BMS group. Reference [77] identiﬁed these transformations as important for counteracting the observed motion of the c.m. in simulations produced by the Simulating eXtreme Spacetimes (SXS) collaboration. This work expands on that analysis, using the recently updated catalog of 2,018 SXS simulations [71, 90], and investigating possible improvements to the correction method. A translation α~ and a boost ~β will transform the waveform h, measured at some point distant from the source, as

~ ~ ~ 2 2 h(t) → h t + (α~ + βt) · nˆ + O |β|h + O |α~ + βt| ∂t h , (2.1a)

≈ h(t) + ∂th(t)(α~ + ~βt) · nˆ, (2.1b) where nˆ is the direction to the observer from the source [77]. Note that this is independent of the distance to the source; even the asymptotic waveform will exhibit this dependence regardless of any extrapolation, Cauchy- characteristic extraction, or similar techniques that may be applied to the data (see Appendix A). This can be intuitively understood by thinking about a sphere surrounding the source. If the source is displaced away from the center of the sphere, an emitted signal will arrive at the part of the sphere closest to the source before it will arrive at the opposite side of the sphere. The difference in arrival times is independent of the radius of the sphere; it only depends on the size of the initial displacement α~. The term ~β in Eq. (2.1b) can be interpreted as the Doppler effect for such signals, and introduces an angular dependence that is not generally included in waveform models. Figure 2.1 demonstrates the effects of c.m. motion for two systems from the SXS catalog. The most striking example is the upper pair of panels, which show data from a nonprecessing system with mass ratio 1.23. On physical grounds, there is nothing to suggest modulations in the mode amplitudes on the orbital timescale; this is a relatively symmetric system with very low eccentricity. The dominant physical behavior on the orbital timescale is simply rotation, which should have no effect on the amplitudes of the modes, along with a secular increase toward merger. Nonetheless, the raw waveform data from the simulation (thin dark lines in the upper left panel) shows very clear amplitude modulations of the subdominant modes on the orbital timescale. These modulations—like the c.m. trajectories seen in the upper right panel—show no signs of convergence with increasing numerical resolution in the simulation, even though the initial data for each resolution is created from identical high-resolution initial data (see AppendixA). As is discussed below, the c.m. motions found in the SXS catalog are effectively random and apparently independent of any physical parameters of the systems. Therefore, they comprise an essentially random source of unmodelled and unphysical contributions to waveforms from numerical relativity. In particular, they are not systematic; the modulations found in waveforms for one set of physical parameters will be uncorrelated with the modulations in waveforms even for nearly identical physical parameters. Clearly, expecting waveform models such as effective-one-body (EOB) [58, 60, 93–96], phenomenological [97–99], and surrogate models [100] to accurately represent these features across a range of physical parameters is tantamount to expecting them to fit large,

8 Chapter 2. Centre of mass corrections 9

Mode amplitudes |rh l, m/M| COM trajectories

(2, 2) Lev1 1 10 (2, 1) Lev2 0.00 (3, 3) Lev3 10 2 (3, 2) Lev4 0.05 (3, 1) M / 10 3 (4, 4) y (4, 2) 0.10 SXS:BBH:0314 10 4 0.15 10 5 0 1000 2000 3000 0.05 0.00 0.05 0.10

(2, 2) Lev1 0.04 1 10 (2, 1) Lev2 (3, 3) Lev3 0.02 10 2 (3, 2) (3, 1) M

0.00 / 10 3 (4, 4) y (4, 3)

SXS:BBH:0622 (4, 2) 0.02 4 10 (4, 1) 0.04 10 5 0 1000 2000 3000 4000 0.04 0.02 0.00 0.02 (t r )/M x/M Figure 2.1: Centre-of-mass motion and its eﬀect on waveforms. These plots show data for two systems from the SXS catalog. The upper panels correspond to the nonprecessing system SXS:BBH:0314 [91], which has a mass ratio of 1.23, with spins of 0.31 for the larger black hole and −0.46 for the smaller black hole, and are aligned with the orbital angular momentum. The lower panels correspond to the precessing system SXS:BBH:0622 [92], which has a mass ratio of 1.2, with randomly oriented spins of magnitude 0.85. The panels on the right side show the c.m. trajectories in the simulation coordinates calculated from the apparent horizons of each black hole for a variety of resolutions, labelled “Lev1”, “Lev2”, “Lev3” in order of increasing resolution. The panels on the left show the dominant mode amplitudes of each system, both before and after c.m. correction—the thin darker lines being the raw waveform data, and the thicker transparent lines being the corrected data.

discontinuous, random signals.

However, by simply compensating for the inertial part of the measured c.m. motion, the modulations can be almost completely eliminated (thick transparent lines in the upper left panel). It is notable that the c.m. only drifts by roughly 0.1M during almost the entire inspiral for the system shown in the upper panels of Fig. 2.1, but still has such a drastic eﬀect on the waveform’s modes. Even though this is only a gauge choice—which are considered irrelevant in principle—in practice, gauge choices must be made consistently and systematically for the waveforms to be really useful. In this sense, it may be suggested that the c.m.-centred gauge is really an optimal choice.

A more diﬃcult comparison is for the precessing system shown in the lower panels of Fig. 2.1. The precession already mixes the modes drastically, leading to a complicated waveform with pronounced amplitude modulations, even after c.m. correction. Clearly the c.m. correction changed the data, but it is not obvious that it was a change for the better—at least from looking at this plot alone. To make the comparison more quantitative, a new measure of a waveform’s “simplicity” is introduced in Sec. 2.4. Essentially, this quantity measures the residual when the waveform is modeled by simple linear-in-time amplitudes in the co-rotating frame [101]. The value of this residual

9 Chapter 2. Centre of mass corrections 10 is 117 times smaller for the c.m.-corrected data than for the raw data in this precessing system, showing that the corrected waveform is clearly and objectively better in this sense at least. To address the miscalculation of the c.m. and its correction, this chapter is organized as follows:

i In Sec. 2.2,the current deﬁnition of the c.m. and the consequences this deﬁnition and its use have on SXS gravitational waveforms are discussed.

ii In Sec. 2.3, the current methods for correcting waveform data and selecting an optimal gauge are discussed. Any correlations found between simulation parameters and the c.m. correction factors are discussed.

iii In Sec. 2.4, a quantitative method for evaluating the “correctness” of the gauge a waveform is currently in is discussed. A comparison of the waveform data in its original, unoptimized gauge to the c.m.-corrected gauge is described in Sec. 2.3.

iv In Sec. 2.5, we discuss how the definition of the c.m. may be improved to find a better correction, potentially leading to a further optimized choice of gauge. This section also investigates alternative definitions of the c.m. with a focus on potential physical causes of c.m. motion like that seen in Fig. 2.1, including Post-Newtonian definitions and considerations of linear-momentum recoil.

v Findings and results are presented in Sec. 2.6.

2.2 The need for c.m. corrections

One of the primary concerns with BBH simulations with regards to gravitational-wave astronomy is the validity of their gravitational waveforms. Above all, the output from a BBH simulation should result in a reliable, reproducible waveform that can then be released for public usage. In the case of the SXS collaboration, many of the waveforms produced are also compressed into a catalog that is released to LIGO for data analysis and waveform comparisons with their gravitational wave detector data. Gravitational waveforms in the SXS catalog are given in terms of the gravitational-wave strain h, or the Weyl component Ψ4. In regards to detecting gravitational waves, h and Ψ4 contain essentially the same information since

Ψ4 ≈ h¨. The analysis and corrections applied in this work may be applied to either with the same results. For simplicity, this work focuses on h. Waveforms from SXS are represented by mode weights, or amplitudes, for spin-weighted spherical harmonics (SWSHs). The gravitational-wave strain may be represented by the transverse-traceless projection of the metric perturbation caused by the gravitational waves at time t and location (θ, φ) relative to the binary, and can be combined into a single complex quantity, given by

h(t, θ, φ) B h+(t, θ, φ) − ih×(t, θ, φ). (2.2)

10 Chapter 2. Centre of mass corrections 11

For each slice in time, the combined perturbation h(t, θ, φ) is measured on the coordinate sphere. The angular dependence of this measurement can then be expanded in SWSHs. The quantity h(t, θ, φ) has a spin weight of −2 [86], and may be represented as

X l,m h(t, θ, φ) = h (t) −2Yl,m(θ, φ), (2.3) l,m where the complex quantities hl,m(t) are referred to as modes or mode weights, and are much more convenient when analyzing BBH than the total perturbation in any particular direction [102, 103]. Spin-weighted Spherical harmonics are further discussed in AppendixB.

The expansion in Eq. (2.3) depends on orientation of the spherical coordinates θ, φ and their origin. The customary choice places the binary at the origin with the binary’s initial orbital plane coinciding with the equatorial plane θ = π/2. For comparable mass, nonprecessing binaries, the quadrupolar (l, m) = (2, ±2) modes then dominate the waveform. While the h2,±2 modes are dominant, it is important to consider the behavior of the other modes present in the waveform. The other modes are much smaller in magnitude compared to the h2,±2 modes for most systems, but are important for parameter estimation [104–106], and recently have been included in BBH and other binary compact object searches by LIGO and VIRGO [35,36] after several such proposals [107–109]. Higher-order modes are also useful for verifying the reliability and potentially the accuracy of the waveform. If the shape, variability, magnitude, or any other characteristic of the higher order, or subdominant, modes are found to not suitably match with theory, then this could indicate a possible ﬂaw in the simulation.

One clear issue is the coordinate system, or gauge choice, for the simulation, as spherical harmonics and hence SWSHs depend on the coordinates. The center chosen for the simulation is the c.m. of the system, as calculated and set in the initial data. It is expected that the c.m. will move slightly throughout the simulation; however, large movements are not expected and suggest a flaw in the choice of gauge. If the c.m. moves significantly, there is mode mixing [77]. The dominant effect [77] is leaking of the h2,±2 modes of BBH waveforms into the higher modes, and this leakage can be at least partially removed through c.m. drift corrections, as described in Sec. 2.3.

Mode mixing is manifested in the waveforms as oscillating amplitudes, which can clearly be seen in the left column panels of Fig. 2.1, especially for the (2, 1), (3, 1), and (3, 3) modes in the top panel for SXS:BBH:0314. Precessing simulations, like SXS:BBH:0622, are expected to have some amplitude modulations purely due to the orientation of the system. The worse the c.m. calculation is for a simulation, the more altered the SWSH representations are, and the worse the mode mixing becomes.

It is easily seen in Fig. 2.1 that the applied c.m. correction removes unphysical waveform amplitude modulations for nonprecessing systems, however for precessing systems it is not so clear or obvious. Therefore, especially for precessing systems, a quantitative method is required for evaluating the “correctness” of the c.m. and gauge. This is discussed in Sec. 2.4.

11 Chapter 2. Centre of mass corrections 12

The current deﬁnition used during BBH simulations for the c.m. is the usual Newtonian deﬁnition:

m m ~x = 1 x~ + 2 x~ , (2.4) c.m. M 1 M 2 where M = m1 +m2 is the total mass of the system, m1 and m2 are the Christodoulou masses [110] of the primary and secondary black holes respectively, and ~x1 and ~x2 are the coordinates of the centers for the primary and secondary black hole respectively. This is a Newtonian expression for the c.m., and from output of the simulations like in Fig. 2.1, it is not a perfect description of the optimal c.m.. The tracking of the c.m. throughout the simulations can be seen for SXS:BBH:0314 and SXS:BBH:0622 in the right column panels of Fig. 2.1. The c.m. motion is an effect of the initial data. One aspect of the initial data construction method proposed in Ref. [111] is the elimination of Arnowitt-Deser-Misner (ADM), or spatial, linear momentum in the initial data for precessing systems, namely enforcing P~ADM = 0. The work done in Ref. [111] proposed a new method for calculating and constructing the initial data for BBH simulations, which has since been adopted. The improved method for calculating initial data has far-reaching effects in Spectral Einstein (SpEC) simulations and most of the simulations in the SXS simulation catalog were completed using this relatively new method (see Appendix A). This had the effect of reducing specific components of mode mixing as seen in the gravitational waveforms, however as showcased in Fig. 2.1, significant mode mixing is still present. As is further discussed in Sec. 2.5.2, linear-momentum recoil is an expected physical contribution to the motion of the c.m.. However, unphysical contributions to the linear momentum in the initial data of simulations introduce unphysical motion in the c.m., essentially imparting spurious linear-momentum kicks. By controlling the linear momentum and removing it, this effect from the initial data is removed. However, even for simulations with initial data constructed using the method described in Ref. [111], significant translations and boosts, and the resulting mode mixing, are still present in the gravitational waveforms. This warrants further investigation into the c.m. motion and the application of a c.m. correction. It had been suggested in Refs. [77,111] that much of the c.m. motion depicted in the right column panels of Fig. 2.1, and seen in all SXS simulations, was largely unphysical and could be removed from the data. The c.m. correction used to remedy the unphysical c.m. motion is discussed in the following section. Additionally, there are alternative definitions of the c.m. and physical effects that are expected to cause the c.m. to move, or imply that the c.m. is not moving at all. The more obvious of these physical effects are Post-Newtonian (PN) corrections for the c.m. which may include effects explaining the c.m. motion, and linear-momentum recoil from the system. PN and linear momentum contributions are examined in Secs. 2.5.1 and 2.5.2.

2.3 Centre-of-mass correction method

Previous work [77, 111] suggests that the c.m. motion is largely a result of gauge choice. Therefore, understanding the c.m. correction begins with understanding the permissible gauge transformations. More speciﬁcally, the gauge transformations that will aﬀect the waveform measured by distant observers. Because a gravitational-wave detector

12 Chapter 2. Centre of mass corrections 13 will typically be very far from the source, only the asymptotic behavior of the waves is generally considered relevant—specifically at future null infinity, I +. While the asymptotic gauge of waveforms from numerical relativity has not been extensively investigated, it is certainly fair to say that no results in the literature thus far have been in a completely specified gauge. Even the strongest claims of “gauge-invariant” asymptotic waveforms [112] are only invariant modulo the infinite-dimensional Bondi-Metzner-Sachs (BMS) gauge group [79,113], which is the asymptotic symmetry group corresponding to the Bondi gauge condition. An important feature of Bondi gauge is that the gravitational waves measured by any distant inertial observer (at least over a duration short compared to the distance to the source) are approximately given by the asymptotic metric perturbation at fixed spatial coordinates as a function of retarded time in some member of this gauge class—and conversely, any such function corresponds to a signal that could be measured by some distant inertial observer [114]. Essentially, the Bondi gauge may be considered to be “as simple as possible, but not simpler” for the purposes of gravitational-wave detection. Because the BMS group alters the waveform while preserving Bondi gauge, it is considered here to be the fundamental symmetry group relevant to gravitational-wave modeling.

It should be noted that other possible gauge choices exist. For example, Newman-Unti gauge [80] is closely related to Bondi gauge, and is invariant under the same asymptotic symmetry algebra [115]. More generally, it is not even clear that waveforms from numerical relativity are actually expressed in either of these well-deﬁned gauge classes, in which case more general gauge transformations may be of interest. Ultimately, the gauge freedoms relevant to counteracting c.m. motion are simply space translations and boosts. As long as these transformations are allowed, this discussion of c.m. motion remains relevant. Previous work [116] suggests that SXS waveforms are consistent with waveforms in Bondi gauge, though further research is warranted.

Because BMS transformations preserve the inertial property of observers, it is not possible to counteract all of the c.m. motion seen in Fig. 2.1—particularly the cyclical behavior. However, in addition to the cyclical behavior, these coordinate tracks begin with some overall displacement from the origin, and then drift away from that initial location over the entire course of the inspiral. Thus, it is expected that a space translation and a boost are needed to negate the eﬀects of some of the c.m. motion. In particular, the translation α~ and boost ~β will be chosen to minimize the average of the square of the distance between the measured c.m. and the origin. This measure will be invariant under time translation and rotation, which are generally dealt with separately during gravitational-wave analysis, so those degrees of gauge freedom are simply ignored. Furthermore, it is not at all clear how a higher-order supertranslation should aﬀect the coordinates close to the center of a simulation, and so discussion of more general supertranslations is left to future work.

2.3.1 Choosing the translation and boost

Appendix E of Ref. [77] is followed for choosing the translation α~ and boost ~β to minimize the average square of the distance between the c.m. measured in the raw data and the origin of the corrected frame. That is, choose α~ and

13 Chapter 2. Centre of mass corrections 14

200

150

100

50 Non-precessing simulations 0 10 8 10 6 10 4 10 2 100 10 10 10 8 10 6 10 4 10 2 10 8 10 6 10 4 10 2 100

600

400

200 Precessing simulations

0 10 8 10 6 10 4 10 2 100 10 10 10 8 10 6 10 4 10 2 10 8 10 6 10 4 10 2 100 | |/M | | | |/M Figure 2.2: Magnitudes of c.m. oﬀsets and drifts for all simulations in the SXS catalog. The top row shows values for nonprecessing systems (i.e., nonspinning, spin aligned, and spin antialigned) and the bottom row shows values for precessing systems. The horizontal axis for each plot is the magnitude of the c.m. value shown (|α~|, |~β|, or |~δ| = |α~ + ~βtmerger|, where tmerger is the ﬁrst reported instance of a common apparent horizon found between the two BHs) and the vertical axis is the number of simulations that have c.m. values of that bin magnitude. Note that typical values of |~β| are quite small, but accumulate over the course of a simulation to cause a large overall displacement by merger. Blue indicates runs using the initial-data method described in Ref. [111]; orange indicates runs using the previous initial-data method. These results suggest that this procedure improves the initial location of the center of mass, but does little to improve its drift.

~β to minimize the function Z t f 2 Ξ(α,~ ~β) = |~xc.m. − (α~ + ~βt)| dt. (2.5) ti It is not hard to ﬁnd the minimum of this quantity analytically. Two moments of the c.m. position are deﬁned as

Z t 1 f h~x i = ~x (t) dt, (2.6a) c.m. − c.m. t f ti ti Z t 1 f ht~x i = t ~x (t) dt. (2.6b) c.m. − c.m. t f ti ti

Then, the minimum of Eq. (2.5) is achieved with

2 2 4(t f + t f ti + ti )h~xc.m.i − 6(t f + ti)ht ~xc.m.i α~ , = 2 (2.7a) (t f − ti)

12ht ~xc.m.i − 6(t f + ti)h~xc.m.i ~β . = 2 (2.7b) (t f − ti)

14 Chapter 2. Centre of mass corrections 15

100

10 2 M / |

k 4 j 10 0 0

| 6 x 10 a k j 100% change m 10% 10 8 q = 1, 1 = 2 = 0 1% q = 1, 1 = 2 0 Everything else 10 10 10 7 10 5 10 3 10 1 101

| 00|/M

Figure 2.3: Comparing the size of c.m. corrections in the SXS catalog, |~µ00|, to how much those corrections change under small variations in the end points of integration used to compute the c.m. correction. The vertical axis shows the largest change in the c.m. correction if ti and/or t f is shifted later by half an orbit. The systems with the largest c.m. corrections—where these corrections are presumably the most important—change by small fractional amounts. On the other hand, there are several systems in which the c.m. correction changes by more than the original correction; those systems also have some of the smaller c.m. corrections in the catalog. The median percentage change is 4% of the original correction, and even the largest individual change is smaller than the median value of |~µ00|.

This transformation is then applied to the asymptotic waveform using the method described in Ref. [77]. Note that this rests on implicit assumptions about how directly comparable the coordinates of the apparent-horizon data and the asymptotic coordinates are. For example, this assumes that the time coordinate of the apparent-horizon data and the asymptotic retarded-time coordinate are equal. While there is no rigorous motivation for this assumption, the results of Sec. 2.4 bear out its approximate validity. Using this minimization method, the c.m. offsets for every public waveform in the SXS catalog have been corrected in the waveform data. The first instance of the c.m. corrections to waveforms in the SXS public waveform catalog was in January of 2017. Center-of-mass corrected waveform data is recommended over non corrected data in all cases, and corresponding files are listed in the SXS public waveform catalog as files ending in CoM.h5. An overview of the c.m. correction values is shown in Fig. 2.2. It is clear from the upper-left panel of Fig. 2.1 that c.m. removal is “helpful” in the sense that it reduces the amplitude oscillations, which are not expected on physical grounds. Unfortunately, this by-eye analysis is not quantitative, and it is not clear how it would apply to a precessing system, as seen in the lower-left panel of the same figure. A better measure of how the waveform quality is impacted by c.m. corrections is discussed in Sec. 2.4.

2.3.2 Choosing the integration region

The determination of α~ and ~β is made over a subset of the total simulation time, from ti to t f [see Eq. (2.5)].

Choosing diﬀerent values of ti and t f may aﬀect the resulting α~ and ~β values. For the corrections performed on the SXS catalog, a standard subset of the simulation time was chosen. All waveforms had their c.m.-correction values

15 Chapter 2. Centre of mass corrections 16

calculated from ti = trelax, the “relaxation” time after which the initial transients have dissipated, to 10% before the end of the inspiral: t f = 0.9tmerger. These time bounds were chosen to avoid including periods of junk radiation as well as the merger and ringdown stages (see AppendixA).

However, changing ti, t f by small amounts could change the c.m. correction values. As there is epicyclical motion of the c.m. (as seen in Fig. 2.1, for example), changing the beginning or ending time may cause the resulting

α~ and ~β to change, depending on where ti and t f fall on an epicycle. For example, if ti and t f are separated by an integer number of epicycles, then it might be expected that any effect from the epicycles on the calculation of α~ and ~β are cancelled out. However, if ti and t f are separated by a noninteger number of epicycles, especially by a half-integer number of epicycles, the epicycles may induce significant bias in α~ and ~β. The overall number of epicycles included in the calculation of α~ and ~β may also affect how sensitive they are to this bias.

Here, the size of the c.m. correction is compared using the standard prescription to the size of the correction when ti and/or t f are changed by half an orbit. This will give us some idea of the stability of the c.m.-correction procedure. However, it must be noted that, at a larger scale, the choices of ti and t f are quite arbitrary. For some purposes, it may be preferable to choose those values to range over only the ﬁrst half of the inspiral, or even just the ringdown stage. The values used in the SXS catalog were chosen for robustness and easy reproducibility.

To simplify the comparison, the c.m. motion is described using the quantity ~µ, which gives the most distant position of the corrected origin of coordinates throughout the inspiral, relative to the origin used in the simulation. Speciﬁcally, ~µ can be deﬁned according to

  α~ if |α~| > |α~ + ~βt |,  merger ~µ =  (2.8)  ~ ~ α~ + βtmerger if |α~| ≤ |α~ + βtmerger|.

For 96% of the simulations in the SXS catalog, it was found that ~µ = α~ + ~βtmerger. The 4% of simulations with ~µ = α~ have no apparent correlations with system parameters, and are eﬀectively random. Subscripts are introduced, so that ~µ00 is the result of this calculation when using the original values of ti and t f ; ~µ10 is the result when moving ti later by half an orbit; ~µ01 is the result when moving t f later by half an orbit; and ~µ11 is the result when moving both ti and t f later by half an orbit.

In Fig. 2.3, max jk |~µ00 − ~µ jk| is examined as a measure of how robust the c.m. corrections are with respect to these small adjustments in the choices of ti and t f . In the great majority of systems the c.m. changes by less than 10−2 M. This is, for example, just one tenth the size of the displacements seen in the upper panels of Fig. 2.1. The −3 −2 median change is 3.1 × 10 M, and in all cases is smaller than the median value of |~µ00| itself, which is 6.9 × 10 M. The systems with the largest c.m. corrections in the SXS catalog change by fractions of a percent, suggesting that the results are certainly stable in the cases where applying a c.m. correction is most important. There are several cases where the fractional change is greater than 100%, though these are systems with relatively small values of ~µ00. The median fractional change is 4.3%. It is also notable that the data points separate roughly into three groups. The group in the lower left corner of Fig. 2.3 is comprised exclusively of equal-mass nonspinning simulations

16 Chapter 2. Centre of mass corrections 17

+0.35 0.00 0.35

+0.40 0.40 0.40 1.0 p 0.5 +1.63 2.00 0.99 10

q 5

+2.76 4 (3.48 2.34) × 10

0.0030 e

0.0015 +3.08 19.63 1.91 60

30 Orbits +0.01 0.01 0.00

0.050 M / | | 0.025 +1.88 5 (1.34 1.15) × 10

| 0.00010 |

+0.09 0.00005 0.07 0.06 0.8 M / |

| 0.4

0 1 5 0.5 1.0 10 30 60 0.4 0.8 0.025 0.050 0.0015 0.0030 0.00005 0.00010 Orbits eff p q e | |/M | | | |/M

Figure 2.4: Center-of-mass correction values and relevant simulation parameters. χeff is the effective spin, χp is the effective precessing spin, q is the mass ratio, e is the eccentricity, Orbits represents the total number of orbits the simulation had at tmerger, and α~, ~β, ~δ = α~ + ~βtmerger are the c.m. correction values representing the spatial translation, boost, and total c.m. displacement respectively. Red represents spin-aligned simulations, and teal represents precessing simulations. The numbers above each column represent the median of each variable over all simulations, with superscripts and subscripts giving the offset (relative to the median) of the 84th and 16th percentiles, respectively. with various eccentricities, though several of these are also found in the central group. The central group is where all equal-mass simulations with equal but nonzero spins are found, which includes ten systems with significant precession. Every other type of system is in or near the largest group, on the upper right.

2.3.3 Correlations between c.m. correction values and physical parameters

Along with having c.m. corrected the waveforms, an analysis of the values of the boosts and translations needed by each simulation in the SXS public waveform catalog has been performed. No obvious correlations can be seen in Fig. 2.2 between spin aligned and precessing systems. A more in-depth correlation plot is shown in Fig. 2.4, taking more of the simulation parameters into consideration. It can be seen

17 Chapter 2. Centre of mass corrections 18

that typically precessing simulations may have larger overall c.m. displacement, ~δ = α~ + ~βtmerger, and that larger boost values ~β correspond with larger overall displacement values ~δ for both spin aligned and precessing systems. Outside of the correlations between the boost ~β and total displacement ~δ of the c.m., there does not appear to be any other strong correlations present for the current SXS simulation catalog. It was expected that precessing, high mass ratio, and eccentric systems should have vastly diﬀerent c.m. correction values than spin aligned, low mass ratio, and more circular systems, however no such correlations are present with this data set. For Fig. 2.4, the eccentricity e, number of orbits, and mass ratio q are reported by SpEC at the end of the simulation. The eﬀective spin [31, 117, 118] is calculated by

 ~ ~  ~ ~ c  S 1 S 2  L ~χam1 + ~χbm2 L χeﬀ =  +  · = · , (2.9) GM m1 m2 |~L| M |~L| and an eﬀective precession parameter [31, 48]

c χ B |S~ |, B |S~ | . p = 2 max( 1 1,⊥ 2 2,⊥ ) (2.10) B1Gm1

~ 2 Here, M = m1 + m2 is the total mass of the system, S i = G/c ~χimi is the angular momentum of the i-th black hole and ~χi its dimensionless spin, B1 = 2 + 3m2/2m1, B2 = 2 + 3m1/2m2, and the subscript ⊥ indicates the quantity perpendicular to the orbital angular momentum ~L, e.g., S~ 1,⊥ = S~ 1 − (S~ 1 · Lˆ)Lˆ. Note that χp gives a measure of how much a system is precessing during a simulation.

2.4 Quantifying c.m. Correction Using Waveforms Alone

Any discussion of c.m. based on the positions of the individual black holes will suffer from the same fundamental ambiguity: reliance on coordinates—specifically in the highly dynamical region between the two black holes—that are subject to unknown gauge ambiguities. The only region of the spacetime where the gauge freedom is limited in any useful sense is the asymptotic region, in which it is assumed the only freedom is given by the BMS group (described in Sec. 2.3). While there are many suggestions in the literature [82, 119–127] for using asymptotic information to specify the asymptotic gauge more narrowly, they all require more information than is available from most catalogs of numerical-relativity waveforms—such as additional Newman-Penrose quantities or more precise characterization of the asymptotic behavior of the various fields. Here is presented a simplistic but effective measure of c.m. effects that can be applied exclusively to asymptotic waveform data h or Ψ4. The basic idea is that it is possible to decompose a waveform measured in c.m.-centered coordinates into modes that are, at least for small portions of the inspiral, given by a slowly changing complex amplitude times a complex phase that varies proportionally with the orbital phase. When the waveform is decomposed in off-center coordinates, those well-behaved modes mix, so that the amplitude and phase do not behave as expected. Therefore, it was attempted to model a given waveform in a sort of piecewise fashion that assumes the expected behavior, and simply measure the residual between the model and the waveform itself. For a

18 Chapter 2. Centre of mass corrections 19 given transformation applied to the waveform, the residual is minimized by adjusting the parameters to the model while keeping the waveform ﬁxed. The smaller the minimized residual, the more accurately the waveform with that transformation can be modeled in this simple way, and the more nearly it can be expected that the waveform is decomposed in c.m.-centered coordinates. Roughly speaking, this can be thought of as a measure of the “simplicity” of the waveform, which is not only in line with our basic expectations for waveforms in the appropriate coordinates, but also a useful measure of how accurately simple waveform models (EOB, surrogate, etc.) will be able to capture features in the numerical waveforms. This criterion is obviously totally distinct from any criteria involving the BH positions, but is important precisely because it provides a complementary way of looking at the data. Finding agreement between the results of this method and another will lend support to the idea that the other method is suitable.

2.4.1 Deﬁning the method

The initial inputs are some translation α~ and boost ~β that are to be evaluated. The waveform is transformed by those inputs and denotes the result Tα,~ ~β[h]. Then transform to a “co-rotating frame”, which is a time-dependent frame chosen so that the waveform in that frame is varying as slowly as possible [101]. Only the angular velocity of this frame, Ω~ , is determined by the condition that the waveform vary slowly; it is integrated in time to obtain one such frame [128], but the result is only unique up to an overall rotation. That overall rotation is chosen so that the ~z 0 axis of the ﬁnal co-rotating frame is aligned as nearly as possible throughout the inspiral portion of the waveform with the dominant eigenvector [129, 130] of the matrix

Z n o∗ n o hL(1L2)i B L(1 Tα,~ ~β[h] L2) Tα,~ ~β[h] dA, (2.11) S 2 where La is the usual angular-momentum operator. This still leaves the frame defined only up to an overall rotation about ~z 0, but such a rotation will have no effect on our results. The transformed waveform in this co-rotating frame n o will be denoted R Tα,~ ~β [h] (t, θ, φ), though the parameters will usually be suppressed, and may decompose the angular dependence in terms of SWSH mode weights as usual. This is the quantity that will be modelled. For the model itself, the inspiral is broken up into smaller spans of time; the waveforms will be modelled using simple linear-in-time approximations, so it is not expected to accurately reproduce the nonlinear evolution over a very long portion of the inspiral using just one such model. The relevant measurement of the waveform’s dynamical 0 behavior is the angular velocity of the co-rotating frame. More specifically, Ωz0 = Ω~ · ~z , and used to determine a 1 R phase Φz0 = Ωz0 dt. An obvious span of time would be a single cycle of this phase, which would include enough data so that the fit would actually reflect the behavior of the waveform, but not so much that a poor fit due to evolution on the inspiral timescale is expected. However, this model means fitting oscillatory terms with linear models. In the simple

1This phase is loosely related to the orbital phase of the binary. The angular velocity Ω~ , however, is deﬁned solely with respect to the waveform at I +, and entirely without reference to any quantities at ﬁnite distance in the system. Nonetheless, for reasonably well-behaved coordinate systems, it is expected to agree roughly with the orbital phase deduced from the trajectories of the black holes, especially during the early inspiral regime.

19 Chapter 2. Centre of mass corrections 20 case of fitting a line to a basic sine function, it is not hard to see that the optimal line has the expected slope of zero—independent of the phase of the sine function—when the fit region is such that the argument of the sine function goes through a phase change of ϕ ≈ 8.9868 [or other solutions of ϕ = 2 tan(ϕ/2)]. Therefore, each span of time is selected so that it extends over a phase Φz0 of approximately ϕ, thereby determining the difference in time between ti,1 and ti,2 so that they satisfy

Φz0 (ti,2) − Φz0 (ti,1) = ϕ. (2.12)

This choice does drastically reduce the oscillations in the optimal fit parameters as the fitting window is shifted. While the individual time spans extend over this range, the remaining effects from oscillation are minimized by selecting successive time spans to be separated by half of a period—so that Φz0 changes by exactly π between ti,1 and ti+1,1:

Φz0 (ti+1,1) − Φz0 (ti,1) = π. (2.13)

So that the model may be reasonably accurate, without encountering excessive numerical noise or excessively dynamical behavior at merger, the region over which these time spans are chosen is the central 80% of time between the “relaxation time” listed in the waveform metadata and the time of maximum signal power in the waveform.

This establishes t0,1, and all successive times can be computed from that using Eqs. (2.12) and (2.13).

Now, the waveform is modelled “piecewise” on these spans of time, though the pieces are overlapping. The advantage of transforming the waveform as described above is that each mode separates [102] into two parts that are symmetric and antisymmetric under reﬂection along the z axis. The symmetric part varies on an inspiral timescale because the primary rotational behavior has been factored out by transforming to the co-rotating frame; the antisymmetric part is mostly due to spin-orbit coupling and therefore varies most rapidly by a complex phase with frequency equal to the rotational frequency of the frame itself, though possibly with opposite sign. These two parts are modelled separately as simple linear-in-time complex quantities, with an additional phase-evolution term for the antisymmetric parts. For each time span i,

X h i l,m l,m l,m l,m iσ(m,l) Φz0 (t) µi(t, θ, φ) = si + s˙i (t − ti,c) + ai + a˙i (t − ti,c) e −2Yl,m(θ, φ). (2.14) l,m

l,m l,m l,m l,m Here, each of the coeﬃcients si , s˙i , si , and a˙i is a complex constant, ti,c = (ti,1 + ti,2)/2 is used to mitigate degeneracy between the constant-in-time and linear-in-time terms, and the function σ is given by

  1 |m| < l, σ(m, l) =  (2.15)  −1 |m| = l.

These signs are chosen because they represent the dominant behavior of the corresponding terms in the data. Note l,m that the symmetry properties imply that once the quantities si , etc., are chosen for positive m, they are automatically

20 Chapter 2. Centre of mass corrections 21 known for negative m from the relations

l,m l l,−m l,m l ¯l,−m si = (−1) s¯i , s˙i = (−1) s˙i , (2.16a) l,m l+1 l,−m l,m l+1 ¯l,−m ai = (−1) a¯i , a˙i = (−1) a˙i . (2.16b)

Because the m = 0 modes of the SXS waveforms are not considered reliable [74, 90], those modes in both the model and the data are ignored. That is, the sum in Eq. (2.14) does not include any m = 0 modes. If the sum over modes extends from l = 2 to some maximum l = L, the total number of (real) degrees of freedom in this model is 4L(L + 1) − 8 for each span of time. While the data used contains up to l = 8, the highest-order modes contribute little to the result, and drastically increase the number of degrees of freedom in the problem (and therefore the time taken to optimize the model). Therefore, only up to l = 6 is used in constructing the model and evaluating the residual, reducing the degrees of freedom from 280 to 160 per time span. Finally, because this is still such a large number of degrees of freedom, the evaluation is limited to only the first two and last two time spans; including the rest has no significant effect on the result, but vastly increases the amount of processing time required. This results in a manageable 640 degrees of freedom in this model.

Now, using this model, the objective function is deﬁned as

X Z ti,2 Z 2 ~ n o Υ α,~ β = min R Tα,~ ~β [h] − µi dA dt s,s˙,a,a˙ 2 i ti,1 S X Z ti,2 X 2 n ol,m l,m = min R T ~ [h] − µ dt. (2.17) s,s˙,a,a˙ α,~ β i i ti,1 l,m

This function is used in two ways: ﬁrst, to simply evaluate Υ for given values of (α,~ ~β), where those values are obtained from the methods described in other sections; second, to minimize Υ over possible values of (α,~ ~β) to ﬁnd the optimum c.m. correction.

2.4.2 Results for the standard c.m.-correction method

The value of Υ deﬁned in Eq. (2.17) for all the waveforms discussed in the previous sections is then compared.

First, its value Υraw in the raw data is compared to its value Υ0PN using α~ and ~β as given by Eqs. (2.7), where ~xc.m. is given by the Newtonian (0PN) formula. The latter corresponds to the technique actually used in the current SXS data, for waveforms found in the SXS simulation catalog with file names ending in CoM. The results of this comparison are shown in Fig. 2.5. One unusually short simulation (SXS:BBH:1145 [131]) in the SXS catalog did not have enough GW cycles to evaluate Υ properly, leaving a total of 2,017 systems shown in these figures. The vast majority of systems improve significantly by this measure. The notable feature is that even though the naive 0PN c.m. trajectory is so fundamentally different from Υ, this plot suggests that they agree in the sense that the 0PN

21 Chapter 2. Centre of mass corrections 22

2014 positive 3 negative

102

101 Number of systems

100 10 8 10 6 10 4 10 2 100

( raw 0PN)/ raw

Figure 2.5: Relative difference between Υraw evaluated on the raw waveform data and Υ0PN evaluated using the values α~ and ~β given by the simplest Newtonian (0PN) approximation of Eqs. (2.7)—the same c.m. correction used in the current SXS catalog. In the vast majority of cases, the value of Υ decreases substantially from the raw waveform data to the 0PN c.m. corrected waveform data (though it actually increases very slightly in three cases with significant eccentricity). This suggests that even though Υ and the coordinate-based c.m. are such entirely different measures and based on completely different data, they agree that the changes introduced by naive c.m. corrections are generally improvements. correction improves the waveforms for all but three systems—and even for those three the change is very small.2 Υ can be actively optimized over the values of α~ and ~β. The results are shown in Fig. 2.6. Naturally, Υ improves in every case because it is specifically being optimized. In Fig. 2.6, the pattern that the vast majority of systems are changing by small fractions. In this case, there are just three systems in which Υ changes at the percent level. These are some of the same systems that changed the most in going from the raw data to the 0PN-corrected data. These particular systems also happen to be extremely long, with significant overall accelerations during the inspiral. This suggests that the corrections will be sensitive to the precise span of times over which the corrections are being made, which may explain why they continue to change so much by optimization. However, as discussed in Sec. 2.3.2, changing the beginning and ending fractions does not significantly change α~ and ~β. Nevertheless, the overall scale of the changes seen in this plot suggests once again that the naive 0PN c.m. correction is achieving near-optimal results in the vast majority of cases.

2.5 Improving the c.m. correction

2.5.1 Post-Newtonian c.m. deﬁnition

To characterize the motion seen in the c.m. in the raw simulation data, an obvious first step to finding a more accurate definition of the c.m. during the simulation is to try low orders of Post-Newtonian (PN) corrections. Note

2These three systems are unusual, in that they are quite short (having 13 to 15 orbits before merger, compared to an average of 22), and have eccentricities (0.215 and higher) that place them among the 12 most eccentric in the SXS catalog. Furthermore, the magnitude of the change in Υ for each of them is very small—in the lowest percentile for the entire catalog—which suggests that the negative results may be consistent with numerical error, and in any case are not cause for much concern.

22 Chapter 2. Centre of mass corrections 23

100

10 2 N P 0

/ 4

) 10 t p o

N 6

P 10 0 ( 10 8

10 10 10 8 10 6 10 4 10 2 100

| raw 0PN|/ raw

Figure 2.6: Comparison between the value Υopt for which α~ and ~β are optimized, and Υ0PN evaluated using the values α~ and ~β given by the simplest Newtonian (0PN) approximation—the same c.m. correction used in the current SXS catalog. The vertical axis shows the relative improvement in going from the Newtonian correction to the optimized correction. The dashed diagonal line represents where the comparisons are equal—the “x = y” line. Optimization improves the results for the great majority of systems by less than 1%. The three exceptions to this rule are particularly long systems.

100

10 2 M /

| 4

N 10 P 0

N 6 P 10 n 100% change | 10% 10 8 1% 1PN 2PN 10 10 10 7 10 5 10 3 10 1 101

| 0PN|/M Figure 2.7: Differences in the maximum displacements between the c.m. correction computed using the Newtonian (0PN) c.m. formula and the 1PN or 2PN c.m. formulas. The quantity ~µ, defined in Eq. (2.8), is the largest displacement between the origin of coordinates in the simulation and the corrected origin. The Post-Newtonian corrections change the c.m. correction values by roughly 1% in the majority of cases. Systems with larger changes are consistent with systems that are sensitive to small changes in the end points of integration used to find the c.m. correction, as seen in Fig. 2.3.

23 Chapter 2. Centre of mass corrections 24

100

10 2 M / | N P 0 10 4 N P n | 10 6

1PN 10 8 2PN

10 5 10 4 10 3 10 2 10 1 100 Eccentricity

Figure 2.8: Diﬀerences between the Newtonian c.m. and the 1PN corrected c.m. and 2PN corrected c.m. correction values versus the eccentricity of the simulation at relaxation time. No correlations are evident between either the 1PN or 2PN correction and eccentricity values.

100 100 1479 negative 1376 negative 10 2 538 positive 10 2 641 positive

N 4 N 4

P 10 P 10 0 0 / / ) 6 ) 6 N 10 N 10 P P 1 2 10 8 10 8 N N P P 0 0 10 10 ( 10 ( 10

10 12 10 12

10 14 10 14 10 8 10 6 10 4 10 2 100 10 8 10 6 10 4 10 2 100

| raw 0PN|/ raw | raw 0PN|/ raw Figure 2.9: Comparison of Post-Newtonian contributions for determining the center of mass. These plots show the difference between the value of Υ [Eq. (2.17)] resulting from the naive 0PN method based on the coordinate trajectories of the apparent horizons and the value of Υ when incorporating 1PN and 2PN effects, as described in the text. The horizontal axes show the relative magnitude of the change when going from the raw data to the corrected waveform. The dashed diagonal line in both plots represents where the comparisons on the horizontal and vertical axes are equal—the “x = y” line. In most cases, the values actually become significantly larger when going from the 0PN value to other values. Those systems are shown as crosses, while systems with smaller values are shown as circles.

24 Chapter 2. Centre of mass corrections 25 that the c.m. should be near the origin of the coordinate system, with minimal motion around the origin from linear-momentum recoil, as discussed in Sec. 2.5.2. 1PN order corrections are analytically trivial, and are zero for circular systems. However, SXS simulations are not perfectly circular and so the 1PN and 2PN order corrections to the c.m. are investigated. The PN corrections given by Eq (4.5) in [132] are implemented. This formalism goes up to 3.5 PN for the c.m. vector in time. The 2PN version of the expression used can be found in AppendixC. Using this formalism, the effects of the correction on the c.m. are small but measurable at 1PN and significantly different at 2PN for many systems. The coordinates of the c.m. itself changing are not of interest, but of the c.m. correction values changing, as discussed in Sec. 2.3. In general, it is assumed that the c.m. drifts in a linear fashion and can be corrected with a translation α~ and boost ~β. The results from the 1PN and 2PN analysis are shown in

Fig. 2.7, which shows the relative difference between α~, ~β, and total c.m. displacement ~δ = α~ + ~βtmerger for the 1PN (top panels) correction to the c.m. and the 2PN (bottom panels) correction to the c.m.. The 1PN corrections show small changes for most simulations. However, the 2PN correction shows more sizable changes in the c.m. correction values, which may indicate that including at least up to 2PN corrections to the c.m. will give better accuracy either to the c.m. itself or to the correction factors. To see any potential correlations with large 1PN or 2PN corrections and simulation parameters, the relative difference in the c.m. corrections to the eccentricity e of the system is compared. As shown in Fig. 2.8, no correlations between the magnitude of the 1PN and 2PN corrections to the c.m. and the eccentricity are apparent, despite the definition of the 1PN contribution to the c.m. being dependent on e. Using the method described in Sec. 2.4, comparisons of Υ, as defined in Eq. (2.17), between 1PN, 2PN, and the original c.m. correction method (dubbed 0PN), can be found in Fig. 2.9. The striking feature of these plots is that a significant number of systems actually have larger values of Υ when including either of these corrections. While it is reassuring that the majority of systems in each case only exhibit quite small changes—changes of order 10−4 or less in a quantity that already improved significantly from the raw data—the 1PN and 2PN corrections plots include a large group of systems that change at the percent level. These systems also happen to be the same systems that changed most drastically in going from the raw data to the 0PN-corrected data (found near the upper-right corner of the plots), and are particularly biased towards increasing values of Υ. That is to say, it appears that the 1PN and 2PN corrections do worst for the most extreme systems. This should not come as a great surprise, since those systems tend to be the ones with the most extreme mass ratios and precession, so that Post-Newtonian analysis is also expected to be at its least accurate.

2.5.2 Linear-momentum recoil

Any binary with asymmetric components will emit net linear momentum in the form of gravitational waves, which will cause a recoil of the binary itself. As the system rotates, the direction of recoil will also rotate, pushing the c.m. roughly in a circle [133, 134]. In principle, this eﬀect could cause the epicyclical motion apparent in the c.m. trajectories, which is further characterized in Sec. 2.5.4. To see if recoil is responsible, the methods described in Refs. [102, 133–135] are used to investigate the size of the linear-momentum recoil implied by the

25 Chapter 2. Centre of mass corrections 26

1000 q 1.05 q > 1.05

800

600

400 Number of systems 200

0 101 103 105 107 109 101 102 103 104 105 106 101 102 103 104 105 106

rmeasured/rrecoil |aCOM|/|pGW/M| |aCOM|/|pGW/M| Figure 2.10: Comparing measured c.m. motion to motion caused by emission of linear momentum carried away by gravitational waves. The panel on the left shows the average ratio of the measured radius of c.m. motion, given by the time-averaged magnitude of the c.m. epicycles and stated in Eq. (2.20), to the radius expected from leading-order calculations given by Eq. (2.19). The center and right panels show the average ratio of the measured c.m. acceleration, given by the second time derivative of the c.m. coordinate positions and stated in Eq. (2.21), to the acceleration due to asymmetric momentum ﬂux carried by the measured gravitational waves—for near-equal mass ratios and larger mass ratios, respectively. Blue indicates runs using the initial-data method described in Ref. [111], orange indicates runs using the previous initial-data method. In every case, the measured motion is at least an order of magnitude larger than the motion expected from gravitational-wave recoil.

gravitational-wave emission in these systems.

As shown in the right-column panels of Figure 2.1, the c.m. motion follows an overall linear track with additional epicyclical motion. The linear motion of the c.m. is well understood, and discussed in Sec. 2.3. For this analysis, it is assumed that the linear part of the c.m. motion may be removed from the data without loss of information, leaving the epicyclical behavior about the coordinate origin.

Blanchet and Faye [133] consider the motion of the c.m. from linear momentum flux and the flux of the c.m. itself up to 3.5PN order to calculate the instantaneous c.m. motion induced by these effects, finding the position of the c.m. relative to its average location over an orbit to be

48 G4 G~ = − m2 m2 (m − m ) λ,ˆ (2.18) 5 c7r4ω 1 2 1 2

[cf. Eq. (6.9) in Ref. [133]] leading to a circular motion of the c.m. with radius

48 G4 r = |G~| = m2 m2 (m − m ) (2.19) recoil 5 c7r4ω 1 2 1 2 for a system comprised of nonspinning BHs in a circular orbit. Here, r = |~x1 − ~x2| is the distance between the two p black holes, ω = GM/r3 is the Newtonian orbital frequency, and λˆ is the unit vector in the direction of motion of m1 [cf. Eq. (2.25)]. An earlier analysis using a simpliﬁed model and lower-order approximations can be found in Ref. [134].

Subtracting the motion described by Eq. (2.19) out of the c.m. motion and comparing the radius of the measured

26 Chapter 2. Centre of mass corrections 27 motion to that of Eq. (2.19) immediately shows that the measured c.m. motion—speciﬁcally the epicyclical motion— is signiﬁcantly larger than what can be explained by linear-momentum and c.m. reactions for all simulations. Figure 2.10 shows a comparison across the SXS public waveform catalog for the ratio between the measured c.m. radius about the coordinate origin and the estimated c.m. radius given by Eq. (2.19). The measured c.m. radius is calculated by averaging the distance of the c.m. at time t away from the line α~ + ~βt between times ti and t f :

Z t 1 f r = |~x − (α~ + ~βt)|dt. (2.20) measured − c.m. t f ti ti

The results show that the measured radius is typically hundreds of times larger than the radius implied by Blanchet and Faye’s analysis.

It is also possible to go beyond the analysis in Ref. [133] by using the measured gravitational waves to compute the linear-momentum ﬂux, and compare the acceleration that would cause to the measured acceleration of the c.m. in the simulation. The acceleration of the c.m., ~ac.m. was calculated by taking two time derivatives of the coordinate position of the c.m. after removing the linear motion:

d2 ~a = ~x − (α~ + ~βt) . (2.21) c.m. dt2 c.m.

The linear-momentum ﬂux may be calculated from the gravitational radiation as [136]

d~p c2 R2 dh 2 = rˆ, (2.22) dΩ dt G 16π dt where rˆ is the direction from the source to the point in question, R is the distance from the source to the observation sphere, and Ω represents all angles on the sphere. Integrating over all angles to ﬁnd the total linear momentum ﬂux d~p/dt = ~p˙, h is expanded into SWSHs as done in Eq. (2.3). rˆ can also be decomposed accordingly as

r 2π √ rˆ = (Y − Y , iY + iY , 2Y ). (2.23) 3 1,−1 1,1 1,−1 1,1 1,0

Then integrating over all angles gives

    2 2 r 0 0 0 X 0 0 0 0  l l 1  l l 1 c R 1,m −m l,m ¯ l ,m 3(2l + 1)(2l + 1) m     p˙ j = rˆ h˙ h˙ × (−1)     , (2.24) G 16π j 4π  0 0    l,l0,m,m0 m −m m − m 2 −2 0 where the last two factors are Wigner 3-j symbols. The sum over m0 and most terms in the sum over l0 can be eliminated using properties of the 3-j symbols. Explicit expressions for the calculation of the linear momentum ﬂux are given in Appendix D. Our calculation of the linear momentum ﬂux can be found in the open-source package spherical_functions [137].

Of course, there are multiple methods for calculating the linear momentum for a BBH system. The method presented here was compared with the method proposed in Ref. [135] and originally stated in Ref. [136], and found

27 Chapter 2. Centre of mass corrections 28 that the two derivations give the same results to within numerical accuracy. This is unsurprising given that the methods are identical up to the choice of notation. In particular, the core definition of the total linear momentum flux given in Eq. (2.22) is the same as that given in Eq.(2.11) of Ref. [136]. The linear-momentum flux divided by the total mass of the system was compared with the measured acceleration of the c.m. and it was found that the values for most of the SXS public waveform catalog do not agree. It was further found that the c.m. acceleration for nonequal mass systems is consistently larger than the acceleration found through the linear momentum flux, confirming that most of the c.m. motion is not due to linear-momentum recoil. ˙ An overview of the ratios between ~ac.m. and ~p/M can be seen in Figure 2.10. Note that linear-momentum recoil in equal mass or near-equal mass BBH systems is expected to be very small, and so these systems are isolated as a separate case in the middle panel of Figure 2.10.

2.5.3 Causes of unphysical c.m. motion

If the c.m. motion seen in the SXS BBH simulations cannot be explained by physical processes such as inclusion of PN terms or linear-momentum recoil, then why is it there? There are two potential causes for the appearance of unphysical or erroneous c.m. motion: (i) the presence of uncontrolled residual linear momentum in the initial data, and (ii) the emission of unresolved junk radiation at the beginning of the BBH simulation causing effectively random and unphysical coordinate kicks to the system. The presence of uncontrolled residual linear momentum was addressed and partially rectified in Ref. [111], leading to a new method of creating initial data for the BBH simulations. However, this method was not used for all systems in the SXS catalog, and does not completely resolve the issues of spurious translations and boosts even when it is applied. Another factor that seems to cause c.m. motion starting from early in the simulations appears to be junk radiation, which is an inherent part of BBH simulations. It is the radiation emitted when a BBH relaxes from its initial-data “snapshot”, which is only an approximation to the true state of a long binary inspiral at the time the simulation starts. Junk radiation is physical, in the sense that if the entire spacetime were actually in the configuration given by the initial data, it would indeed emit this radiation as the system evolved. However, it is not astrophysical, in the sense that no real system in the universe is expected to contain this type of radiation since real systems would go through long binary inspirals instead of two compact objects suddenly appearing in each others vicinity. Junk radiation contains high frequencies that are largely unresolved in BBH simulations because of limits of computational power and time. As the resolution increases in the simulations, more of the junk radiation is accurately treated. This is a potential cause for the difference in initial kicks of the c.m., as seen in Fig 2. Even in our higher-resolution simulations, not all of the junk radiation is accounted for. This failure to resolve all of the junk radiation possibly contributes to the c.m. kicks observed in SpEC BBH simulations. Fortunately, kicks from the emission of unresolved junk radiation can be corrected using the gauge transformations discussed in the previous section. The large epicyclical motion in the c.m. cannot be accounted for using a BMS gauge transformation like the linear motion of the c.m.. The cause for such large, seemingly unphysical epicyclic motion is unknown, and is left for future work.

28 Chapter 2. Centre of mass corrections 29

2.5.4 Epicycle quantiﬁcation

As seen in the right column panels of Figure 2.1, the c.m. motion has both a linear and epicyclic component. The linear component of the c.m. motion has been discussed and is already considered in the current c.m. correction technique. The size of the epicyclic motion in the c.m. cannot be solely explained by linear-momentum recoil, as discussed in Sec. 2.5.2. The leftmost plot in Figure 2.10 illustrates that the expected radius of the epicycles from linear- momentum recoil, even on an approximate basis, is orders of magnitude smaller than what is actually seen, given by rmeasured defined in Eq. (2.20). The actual size of rmeasured is fairly consistent across the SXS simulation catalog regardless of simulation parameters or initial data construction, and tends to be between 0.01 and 0.1 with an average value of 0.026 in simulation units. There is not a significant change in the distribution or magnitude of rmeasured between the 0PN, 1PN, or 2PN representations of the c.m.. It is assumed that the epicycle motion seen in the c.m. after the translation and boost are applied is from the calculated c.m., ~xc.m., being displaced from the optimal c.m. by a small amount. “Displaced” here means that ~xc.m. is displaced from the optimal c.m. along the separation vector ~rab = ~x1 − ~x2, and not in any other direction. Regardless of the origin of these epicycles, removing them to calculate the linear c.m. correction should improve the quality of the waveforms. Their removal should reduce the error associated with the averaging done to calculate the translation and boost correction values. As discussed regarding the choice of beginning and ending times in Sec. 2.3.2, the presence of large epicycles has the potential to affect the reproducibility and reliability of the calculation of α~ and ~β. Removing the epicycles, or at the very least minimizing them, to calculate a more accurate c.m. correction should further reduce mode mixing. Of course, epicycle motion cannot be completely subtracted from the waveform itself as this would require an acceleration correction, which is not an allowed BMS transformation. To accurately describe the epicycles, we need to define the co-rotating coordinate frame. For our simulations, we have three unit vectors that describe the rotating coordinate frame:

where nˆ points along the separation vector ~rab, kˆ points along the orbital angular velocity ω~ , and λˆ points along the direction of rotation. This leads us to a potential method for epicycle removal. First, the estimated spatial translation α~ and boost ~β from the original c.m. ~xc.m. is calculated using the current averaging method

~c1 = ~xc.m. − (α~ + ~βt). (2.26)

29 Chapter 2. Centre of mass corrections 30

Original Epicycle Corrected

0.00 Lev1 COM trajectory 0.00 Lev2 + t Lev3 Lev4 M

0.05 0.05 / y SXS:BBH:0314

0.10 0.10 0.00 0.03 0.06 0.00 0.03 0.06

0.02 0.02 Lev1 COM trajectory M

Lev2 /

+ t y 0.00 Lev3 0.00 SXS:BBH:0622

0.02 0.02 0.04 0.02 0.00 0.04 0.02 0.00 x/M x/M

Figure 2.11: Illustration of epicycle correction for the two simulations already shown in Figure 2.1. The left panels show the Newtonian center of mass ~xc.m., whereas the right panels show the epicycle corrected ~c2. The thick dashed lines indicate linear ﬁts to the respective center of mass trajectories: α~ +~βt (left panels) and α~ epi +~βepit (right panels). Several numerical resolutions are shown (labeled Lev1 to Lev4), and data is plotted only for the time-interval [ti, t f ], which is used for the linear center of motion ﬁts.

Note that h|~c1|i = rmeasured as deﬁned in Eq. (2.20), using the angle-bracket notation to denote averaging over time, as for the moments of the c.m. position in Eq. (2.6). The corresponding co-rotating coordinate frame unit vectors given in Eq. (2.25) are then calculated. These unit vectors are then applied to the original c.m. ~xc.m. as

~c2 = ~xc.m. − ∆nnˆ − ∆λλˆ − ∆kkˆ, (2.27)

where ∆n = h~c1(t) · nˆ(t)i, ∆λ = h~c1(t) · λˆ(t)i, ∆k = h~c1(t) · kˆ(t)i, are the time averaged projections of the linearly corrected c.m. onto the rotational coordinate system unit vectors. The epicycle-corrected c.m. ~c2 is then fed back into the averaging method to get the ﬁnal spatial-translation α~ epi and boost ~βepi values, which can then be applied to the waveform data.

30 Chapter 2. Centre of mass corrections 31

c1(t) x vs. c1(t) y c1(t) n vs. c1(t)

0.02 Lev1 0.02 Lev2 Lev3 Lev4 0.00 0.00 SXS:BBH:0314

0.02 0.02

0.02 0.00 0.02 0.02 0.00 0.02 0.01 0.01 Lev1 Lev2 Lev3

0.00 0.00 SXS:BBH:0622

0.01 0.01

0.01 0.00 0.01 0.01 0.00 0.01

Figure 2.12: Contribution to ~xc.m. which cannot be fitted by a linear drift. Left panel: Deviation of Newtonian center of mass from a linear motion, i.e. the quantity ~c1, plotted in inertial coordinates. Right panel: Projection of ~c1(t) onto the co-rotating basis vectors nˆ and λˆ. The rotation of ~c1 around the origin visible in the left panels is transformed into a nearly constant offset from the origin in co-rotating coordinates of the right panels. Shown are multiple numerical resolutions (Lev1, Lev2, ...) which fall on top of each other, indicating that the epicyclic dynamics is numerically resolved and independent of the direction of the linear drift. Data is plotted only for the time-interval [ti, t f ], which is used for the linear center of motion fits.

A visual representation of our epicycle-correction method is shown in Figure 2.11, which uses the spin aligned system SXS:BBH:0314 and the precessing system SXS:BBH:0622 as sample cases. As seen in the upper two panels on the right, the removal of the epicyclic motion as calculated by the time-averaged values ∆n, ∆λ, and ∆k greatly diminishes the large size of the epicycles and allows for potentially better optimization of the c.m.-correction values α~ and ~β. In these same panels, it can also be seen that not all of the epicycle motion is removed by our method, and in particular there are larger deviations towards the beginning and end of the simulation data, which are a result of our time-averaged method capturing most but not all of the epicycle motion. Speciﬁcally, the epicycle radius tends to grow with time in most (81% of) SXS simulations.

31 Chapter 2. Centre of mass corrections 32

100

10 2 M

/ 4 | 10 N P 0 i

p 6

e 10

| 100% change

10% 10 8 1%

10 10 10 7 10 5 10 3 10 1 101

| 0PN|/M Figure 2.13: Change in the size of the c.m. correction, ~µ as deﬁned in Eq. (2.8), when removing epicycles before ﬁtting for the c.m. correction, as described in Eq. (2.27). These changes are comparable to, but almost always smaller than, the changes due to variations in the end points of integration as seen in Figure 2.3; they are also smaller than most of the Post-Newtonian corrections seen in Figure 2.7, except for systems changing here by more than about 10%.

Figure 2.12 shows ~c1 [Eq. (2.26)] and the projections of ~c1 onto the nˆ, λˆ unit vectors. These panels show very similar behavior between resolutions of the same simulation, implying that the cause of the size of the epicycles is not random, from initial conditions, or the junk radiation phase, and that the randomness of the initial kick has been completely removed by the correction applied in Eq. (2.26).

Applying this method to the BBH simulations in the SXS public simulation catalog, α~ epi and ~βepi c.m. correction values can be calculated. Figure 2.13 compares the usual “0PN” c.m. correction with the epicycle-removed values. The values plotted involve ~µ, defined in Eq. (2.8), which is the largest displacement between the origin of coordinates in the simulation and the corrected origin. It was seen that epicycle removal changes the c.m. correction values at a scale comparable to the changes caused by varying the end points of integration used to determine the c.m., as seen in Figure 2.3. The changes due to epicycle removal are generally somewhat smaller than changes due to varying end points. If the epicycles are removed before applying those variations, the changes seen in Figure 2.3 are reduced by a typical factor of two—though there is no apparent effect on roughly 10% of systems. Systems that change by more than 10% in this figure are also typically changing by more than the Post-Newtonian changes shown in Figure 2.7.

Using the method outlined in Sec. 2.4, it was also found that it cannot be reliably concluded that the epicycle correction actually makes a signiﬁcant improvement in the waveforms. The results of the Υ comparisons between the original c.m. correction method outlined in Sec. 2.3 and the epicycle removal method in this section can be found in Figure 2.14. This plot shows that approximately 30% of simulations improve using the epicycle method, and approximately 70% get worse. This is not enough of a beneﬁt to warrant the use of the epicycle removal step in all simulations, and implies that the epicycle removal, at this stage of BBH simulations, is an unnecessary step in calculating and applying the c.m. correction, a somewhat disappointing conclusion.

32 Chapter 2. Centre of mass corrections 33

100 634 negative 10 2 1383 positive

4 N 10 P 0 /

) 6 i

p 10 e

10 8 N P 0

( 10 10

10 12

10 14 10 8 10 6 10 4 10 2 100

| raw 0PN|/ raw Figure 2.14: This plot shows the diﬀerence between the value of Υ [Eq. (2.17)] resulting from the naive 0PN method based on the coordinate trajectories of the apparent horizons and the value resulting from the epicycle removal method described in Sec. 2.5.4. The horizontal axis shows the relative magnitude of the change when going from the raw data to the corrected waveform, while the vertical axis shows the change when incorporating the epicycle corrections. In most cases, the values of Υ actually become signiﬁcantly larger when going from the 0PN value to the epicycle corrected value. Those systems are shown as crosses, while systems with smaller values are shown as circles.

As mentioned in Sec. 2.3.2, epicycles are also a potential source of instability regarding choice of beginning and ending times ti, t f . Initial investigation into how the epicycle removal method affects changes in the c.m. correction values due to differing ti, t f implies that the epicycle removal method does not diminish changes in the c.m. correction values. This is also an unintuitive and disappointing result, and may imply that other methods are required to calculate the c.m. correction values α~ and ~β after epicycle removal or a different method for epicycle removal entirely. Such an investigation is left to future work.

2.5.5 Position of the c.m.

During the analysis of the epicycles present in the c.m., the position of the c.m. relative to the two black holes was investigated. It was assumed, that ~xc.m. would lie along the separation vector between the two black holes or completely in the rotation direction as given in Eq. (2.18). As seen in Figure 2.15, this is not the case. The c.m. deviates signiﬁcantly between the rotation vector and the separation vector between the two black holes but typically lies in the negative rotation direction −λˆ, as predicted by Eq. (2.18) and the analysis in Ref. [133]. The projection in the ±λˆ direction, ∆λ, averages at −0.44rmeasured when considering all simulations in the SXS Catalog.

On average, the projection of ~c1 into the ±nˆ direction, ∆n, is smaller than ∆λ, with an average ratio ∆λ/∆n for spin aligned systems of −1.43 and −2.48 for precessing systems. The projection in the ±kˆ direction, ∆k, is signiﬁcantly 3 smaller than ∆λ, with an average ratio ∆λ/∆k of −3.22 × 10 for spin aligned systems and −2.78 for precessing systems.

Having typically most but not all of the corrected c.m. vector ~c1 in the direction of −λˆ does not have an obvious cause. This behavior could possibly indicate unaccounted spin-orbit eﬀects on the c.m., unknown eﬀects from

33 Chapter 2. Centre of mass corrections 34

+0.35 0.00 0.35

+0.40 0.40 0.40 1.0 p 0.5

+1.63 2.00 0.99 10

q 5

+0.02 0.00 0.02

0.05 n

0.00

+0.01 0.02 0.01

0.05

0.00

+5.55 5 (0.00 7.64) × 10

0.0005 k

0.0000

0 1 5 10 0.5 1.0 0.00 0.05 0.00 0.05 0.0000 0.0005 eff p q n k

Figure 2.15: Comparisons of effective spin χeff as defined in Eq. (2.9), effective precessing spin χp as defined in Eq. (2.10), mass ratio q, and the time averaged projections of ~c1 onto nˆ, λˆ, and kˆ ; ∆n, ∆λ, ∆k. Red represents spin aligned simulations, and teal represents precessing simulations. The numbers above each column represent the median of each variable over all simulations, with superscripts and subscripts giving the offset (relative to the median) of the 84th and 16th percentiles, respectively. unresolved junk radiation, or additional gauge effects that cannot be compensated for with BMS transformations.

Attributes of ~c1 warrant further investigation, and are left to future work.

Correlations between ∆n, ∆λ, and ∆k with pertinent simulation parameters are shown in Figure 2.15. This plot shows the correlations between ∆n, ∆λ, ∆k, χeff, χp, and q. A few notable correlations are apparent, the most obvious being the correlation between ∆n and ∆λ. Most simulations tend to have negative ∆λ values that grow larger in magnitude with increasing ∆n, however there is also a cluster of aligned-spin simulations with ∆n, ∆λ values close to zero. Additionally, ∆λ becomes more negative with increasing q, for q < 5, and there are some weak correlations between both c.m. position offsets ∆n, ∆λ with χeff, but not with χp. ∆k does not appear to have any strong correlations. It is apparent that spin aligned simulations tend to have ∆k values which are much smaller than for precessing simulations. One possible explanation for the c.m. to move out of the orbital plane is momentum flow between the gravitational fields and black holes [138], however further analysis regarding this mechanism

34 Chapter 2. Centre of mass corrections 35

is left to future work. It can also be seen in Figure 2.15 that larger ∆k values cluster around ∆n = 0 and ∆λ < 0, which may only be due to ∆n values being symmetric around 0 and ∆λ values being mostly negative. No apparent correlations are present for other simulation parameters.

2.6 Conclusions

In this work, the effects of c.m. motion waveforms have been investigated, unphysical c.m. motion has been removed through allowed gauge transformations to the waveforms, and methods for improving the c.m. correction have been investigated. Having unphysical motion in the c.m. causes mode mixing in the gravitational waveforms, and thus a power loss from the dominant (2, ±2) modes to the less-dominant, higher-order modes—which is typically visible as amplitude modulation in the higher-order modes. It was found that the c.m. motion observed in the SXS simulations cannot be entirely accounted for by PN corrections or linear-momentum recoil. It was also found that the motion of the c.m. does not lie along any one basis vector describing the rotating coordinate frame as defined in Eq. (2.25), and is offset from the estimated c.m. within and out of the orbital plane—which is not expected on physical grounds. The current method for correcting the c.m. motion uses allowed BMS transformations, namely a spatial translation and boost that counteracts the linear motion from the c.m. and removes a large amount of the mode mixing from the waveform. The translation and boost are calculated for all simulations at all resolutions, as the c.m. motion is not consistent between different resolutions of the same system. We attempted to improve the c.m. correction by developing a method to remove the large epicycles from the c.m. motion before calculating the BMS translation and boost. We found that the resulting changes to the translation and boost values were not significant and did not improve the waveforms compared to the originally calculated values. Last, a complementary method to quantify the effect of the c.m. correction on the waveforms was introduced. This method was used to determine that PN corrections and the epicycle removal technique did not improve the c.m. correction transformations, and thus would not further improve the waveforms or accurately describe the c.m. physically. Future work includes investigating spin-orbit effects on the c.m. and the peculiarity of the c.m. position. Further investigation is required specifically on the unaccounted for size of the radius of the epicycles seen in the c.m. motion, which may be due to unknown spin-orbit or unresolved junk radiation effects, and may be corrected with additional gauge transformations that minimize the epicycles.

35 Chapter 3

Fundamental frequency analysis for precessing systems

This chapter presents work regarding extracting fundamental frequencies from numerical relativity simulations and comparing ratios of these to perturbation theories and analytic results. This project is in collaboration with Aaron Zimmerman at the University of Texas at Austin and is still ongoing. The status of the project is presented here in full. A manuscript of this work is in preparation and planned to be submitted for publication in fall 2020.

3.1 Motivation

As mentioned in Chapter 1, it is important to consider gauge-independent quantities when comparing relativistic methods and simulations. In the case of numerical relativity, investigations across simulations from the same code, diﬀerent codes, and from semi-analytic methods are meaningful when done with gauge-independent quantities. One potential set of gauge-independent quantities are ratios of the fundamental frequencies, which can be extracted from the trajectory and parameter data of numerical relativity simulations. Frequency ratios are particularly good for analysis as they are reasonably simple to extract and understand from simulations and calculations, have meaningful physical implications regarding the system, and are relatively gauge-independent. When considering a two-body gravitational system in which only one body is rotating, there are three frequencies to consider. The most obvious is the orbital frequency, Ωφ, related to successive azimuthal passages. Second is the radial frequency, Ωr, related to successive radial passages. Lastly is the inclination frequency, Ωθ, related to successive polar passages, which can be thought of as a “vertical” motion. The directions of r, φ, θ are showcased for a general BBH in Fig. 3.1. For systems in which only the primary black hole is spinning, the system may be simulated in numerical relativity by setting the spin of the smaller black hole to zero. In general BBHs, there are additional frequencies to consider due to spin-orbit and spin-spin dynamics. Here, we only consider the case of the

36 Chapter 3. Fundamental frequency analysis for precessing systems 37

Figure 3.1: Figure of a general BBH and the various coordinate systems used to describe its evolution. xˆ, y,ˆ zˆ are the Cartesian NR coordinates, in which the motion of the apparent horizons is computed and contains the raw trajectory data centered on the Newtonian centre of mass (see Ch. 2). The co-rotating coordinate system nˆ, kˆ, λˆ (see Eq. 2.25) is also centred on the Newtonian centre of mass, and describes the direction of the separation vector ~r from the primary black hole m1 to the secondary black hole m2 with nˆ, the direction of the instantaneous angular momentum with kˆ, and the direction of rotation with λˆ. Lastly, the coordinates most naturally associated with the fundamental frequencies are r, θ, φ, where r is the direction of the separation vector ~r, θ is the inclination of the orbital plane, and φ is the motion in the azimuthal direction. θ can be taken as the “vertical” motion of nˆ, or the direction of the wobble of the instantaneous orbital plane, and φ represents the motion of λˆ, in this case, as the co-rotating frame moves counter-clockwise.

primary black hole spinning.

In general, the fundamental frequencies Ωφ, Ωθ, Ωr need to be averaged over many thousands of cycles. This is due to any measurable quantity in the system being modulated by at least one, and likely all three of these frequencies. If Ωφ, Ωθ, Ωr are not in a rational ratio with one another, the quantity in question will never repeat and there will never be a “perfect” cycle to average over. Taking an average of such a quantity over many cycles will minimize the inﬂuence of the sub-dominant frequencies and ensures a measurement of the driving frequency for that particular quantity. Unfortunately, dissipation from gravitational radiation and the limits of NR simulations prevent averages being taken over large numbers of cycles. However, there are assumptions and restrictions that may be placed on a system to allow simpler approximations of the fundamental frequencies to be used.

For circular, non-precessing BBHs, the only relevant frequency is Ωφ. When a system is eccentric, Ωr becomes relevant, and in particular the ratio of the two frequencies describes periastron advance. For a two-body system experiencing periastron advance, the orbit completes an azimuthal passage before a radial passage, or vice versa. In the scenario showcased in Fig. 3.2, the orbit comes to the same azimuthal point in the orbit before it returns to its periastron. The diﬀerence in time between the azimuthal and radial passage is Tr − Tφ, where Tr, Tφ are the radial

37 Chapter 3. Fundamental frequency analysis for precessing systems 38

Figure 3.2: Two successive orbits of a binary undergoing periastron advance shown in polar coordinates (r, φ). The ﬁrst orbit is blue and grey with a blue square indicating the “start”, and the second orbit is grey and orange with an orange triangle indicating the “end”. The red circles indicate where the orbits reach the apocentre, or the beginning and end of one complete radial cycle with period Tr. The black dashed lines from the primary black hole to the apocentre in each orbit showcase where the orbits complete one azimuthal cycle with period Tφ. The angular diﬀerence between these cycles is designated ∆Φ = (Tr − Tφ)Ωφ.

and orbital periods, respectively. The angular diﬀerence is then ∆Φ = (Tr − Tφ)Ωφ. From here,

! ! 2π 2π Ωφ ∆Φ = (Tr − Tφ)Ωφ = − Ωφ = 2π − 1 Ωr Ωφ Ωr ∆Φ → = Kφr − 1, (3.1) 2π where Kφr = Ωφ is a measure of the perisastron advance. Ωr

While the fundamental frequencies themselves are gauge dependent, their ratios are pseudo-gauge-independent, or rather are resiliant to a larger set of gauge transformations than the fundamental frequencies. The ratios of the fundamental frequencies may be considered gauge-invariant in a restricted set of coordinates [139], and may be compared across simulations, methods, and observations [23, 53, 140–142]. This is easily seen by applying a 0 common gauge transformation, such as a time rescaling, to the fundamental frequencies. For example, Ωa = AΩa where a = (r, θ, φ). Then any of the ratios would be transformed as:

0 0ab Ωa AΩa Ωa ab K = 0 = = ≡ K . (3.2) Ωb AΩb Ωb

38 Chapter 3. Fundamental frequency analysis for precessing systems 39

Just as we can define the periastron advance with a ratio, we can likewise create the ratios Krθ = Ωr , Kθφ = Ωθ . Ωθ Ωφ All ratios may be used to describe all types of orbits. Krθ describes the relationship between eccentricity and inclination, and Kθφ describes how inclination and azimuthal frequency relate, and more specifically the precession of the orbital plane. Some of these ratios can be extracted from gravitational wave data, but more pertinently, if it is a small integer ratio, the orbit is in a resonance [142]. If this is the case, analyzing the moments that BBHs go through resonance should coincide with when the system undergoes otherwise unpredictable disturbances, such as velocity kicks [143, 144]. For this work, we consider the fundamental frequency ratio Kθφ. We consider the uninclined, equatorial limit, where the orbit is parametrized by a single variable such as the orbital frequency. This enables an unambiguous comparison to analytic approximations, without having to focus on the specific method for measuring inclination. As such, the systems considered for this work have small inclination angles, defined as the angle between the normal of the instantaneous orbital plane and the primary spin vector ~χ1.

3.2 Previous work

Much work has been done investigating the periastron advance of eccentric, non-precessing systems [53, 140, 141], and of characterizing the fundamental frequencies and related ratios in an analytic and numerical manner [23, 142]. In the most recent work focusing on periastron advance [53], a model for how to investigate and compare fundamental frequency ratios Kab was established. The goals of Ref. [53] were to extract the periastron advance Krφ from numerical relativity simulations and compare this value to the PN and SF values for the same set of input parameters, as well as for the case of a test mass in a Kerr background spacetime, or rather a Kerr geodesic. This comparison of numerical results to analytic rφ and semi-analytic results was done by investigating the relative difference between KNR and each analytic and rφ rφ rφ semi-analytic method, KPN, KSF, Kgeo. The method of extraction in Ref. [53] allowed for error analysis on the relative difference for varied extraction parameters. The results of this analysis showed that for a wide range of mass ratios and spins, the periastron advance extracted from numerical relativity simulations agrees to within a few percent with the same value computed from 3.5 order PN and first order SF approximations. In addition, it was found that resummed SF periastron advance was closer to the numerical relativity values than the 3.5 order PN values, suggesting that resummed SF approximations, at least for periastron advance, are more reliable than high order PN approximations. Here, resummed SF expressions are those expanded using the symmetric mass ratio, as opposed to the traditional expansion in mass ratio q = m1 . m2 The work presented in this chapter is an extension to the analysis and methods presented in Ref. [142]. In Ref. [142], the focus resided on establishing methods for extracting the fundamental frequencies from numerical data reliably and comparing them with first order SF approximations, in particular for Kφr and Krθ.

In this chapter, an analysis of extraction methods for the inclination frequency Ωθ is showcased and tested. Comparisons of the precession of the orbital plane, Kθφ is made across numerical relativity and Kerr geodesics.

39 Chapter 3. Fundamental frequency analysis for precessing systems 40

0.015

0.010

0.005

s 0.000 o c

0.005

0.010

0.015

0 1000 2000 3000 4000 5000 6000 7000

t r * Times ( M )

Figure 3.3: cos θ calculated with Eq. 3.3 from the highest resolution (Lev3) of a BBH with mass ratio q = 8 and anti-aligned primary spin |~χ1| = −0.9. The horizontal axis shows retarded time, measured in M = m1 + m2.

We assume that the inspiral is slow and that the adiabatic assumption holds, or rather that energy loss due to gravitational radiation can be ignored (i.e. TRR >> Torb). We further assume circularized systems with eccentricity so small that the ratio Kθφ is the dominant aﬀect and other ratios of fundamental frequencies can be ignored.

3.3 Methods and Meaning

For this analysis, only the trajectory and spin data as a function of time were needed. The 3D trajectory data for each black hole is ~x1(t), ~x2(t), and the 3D dimensionless spin for each black hole is ~χ1(t), ~χ2(t), where the subscripts 1, 2 designate the primary and secondary black hole respectively. The inclination frequency is calculated in the precessing frame and averaged out over subsets in time. Taking a time average of the inclination frequency is done to mitigate short-term eﬀects, such as oscillations from residual eccentricity (i.e. Ωr) and other frequencies that may be relevant to the dynamics of the system on short time scales. The precessing frame is considered here to be the frame in which the spin of the primary BH is held ﬁxed. When only the primary black hole is spinning, i.e. ~χ2 = 0, the angle θ between the spin of the large black hole and the orbital plane is given by ~χ · ~r cos θ = 1 . (3.3) |~χ1||~r| θ will oscillate between maximum and minimum values during an orbit, and the frequency of this oscillation is the inclination frequency Ωθ. An example of θ as given by Eq. 3.3 is shown in Fig. 3.3.

The simplest method for extracting Ωθ from Eq. 3.3 is a peak to peak ﬁtting strategy, where the locations of the

40 Chapter 3. Fundamental frequency analysis for precessing systems 41 maximums and minimums are located in time and then the averaged frequency for each half cycle is determined by

ti,ti+1 π Ωθ = , (3.4) ti+1 − ti where i represents the number of maximums and minimums. The frequency calculated from Eq. 3.4 is only 1 considered for the midpoint of the time bounds, and so is assigned to the time point 2 (ti + ti+1).

With trajectory data available for all time during a numerical relativity simulation, the orbital frequency can be calculated for each time step using the 0PN, or Newtonian, expression:

~r(t) × ~v(t) Ω(t) = , (3.5) r2(t)

˙ where ~r = ~x2 − ~x1 is the separation vector between the black holes, ~v = ~r is the diﬀerence in velocities of the black holes, and r = |~r|. In this way, the orbital frequency of the BBH is known at every time that the trajectories are known. For the simulations considered here, time is measured in total BBH mass M = m1 + m2 and increases monotonically in steps of 0.5M. In general, the orbital frequency as given in Eq. 3.5 is not equivalent to the azimuthal frequency Ωφ. Ωφ would be calculated by projecting the position vector of the secondary BH into an inertial reference frame, say the x-y or initial orbital plane, and averaging this motion. In the approximation of circularized, nearly equatorial orbits, Ωφ may be approximated as a time average of Ω over some integer or half integer of gravitational wave cycles and translated into the precessing frame. This translation is done by subtracting the spin precession frequency from the orbital frequency at each time step and then taking a time average to obtain

Ωφ, as seen in Eq. 3.8 below.

The magnitude of the spin precession vector for the primary black hole Ω~ 1 is given by

dS~ a = Ω~ × S~ , ω ≡ |Ω~ |. (3.6) dt a a p 1

To 3.5PN order, the precession vector for the primary black hole is given by Eq. 394 in Ref. [145], which reads:

" !# 3 1 3 9 5 1 9 5 Ω~ = x5/2kˆ + ν − ∆ + x + ν − ν2 + ∆ − + ν 1 4 2 4 16 4 24 16 8 " !#! (3.7) 27 3 105 1 27 39 5 + x2 + ν − ν2 − ν3 + ∆ − + ν − ν2 32 16 32 48 32 8 32

2/3 where x = (MΩφ) is the PN ordering parameter, kˆ is the rotating frame unit vector in the direction of rotation

m1m2 m1−m2 deﬁned in Eq. 2.25 and shown in Fig. 3.1, ν = M2 is the symmetric mass ratio, ∆ = M , and maintaining the G = c = M = 1 convention. Note that in this convention, the units of M are time, or equivalently, length.

The azimuthal frequency Ωφ can then be approximated by averaging Ω − Ω1 over the speciﬁed time interval given by Eq. 3.4. If the width of the window in question is changed from a half cycle, the resulting Ωθ and Ωφ may

41 Chapter 3. Fundamental frequency analysis for precessing systems 42 be calculated with nπ ti,ti+n ti,ti+n Ωθ = , Ωφ = hΩ − Ω1i , (3.8) ti+n − ti where n ∈ N and represents the number of half cycles to include in the window. To improve the precision of extracting the times for Eqs. 3.4, 3.8, a cubic spline is applied to the curve of cos θ near the maximum or minimum. The time derivative of the cubic spline is computed and the time at which the derivative is zero is taken as the maximum or minimum point on the cubic spline. Once the inclination and orbital θφ frequencies at each midpoint have been found, their ratio may be computed to give KNR.

3.3.1 Testing the numerical relativity methods

The peak to peak method was tested using a series of models that mimicked the evolution of cos θ, including scale, length, and increasing frequency. Toy models were made knowing that the frequencies that needed to be extracted would sweep upward with time, which would introduce averaging bias. These tests were to measure how large the bias in the peak-to-peak method is for various toy models with similar frequency proﬁles as Ωθ.

The toy models were constructed by making a frequency function that matched the expected shape of Ωθ and then feeding the time integrals of the frequencies into a sine function. It was found that frequencies of the form

!−m/n t f − t Ωtoy = Ω0 (3.9) t f − t0 were the most similar, with t0, t f marking the time bounds of the function, Ω0 representing the frequency at t = t0, and m, n as integers. For the toy model shown in Fig. 3.4, m/n = 1/2, Ω0 = 0.01745, t0 = 0, t f = 5600. The peak to peak method performed well, and even for extreme models with varying frequency, was within a percent of the true frequency value. Only for the last few points just before the end of the model data did its accuracy diverge, and it is well established that no extraction methods can hold up when the change in frequency accelerates significantly, similar to when methods fail for a simulation that is close to merger due to the significant deviation from the adiabatic approximation [53,142]. An example of one of the stress tests performed on the peak-to-peak method is shown in Fig. 3.4. Other methods besides peak-to-peak averaging were tested, including using the Hilbert transform or integrating to find the inclination frequency at all times. The Hilbert transform assumes that an analytic signal can be written as iΦ(t) 1 R ∞ s(τ) sA = s(t)+isˆ(t) = A(t)e , where sˆ(t) = H(s(t)) = π −∞ t−τ dτ. In this case, the original signal is s(t) = cos θ from Eq. 3.3. The instantaneous amplitudes A(t) can be found from the envelopes of cos θ for each simulation and divided out. Finding the polar phase Φ(t) in this way lead to significant agreement with that found from the peak-to-peak θφ method, but led to oscillations in the KNR that are small for early times but quickly become on the order of 10%, which may be due to the ratio then being defined at every time step and not averaged as the original definition of the fundamental frequency ratios assume. Integrating to find the inclination frequency at all times assumes that the vertical motion due to inclination, i.e. what is given in Eq. 3.3, can be represented as cos(θ) = A(t) sin(Φ(t)), 1 dΦ(t) where then 2π dt is the inclination frequency at every time step, or can be interpolated as a continuous function.

42 Chapter 3. Fundamental frequency analysis for precessing systems 43

0.015

0.010 ) )

t 0.005 ( y o t , 0.000 ( s o

c 0.005 A

0.010

0.015

0 1000 2000 3000 4000 5000

10 2 y o t , / | p p , 10 3 y o t , | 10 4

0 1000 2000 3000 4000 5000

t r * Time ( M )

Figure 3.4: The top panel shows a toy model used for stress testing the peak to peak method for extracting the inclination frequency Ωθ, constructed using the toy frequency described in Eq. 3.9. The parameters used for this model are m/n = 1/2, Ω0 = 0.01745, t0 = 0, t f = 5600. The bottom panel shows how the frequency estimated by the peak to peak method (Ωθ,pp) compares with the true frequency (Ωθ,toy) via relative diﬀerence.

43 Chapter 3. Fundamental frequency analysis for precessing systems 44

Therefore, since cos θ is known at every time step, one could theoretically extract Φ(t) by taking the arcsin of cos θ A(t) and then compute the derivative to find the instantaneous inclination frequency. While possible, this method emphasized the difficulty of working with inverse trigonometric functions for very small oscillating quantities. The results proved to be mostly similar to those found with the peak-to-peak averaging method, but were not necessarily monotonically increasing or smooth, which are required properties for this analysis. A similar integration method for extracting the inclination frequency was presented in Ref. [142], from the θ definition cos θ = cos θmin cos χ , where θmin = π/2 − ι is the smallest polar angle the secondary BH reaches at the height of its vertical motion and χθ is the polar phase, whose time derivative should give rise to the instantaneous inclination frequency. This method likewise falls victim to the sensitivity of the inverse trigonometric functions, and computes a polar phase that is neither smooth nor monotonically increasing. Despite the promise and witnessed effectiveness of these methods in other works, the changing frequency and amplitude of our numerical data proved to highlight their weaknesses and they were ultimately less reliable than the simple peak-to-peak windowing method.

3.3.2 Ratios from Analytic and semi-Analytic methods

In this work, we compare the fundamental frequency ratio Kθφ calculated from numerical relativity, represented as θφ KNR, to that from analytic methods. We calculate the precession of the orbital plane as described by Kerr geodesics, θφ represented as Kgeo. In the Kerr geodesic case, the assumption is that the primary, spinning black hole creates the Kerr spacetime and the secondary black hole can be approximated by a Kerr geodesic with SF corrections. In this picture, Kθφ is related to polar motion, and is more accurate for large mass ratios. It may also be assumed in this formalism that θφ θφ KNR, Kgeo are related via the SF approximation. θφ The computation of Kgeo is largely based on the discussion in Ref. [23], and more speciﬁcally on the expressions for the fundamental frequencies found in Eqs. 33, 34, 35 in Section 3. In short, the fundamental frequencies may be calculated by taking the partial derivatives of the action-angle Hamiltonian with respect to the action variables, which are in turn related to each spatial coordinate and conjugate momenta. This formalism assumes that the orbit of the black hole and test mass is bound and non-plunging. The motivation for these assumptions is that the time scale of the radiation reaction for the BBH is much larger than the radial, orbital, or inclination periods. A more in-depth presentation of how to calculate the fundamental frequencies for Kerr geodesics is given in AppendixE.

3.4 Data and Analysis

The data considered for this analysis was computed using SpEC [69] on SciNET computing clusters [146] in Ontario, Canada, and the Caltech HPC center [147] in California, USA. A detailed list of the simulations used is presented in Table 3.1. We considered multi-resolution simulations with mass ratios q = 1.5, 3, 5, 8 and primary spins of magnitude |~χ1| = 0.5, 0.9, which are either mostly aligned or mostly anti-aligned with the orbital angular

44 Chapter 3. Fundamental frequency analysis for precessing systems 45 momentum. The spin is slightly tilted away from the angular momentum vector to create inclination of the orbital plane. It should be noted that for high spin, high mass ratio combinations, there are fewer resolutions available as these combinations push the current limits of SpEC (see Appendix A). I computed simulations with spins

|~χ1| = −0.9, −0.5, 0.9 using the same instance of SpEC on the Wheeler computing cluster at Caltech. Simulations with |~χ1| = 0.5 were completed using an earlier instance of SpEC using the SciNET computing clusters by Aaron Zimmerman.

−2 −4 q ~χ1 Ω0(10 ) ι e(10 ) N Lev 1.5 0.9 1.5676948862 1◦ 1.552 21 1, 2, 3 3 0.9 1.8663239213 1◦ 0.947 21 1, 2, 3 5 0.9 2.2526843521 1◦ 2.99 21 2, 3 8 0.9 2.3894123285 1◦ -- 2, 3 1.5 −0.9 1.5619960099 1◦ 6.088 14 1, 2, 3 3 −0.9 1.5764399654 1◦ 2.175 15 1, 2, 3 5 −0.9 1.5839273511 1◦ 2.179 19 1, 2, 3 8 −0.9 1.5884031126 1◦ 1.233 25 1, 2, 3 1.5 0.5 1.7367158274 1◦ 1.331 17 1, 2, 3 1.5 0.5 1.7367158274 0.5◦ 1.377 17 1, 2, 3 3 0.5 2.1508210732 1◦ 2.783 14 1, 2, 3 3 0.5 2.1508210732 0.5◦ 3.034 14 1, 2, 3 5 0.5 1.920117 1◦ 6.061 24 2, 3 5 0.5 1.920117 0.5◦ 6.242 24 2, 3 8 0.5 1.9193267833 1◦ 1.745 33 1, 2, 3 8 0.5 1.19194466850 0.5◦ 1.906 33 2, 3 1.5 −0.5 1.5323756041 1◦ 0.902 19 1, 2, 3 3 −0.5 1.7547328706 1◦ < 0.25 15 1, 2, 3 5 −0.5 1.91428504 1◦ 1.632 14 1, 2, 3 8 −0.5 1.9107345274 1◦ 1.339 18 1, 2, 3

~ 2 Table 3.1: Simulation parameters , including mass ratio q = m1/m2, dimensionless spin ~χ1 = S 1/m1 of the larger black hole, initial orbital frequency Ω0, initial inclination ι, estimated eccentricity e, number of completed orbits before merger N, and the available resolution levels numbered 1, 2, 3 from lowest resolution to highest. N and e information are taken from the highest resolution available.

For the simulations used in this work and showcased in Table 3.1, the runs were intentionally set up to have approximately 20 orbits and low eccentricity. Runs with positive primary black hole spins have the spin vector mostly pointing in the direction of the orbital angular momentum, and runs with negative primary black hole spins

45 Chapter 3. Fundamental frequency analysis for precessing systems 46 have the spin vector mostly pointing opposite to the orbital angular momentum vector. To introduce inclination, the cosine of the inclination angle was multiplied by the total spin value and set as the z component of the spin. The sine of the inclination angle was likewise multiplied by the total spin value and set as the x component of the spin, referencing the unit vectors speciﬁed in Fig. 3.1. In this way, the spin vector of the primary black hole would be mostly perpendicular to the orbital plane but inclined by the degree value of the inclination.

3.4.1 Results from numerical relativity

θφ The results of KNR are shown in Fig. 3.5. The ﬁrst row shows data for all mass ratios for the anti-aligned spin runs, θφ and conﬁrms that the inclination frequency is larger than the orbital frequency as KNR > 1. Physically, this implies that the direction of precession is counter-rotating with the direction of the orbits. The bottom row shows data for all the mass ratios for the aligned spin runs. In particular, all resolutions are shown for each set of simulation parameters, and there is strong agreement between resolutions for each simulation. The space between the curves θφ decreases with increasing mass ratio, and it appears that KNR is approaching the test-mass limit [142], such that " !# 1 1 Kab (m Ωb) = Kab (m Ωb) 1 + ∆Kab + O . (3.10) NR 1 geo 1 q SF q2

This relation is known to hold for periastron advance, or rather for Krφ, and for Krθ [142], but it is not conﬁrmed if this relation holds for precession of the orbital plane.

m1m2 In the SF literature, one can scale with mass ratio q or with the symmetric mass ratio ν = M2 , leading to different expansion frequency scalings using m1 and M respectively [139, 148, 149]. For brevity, we choose to display most of our results using only the q, m1 scaling. θφ Following the procedure from [53], we used a windowing technique to calculate and refine KNR. Here, θφ “windowing” means using more than one set number of cycles to calculate KNR. Using the values found at θφ θφ the same frequencies but different windows, we can estimate the instantaneous KNR, or rather, KNR at a window size of zero.

As seen in Eq. 3.8, any estimate for Ωθ depends on the width ti+n − ti, where n is the window size in half-cycles. Ideally, instantaneous frequency estimates would be obtained by taking n → 0, however this is not possible to compute directly. Since taking n = 0 directly is not possible, we estimate the relevance of n in the frequency estimates by repeating θφ our analysis for different window sizes. Fig. 3.6 showcases the results of KNR using different windows for two systems. These two systems showcase the extremes in mass ratio and in spin magnitude, and as can be seen, the θφ calculated value for KNR is insensitive to changing window sizes at constant frequencies. θφ For the KNR used throughout this analysis, we use the estimated instantaneous values by extrapolating to n → 0. θφ We do so by taking the KNR calculated at n = 1, 2, 3, 4 and fit a polynomial in n using non-linear least squares. We θφ take the value of this polynomial at n = 0, or the zeroth order parameter, as the best estimate of the true KNR. The standard deviation, or 1σ variance, of the zero order parameter is the error associated with the fitting across windows.

46 Chapter 3. Fundamental frequency analysis for precessing systems 47

| 1| = 0.9 | 1| = 0.5 1.08 1.05

1.07 1.04 1.06

1.05 1.03 R

N 1.04 K 1.03 1.02

1.02 1.01 q=1.5, Lev1 1.01 q=1.5, Lev2 q=1.5, Lev3 1.00 1.00 q=3, Lev1 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 0.05 q=3, Lev2

| 1| = 0.9 | 1| = 0.5 q=3, Lev3 q=5, Lev1 1.00 1.00 q=5, Lev2 0.98 q=5, Lev3 0.99 q=8, Lev1 0.96 q=8, Lev2 0.98 q=8, Lev3 0.94 R

N 0.97

K 0.92

0.90 0.96

0.88 0.95

0.86 0.94

0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08

m1 m1

θφ Figure 3.5: Comparisons of KNR = Ωθ/Ωφ versus m1Ωφ for all available simulation data sets with inclination angle of 1◦.

The resulting error estimates are shown in Fig. 3.7, and are consistently smaller than 0.25%. The extrapolated, θφ instantaneous KNR used in the rest of this analysis are shown in Fig. 3.8, with error bounds for reference.

It is clear from Figs. 3.7 and 3.8 that for early times in the simulation, or rather low Ωφ, any window size would give a similar result. Closer to merger, there are larger deviations in the results from different values of n. θφ In Fig. 3.6, 3.7, 3.8, as well as for the KNR used throughout the rest of this analysis, there are points removed from the beginning of the simulation and one point removed from the end. The peak-to-peak method is not without flaws, and will interpret insignificant "wiggles" in motion from the junk-radiation phase of a simulation as real minimums and maximums to include. This leads to an artificially high reported Ωθ value at early times, which is removed accordingly. The last point is removed as this is where our adiabatic assumptions have long since failed and the BBH is entering merger. Additionally, we also know that the peak-to-peak method itself breaks down for the last point due to the quickly changing frequency and amplitude of cos(θ). The inner-most stable circular orbit (ISCO) frequency for Schwarzschild geodesics is shown in Fig. 3.9. The ISCO is a useful guide in regards to where we would expect the adiabatic approximation to fail, but does not

47 Chapter 3. Fundamental frequency analysis for precessing systems 48

0.994

0.993

0.992 R N m = 0.010452 K 0.991 1 m1 = 0.01214

0.990 m1 = 0.016877 m1 = 0.021185 0.989

0.988 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n

1.0425

1.0400

1.0375

R 1.0350 N m = 0.014128

K 1

1.0325 m1 = 0.015861

m1 = 0.020085 1.0300 m1 = 0.024009

1.0275

1.0250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n

θφ Figure 3.6: Each data point shown here is the KNR estimated using Eq. 3.8 with the number of half-cycles n along the horizontal axis. The different colours represent the sampled scaled frequency. The dashed lines connecting the θφ data points are linear polynomial fits, which are then used to estimate the n → 0 instantaneous KNR, shown here as black crosses. The top plot shows data for |~χ1| = 0.5, q = 1.5, and the bottom plot shows data for |~χ1| = −0.9, q = 8. θφ For all runs, window size affects KNR estimates at higher frequencies moreso than those at lower frequencies.

48 Chapter 3. Fundamental frequency analysis for precessing systems 49

| 1| = 0.9 | 1| = 0.5

10 1 10 1

2 10 2 ) 10 % (

1 3 , 10 R N K 10 3 10 4 q = 1.5 q = 3 5 10 10 4 q = 5 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 0.05 q = 8

| 1| = 0.9 | 1| = 0.5

1 10 1 10

2 ) 10 2 10 % (

1 , R

N 3 3 K 10 10

4 10 4 10

0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08

m1 m1

Figure 3.7: The windowing errors associated with KNR|n→0, represented as the 1σ variance translated to a percentage of the results. The error due to windowing is less than 1% for all runs for all frequencies considered in this analysis, θφ and is typically less than 0.25%. This implies that our peak-to-peak calculation for KNR is robust against changes in window size. These errors in KNR|n→0 are assumed in further analysis using this quantity.

necessarily indicate where the plunge begins in each simulation. The Schwarzschild ISCO is chosen here to show where the behaviour of Kθφ may change using a simple, familiar quantity. The scaled Schwarzschild ISCO frequency is given by [150] !3/2 q 6GM m , r ≡ M. 1ΩISCO,Sch = ISCO,Sch = 2 6 (3.11) (q + 1)rISCO,Sch c It should be noted that for this analysis and in SpEC in general, it is assumed that G = c = 1, and that length units are represented in M. In this scaling, rISCO,Sch = 6M, or simply rISCO,Sch = 6. Our results for KNR compared to the Schwarzschild ISCO frequency are shown across spin in Fig. 3.9 and across mass ratio in Fig. 3.10. In either split, it is clear that the data range we consider falls within the ISCO frequency for all anti-aligned spin runs, and a behaviour change can be seen in the spin aligned runs around where they pass the Schwarzschild ISCO. This change is a slight upward curve in the data.

49 Chapter 3. Fundamental frequency analysis for precessing systems 50

| 1| = 0.9 | 1| = 0.5 1.08 1.05

1.07 1.04 1.06

1.05 1.03 R

N 1.04 K 1.03 1.02

1.02 1.01 1.01

1.00 1.00 q=1.5 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 0.05 q=3 q=5 | 1| = 0.9 | 1| = 0.5 q=8 1.00 1.00

0.98 0.99 0.96 0.98 0.94 R

N 0.97

K 0.92

0.90 0.96

0.88 0.95 0.86 0.94

0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08

m1 m1

θφ Figure 3.8: This plot shows the KNR data for the highest resolution available for all simulations. The different colours represent different mass ratios, and the dotted lines showcase the error associated with each calculation. θφ These error bounds were found using windows of 0.5, 1.0, 1.5, and 2.0 cycles, or n = 1, 2, 3, 4 to calculate KNR θφ using Eq. 3.8 and fitting to find an estimate for KNR as the window size goes to zero.

It is expected that the Schwarzschild estimate for the ISCO frequency would get increasingly worse with increased spin magnitude in any alignment. Thus, the Kerr ISCO or ﬁrst order SF ISCO could be used instead. The Kerr ISCO radius and frequency is well known, and given by [151]

1 p m , r Z ∓ − Z Z Z , 1ΩISCO,Kerr = 3/2 ISCO,Kerr = 3 + 2 (3 1)(3 + 1 + 2 2) (3.12) (q+1)rISCO,Kerr q + χ

1/2 2 1/3 1/3 1/3 2 2 where Z1 = 1 + (1 − χ ) (1 + χ) + (1 − χ) , Z2 = 3χ + Z1 , and χ is the spin magnitude of the primary mass, in our notation |~χ1|. Setting χ → 0 will reduce Eq. 3.12 to Eq. 3.11, as expected. The ﬁrst order SF ISCO is given by [55] !! 1 1 MΩ = m Ω 1 + C (χ) + O , (3.13) ISCO,SF 1 ISCO,Kerr q Ω q2

50 Chapter 3. Fundamental frequency analysis for precessing systems 51

| 1| = 0.9 | 1| = 0.5 1.08 1.05

1.07 1.04 1.06

1.05 1.03 R

N 1.04 K 1.03 1.02

1.02 1.01 q=1.5, Lev1 1.01 q=1.5, Lev2 q=1.5, Lev3 1.00 1.00 q=3, Lev1 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 q=3, Lev2

| 1| = 0.9 | 1| = 0.5 q=3, Lev3 q=5, Lev1 1.00 1.00 q=5, Lev2 0.98 q=5, Lev3 0.99 q=8, Lev1 0.96 q=8, Lev2 0.98 q=8, Lev3 0.94 R

N 0.97

K 0.92

0.90 0.96

0.88 0.95

0.86 0.94

0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08

m1 m1

θφ Figure 3.9: The fundamental frequency ratio KNR versus the averaged azimuthal frequency Ωφ, scaled with the primary mass m1. Grouped by the primary spin ~χ1, and the data is coloured according to mass ratio q. Resolution is indicated by line style, with Lev1 being the lowest resolution. The coloured vertical lines represent the ISCO frequency calculated using the ﬁrst order Schwarzschild corrections as given in Eq. 3.11, with colours likewise representing q.

where CΩ(χ) are shifts in the ISCO frequency given in Table 1 of Isoyama et al. [55]. The Kerr and SF shifted ISCO frequencies occur significantly past merger for the majority of the simulations we consider here, and thus are not shown in the figures presented in this work. The Schwarzschild ISCO may not be a good choice for the simulations presented due to the relatively high spins, however as a tool for guiding the eye, it is sufficient for our purposes. θφ The inclination of the runs themselves does not appear to substantially change the values of KNR, as seen in Fig. 3.11. To see how valid our analysis would be for small inclination runs in general and not only for systems with inclinations of one degree, we compared the fundamental frequency ratio found in simulations with ~χ1 = 0.5, q = 3 with differing inclinations that were all below five degrees. For inclinations this small, it would be expected that the results would be similar. This is confirmed by the errors presented in Fig. 3.11, which are typically less than 0.15%. There is less agreement as the systems progress towards merger, with a modulation in that agreement. The results used for comparisons with Kerr geodesics and extracting the SF perturbations are done using the

51 Chapter 3. Fundamental frequency analysis for precessing systems 52

q = 1.5 q = 3 1.06 1.03 1.04 1.02 1.02 1.01 1.00 1.00 R N K 0.98 0.99 0.96 0.98 | 1| = 0.9, Lev1 0.94 0.97 | 1| = 0.9, Lev2 | | = 0.9, Lev3 0.92 1 0.96 | 1| = 0.5, Lev1 0.01 0.02 0.03 0.04 0.02 0.04 0.06 | 1| = 0.5, Lev2 q = 5 q = 8 | 1| = 0.5, Lev3 1.075 | 1| = 0.5, Lev1 | | = 0.5, Lev2 1.050 1.05 1 | 1| = 0.5, Lev3

1.025 | 1| = 0.9, Lev1 1.00 | 1| = 0.9, Lev2 1.000

R | 1| = 0.9, Lev3 N K 0.975 0.95

0.950 0.90 0.925

0.900 0.85 0.02 0.04 0.06 0.08 0.025 0.050 0.075 0.100 0.125

m1 m1

Figure 3.10: The same information is shown here as in Fig. 3.9, with the only diﬀerence being that the simulations are grouped via mass ratio q. In this plot, the lines are coloured according to their spin, with blue representing a primary spin of -0.9, magenta representing -0.5, red representing 0.5, and orange representing 0.9. The vertical grey line represents the scaled Schwarzschild ISCO frequency.

ι = 1◦ simulations of all available spins and mass ratios. If only one resolution is shown, the highest available resolution is used for that respective parameter set. In addition to the concerns of the numerical simulations, there are also issues inherent to SpEC and numerical relativity that may affect our analysis. In numerical relativity, the spin axis, or the direction of the spin, is undermined by gauge freedom, as discussed in Chapter 1. This causes the definition of the spin axis to be vague and ad-hoc in NR. Using different definitions for the spin axis will give different nutation features, or “wobbles” in the spin direction. The BH spin is imprinted onto gravitational waves, but less so than mass. However, it is a key near-term target for precision measurement for LIGO. While the spin axis is not gauge-invariant, the magnitude of the spin is. Since the direction of the primary spin is used for calculating one of the fundamental frequencies and the inclinations we consider here are small, we must consider the definition of the spin axis in SpEC. As outlined and compared in Ref. [152], the definition of the spin axis in SpEC assumes that the Kerr geometry is a good approximation of the spacetime quasilocal to the apparent horizon of the BH, and that the axisymmetry

52 Chapter 3. Fundamental frequency analysis for precessing systems 53

0.25 Compared with = 0.25° e

c Compared with = 0.5° n

e 0.20

r Compared with = 1° e f f

i Compared with = 2° D

0.15 e v i t a l 0.10 e R

, 0.05 1 0 . 0 K / i

, 0.00 1 0 . 0 K 0.05

0.025 0.030 0.035 0.040 0.045 0.050

Figure 3.11: This plot showcases the relative error in the fundamental frequency ratio Kθφ of four inclinations compared to that found for the very small inclination ι = 0.01◦. The vertical axis shows the percent relative error and the horizontal axis is frequency scaled with the primary mass. Then inclination used for the systems considered throughout this analysis are those with inclinations ι = 1◦.

of the apparent horizon can be used to deﬁne the spin axis. The explicit calculation of the spin axis is through coordinate moment integrals of spin-related quantities on the apparent horizon of the BH. Speciﬁcally, SpEC uses the scalar quantity that represents the imaginary part of the complex curvature of a 2-surface embedding as the default measure of the spin axis and then multiplies it by the approximate Killing vector methods outlined in Ref. [153] to calculate the spin of each BH.

This method of defining the spin axis and overall spin vectors has several benefits, including being centroid invariant, i.e. changes in the coordinates of the apparent horizon do not affect the spin axis, and boost-gauge invariant, i.e. the spin axis is unchanged under slicing transformations that leave the apparent horizon 2-surface fixed. However, it has also demonstrated a surprising incompatibility with relatively high order PN estimates for the spin vectors due to large nutations [152, 154].

In our analysis, the most blatant consequence of large, unphysical nutations in the spin axis would be modulations θφ in Ωθ which would then carry through to KNR. As seen in Fig. 3.5 and demonstrated via error estimates from the windowing technique, any imprinting of nutations from the spin axis onto the quantities considered here play a minor role, if any. Additionally, specifically due to the windowing technique outlined in Eq. 3.8 defining windows as half-integers of cycles, the effect of the nutations on Ωθ are expected to average out due to their elliptical motion completing every cycle. Therefore, the nutation captured in the windowing method would appear more like PN nutations due to discrete sampling, and is best compared to Fig. 7 in Ref. [154]. Lastly, if our discrete sampling did not mitigate the affect of spin nutations, these affects would be averaged out over the time frame that each calculation of the fundamental frequencies are taken.

53 Chapter 3. Fundamental frequency analysis for precessing systems 54

With the aforementioned checks for applicability, reproducibility, and robustness, the comparison with Kerr geodesics, leading to estimates of the SF corrections to Kθφ, are continued below. While it is helpful to split the data according to spin and mass, we choose to represent the comparisons with Kerr geodesics and SF perturbation extractions via grouping by spin.

θφ Looking at any of the ﬁgures presented for KNR, say Fig. 3.5, some generic conclusions can be made about the trajectories of the black holes in each system. The most obvious one being that all spin-aligned systems have θφ θφ KNR < 1 and all spin anti-aligned systems have KNR > 1, which is expected from analytic theory (see Appendix E). This implies that spin-aligned systems have a slower inclination frequency than orbital frequency, or that the wobbling of the orbital plane is not as fast as the completion of an orbit. The reverse is true for spin anti-aligned θφ systems - the orbital plane wobbles faster than orbits are completed. Additionally, the values of KNR appear to be approaching a limit with higher mass ratios, or rather the lines are getting closer together with higher mass ratio. This implies, as expected, that the fundamental frequency ratio is approaching a SF limit with increasing mass ratio.

3.4.2 Comparisons with Kerr geodesics

Comparing the numerical fundamental frequency ratio to that estimated from analytic Kerr geodesics is the ﬁrst step in estimating the the SF perturbations. Following the assumption that the precession of the orbital plane follows the form given in Eq. 3.10, we assume that

" !# 1 1 1 Kθφ (χ, m Ω ) = Kθφ (χ, m Ω ) 1 + Kθφ (χ, m Ω ) + Kθφ (χ, m Ω ) + O , (3.14) NR 1 φ geo 1 φ q SF(q),1 1 φ q2 SF(q),2 1 φ q3 which may also be stated as

θφ θφ h θφ 2 θφ 3 i KNR(χ, MΩφ) = Kgeo(χ, MΩφ) 1 + νKSF(ν),1(χ, MΩφ) + ν KSF(ν),2(χ, MΩφ) + O(ν ) , (3.15)

θφ θφ KNR−Kgeo through a resummation. Rewriting either Eqs. 3.14 or Eq. 3.15 shows that the quantity ∆K/Kgeo = θφ is Kgeo θφ θφ essential in estimating either representation of the SF perturbations KSF,1, KSF,2.

The results of ∆K/Kgeo are shown in Fig. 3.12 for q, m1 scaling and Fig. 3.13 for ν, M scaling. The Kerr geodesic consistently over-estimates the ratio for all spin anti-aligned systems, and consistently underestimates the ratio for θφ all spin-aligned systems. The clustering of the curves with increasing mass ratio shows that KNR is approaching the test-mass limit. There is signiﬁcant agreement between the resolutions for each simulation, except for the last few points which are near merger and are expected to deviate [53, 142]. There also appears to be some small oscillations, which is expected given a similar result was found in Ref. [142] when considering ratios including Ωθ. This potentially hints at needing a diﬀerent method for extracting the inclination frequency.

54 Chapter 3. Fundamental frequency analysis for precessing systems 55

| 1| = 0.9 | 1| = 0.5

0.02 0.01

0.02 0.04 0.03 o e g

K 0.06 / 0.04

K q = 1.5, Lev1 0.08 q 0.05 q = 1.5, Lev2 q = 1.5, Lev3 0.10 0.06 q = 3, Lev1 0.07 q = 3, Lev2 0.12 q = 3, Lev3 0.08 0.01 0.02 0.03 0.04 0.01 0.02 0.03 0.04 0.05 q = 5, Lev1 q = 5, Lev2 | 1| = 0.9 | 1| = 0.5 q = 5, Lev3

0.30 0.10 q = 8, Lev1 q = 8, Lev2 0.25 q = 8, Lev3 0.08 o

e 0.20 g K / 0.06

K 0.15

q 0.04 0.10

0.05 0.02

0.00 0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08

m1 m1

θφ θφ θφ Figure 3.12: Comparisons of the quantity q∆K/Kgeo = q(KNR − Kgeo)/Kgeo versus m1Ωφ for all available simulation data sets with inclination angle of 1◦. The different colours describe different mass ratios, and different linestyles represent different resolutions. There is significant agreement between the resolutions, and as expected, the curves agree more with higher mass ratio.

3.4.3 SF ﬁtting and extraction

The primary focus for extracting SF information from the fundamental frequency comparisons is to utilize that all orders of SF corrections are inherently embedded in NR data. First order SF corrections are relatively well-known and can be calculated for non-spinning BBH systems and for periastron advance [15, 49–53, 55–57, 139, 155–160], but second order SF corrections are not yet as readily available [161]. This work aims to build the framework needed to compare and understand the SF perturbation coefficients for precession of the orbital plane, and specifically to make predictions for the second order SF correction so that it may be compared with theory when available. Additionally, we aim to test the limits of where SF perturbation theory is valid. SF is typically used at large mass ratios, and so testing it with relatively low mass ratios of q = 1.5, 3, 5, 8, as done here, pushes or goes beyond the limit of validity for SF. It has been demonstrated previously that the SF approximation was better suited for a wider range of the BBH parameter space than PN theory [53, 142], however only the first order SF corrections have been calculated thus

55 Chapter 3. Fundamental frequency analysis for precessing systems 56

| 1| = 0.9 | 1| = 0.5 0.04 0.075

0.100 0.06

0.125 0.08 o e g 0.150 K 0.10 / 0.175 q = 1.5, Lev1 K 0.12 0.200 q = 1.5, Lev2 q = 1.5, Lev3 0.14 0.225 q = 3, Lev1 q = 3, Lev2 0.250 0.16 q = 3, Lev3 0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05 0.06 q = 5, Lev1 q = 5, Lev2 | 1| = 0.9 | 1| = 0.5 q = 5, Lev3 0.7 0.25 q = 8, Lev3 q = 8, Lev3 0.6 0.20 q = 8, Lev3 0.5 o e g K 0.4 0.15 / K 0.3 0.10 0.2

0.1 0.05

0.025 0.050 0.075 0.100 0.125 0.02 0.04 0.06 0.08 M M

Figure 3.13: This figure showcases the same information as that in Fig. 3.12, but with frequencies scaled with total mass M and ∆K/Kgeo scaled with ν. This allows comparisons to the resumming of SF perturbations using the symmetric mass ratio as given in Eq. 3.15. Different colours represent different mass ratios and different linestyles represent different resolutions. As in Fig. 3.12, higher mass ratio curves lie on top of each other as expected and there is agreement between resolutions.

far for periastron advance. Gleaning information about higher order SF corrections, even if we do not know their explicit values and forms, prepares the ﬁeld for the eventual use of higher order SF terms.

In this analysis, we assume that the ratios of the fundamental frequencies have a SF expansion as shown in Eq. 3.14 and Eq. 3.15. We have used the comparisons of Kθφ from NR and Kerr geodesics to ﬁt for the ﬁrst and second order SF corrections to the precession of the orbital plane using both q and ν scalings.

Therefore, we have two diﬀerent scaling methods to consider for an overall linear ﬁt, namely

∆K θφ 1 θφ q = KSF(q),1 + KSF(q),2 (3.16) Kgeo q

∆K θφ θφ = KSF(ν),1 + νKSF(ν),2. νKgeo

The same fitting strategies were applied to the q and ν scaled fits. To demonstrate the fitting procedure, the q scaled

56 Chapter 3. Fundamental frequency analysis for precessing systems 57

| 1| = 0.9 | 1| = 0.5 0.012 0.0225

0.0250 0.014 0.0275 0.016 o

e 0.0300 g K / K 0.0325 0.018 ) q

( 0.0350 0.020 0.0375

m1 = 5% 0.0400 0.022 m1 = 15% 0.0425 m1 = 25% 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 m1 = 35%

| 1| = 0.9 | 1| = 0.5 m1 = 45%

m1 = 55% 0.0300 0.050 m1 = 65%

0.0275 m1 = 75% 0.045 0.0250 o e

g 0.040

K 0.0225 / K

) 0.035 0.0200 q (

0.030 0.0175

0.0150 0.025 0.0125

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 1/q 1/q

Figure 3.14: Values of the continuous function q∆K/Kgeo(m1Ωφ) versus 1/q, grouped by spin magnitude and alignment. The colouring of the lines represent the relative values of m1Ωφ as percentages of the maximum scaled frequency considered for each spin group, given as m1Ωφ|q=1.5,last in Table 3.2. The windowing error from the θφ embedded KNR calculation are propagated and shown here as error bars data is presented ﬁrst. All available simulations were used for these ﬁts.

Firstly, a cubic spline without smoothing was taken of q∆K/Kgeo as a function of mass scaled frequency m1Ωφ for all available resolutions in each simulation, with samples from the highest available resolution shown in

Fig. 3.14. This provides a smooth, continuous function in m1Ωφ for each combination of q = 1.5, 3, 5, 8 and spin

|~χ1| = −0.9, −0.5, 0.5, 0.9. The purpose of interpolating and creating a spline function for these ratios is to compare the functions at the same frequency ranges.

The next step in the ﬁtting procedure is to determine where the domains overlap for the q∆K/Kgeo(m1Ωφ) functions, grouped by spin. Looking at the original presentation of this data in Fig. 3.12, it is clear that in all cases, the lowest q system will end at smaller frequencies and the highest q system will start at larger frequencies. Therefore, a safe estimate for the overlapping domain for all systems grouped by spin would be [m1Ωφ|q=8,ﬁrst, m1Ωφ|q=1.5,last].

Using this domain for all q∆K/Kgeo(m1Ωφ) grouped by spin allows the functions to be compared at the same frequencies.

57 Chapter 3. Fundamental frequency analysis for precessing systems 58

|~χ1| m1Ωφ|q=8,ﬁrst – m1Ωφ|q=1.5,last MΩφ|q=8,ﬁrst – MΩφ|q=1.5,last -0.9 0.01435402279921071 - 0.024869811713241083 0.016148073798189572 - 0.04144968618873514 -0.5 0.017659645623293137 - 0.026840602023218574 0.019866852990542395 - 0.044734336705364294 0.5 0.01707499077218339 - 0.04073451170886867 0.019209124504650003 - 0.06789085284811446 0.9 0.021723486109720466 - 0.04175719815881863 0.02443861639073064 - 0.06959533026469772

Table 3.2: The lower and upper limits on the mass scaled frequency domain for the SF extraction models. These mass scaled frequency ranges are given for the q, m1 scaling in the centre column, and for the ν, M scaling in the right-most column. The rows designate spin.

With the overlapping frequency domain known, the q∆K/Kgeo(m1Ωφ) functions can be made to be dependent only on spin and frequency. These functions are sampled at 100 linearly spaced mass scaled frequencies within their respective frequency range for each spin grouping, as given in Table 3.2, such that q∆K/Kgeo(m1Ωφ) are calculated for each q and categorized for each spin. In other words, q∆K/Kgeo(m1Ωφ) has been calculated for each spin, independent of q, as a function of m1Ωφ, and may be rewritten as q∆K/Kgeo||χ˜ 1|(m1Ωφ).

Holding spin and frequency ﬁxed for each of these combined functions q∆K/Kgeo||χ˜ 1|(m1Ωφ) gives data points that coincide with each mass ratio q. For example,

   1.5∆K/Kgeo||χ˜ |=0.9(0.6Ωφ = 0.04)  1     3∆K/Kgeo||χ˜ |=0.9(0.75Ωφ = 0.04) |  1  q∆K/Kgeo |χ˜ 1|=0.9(m1Ωφ = 0.04) =   .  ¯   5∆K/Kgeo||χ˜ 1|=0.9(0.83Ωφ = 0.04)    ¯  8∆K/Kgeo||χ˜ 1|=0.9(0.88Ωφ = 0.04)

θφ θφ This is exactly the form needed to calculate the SF corrections KSF(q),1, KSF(q),2 using a linear fit in 1/q, as given in Eq. 3.16. θφ The SF coefficients are found using a non-linear least squares optimization fit with KSF,2(χ = 0) = 0 included. θφ θφ In the case of no spin, there would be no precession and KNR = Kgeo = 1, implying that q∆K/Kgeo = 0 for all systems.

From this fitting method, there is a first and second order SF coefficient for each point in (χ, m1Ωφ) phase space considered. The SF coefficients are plotted against mass scaled frequency in Fig. 3.15, and as a discrete contour plot in Fig. 3.16 with mass scaled frequency along the vertical axis and spin magnitude alone the horizontal θφ axis. From Figs. 3.15, 3.16, some basic conclusions can be drawn. In particular, the sign of KSF(q),1 changes θφ from negative with anti-aligned spin to positive with aligned spin, and the opposite is seen for KSF(q),2. It is also θφ apparent that the spread of the SF corrections is not symmetric — KSF(q),1 is larger in magnitude for |~χ1| = 0.9 θφ than it is for |~χ1| = −0.9, and the opposite is seen for KSF(q),2. This may be due to the sampled parameter space not being symmetric between anti-aligned and aligned spin systems, where anti-aligned spin systems merge at lower frequencies than their spin-aligned counterparts. From the frequency dependencies shown in Figs. 3.15, θφ 3.16 there is an overall trend for the anti-aligned spin KSF(q),1 to linearly decrease with increasing m1Ωφ, and for

58 Chapter 3. Fundamental frequency analysis for precessing systems 59

0.08 0.08

KSF(q), 1 KSF(q), 2 0.06 0.06

0.04 0.04 | 1| = 0.9, Lev3

| 1| = 0.9, Lev2

| 1| = 0.9, Lev1 0.02 0.02 | 1| = 0.5, Lev3

| 1| = 0.5, Lev2

0.00 0.00 | 1| = 0.5, Lev1

| 1| = 0.5, Lev3

0.02 0.02 | 1| = 0.5, Lev2 | 1| = 0.5, Lev1

| 1| = 0.9, Lev3 0.04 0.04 | 1| = 0.9, Lev2

| 1| = 0.9, Lev1 0.06 0.06 101 102

100 101

10 1 100

KSF(q), 1 1 (%) KSF(q), 2 1 (%)

10 2 10 1 0.015 0.020 0.025 0.030 0.035 0.040 0.015 0.020 0.025 0.030 0.035 0.040

m1 m1

θφ Figure 3.15: The upper left panel shows the values for the first order SF correction KSF(q),1, the upper right shows θφ the second order SF correction KSF(q),2, and the bottom row shows the respective 1σ variances as percentages of their related values. All panels are plotted against mass scaled frequency m1Ωφ, with different spin alignments and magnitudes shown in different colours and different resolutions shown with different line styles. The variance in θφ θφ KSF(q),1 is typically on the order of 1%, while the variance in KSF(q),2 is on the order of 10%. Different resolutions largely agree, and overall it appears that the first and second order SF coefficients have a linear dependence on mass scaled frequency.

0.040 KSF(q), 1 KSF(q), 2

0.035

0.030 1 m 0.025

0.020

0.015

0.9 0.5 0.0 0.5 0.9 0.9 0.5 0.0 0.5 0.9

| 1| | 1|

0.04 0.02 0.00 0.02 0.04 0.06 0.03 0.02 0.01 0.00 0.01 0.02 0.03

Figure 3.16: The left panel shows the first order SF correction KSF(q),1 and the second order SF correction KSF(q),2 is in the right panel, calculated using the linear fit given in Eq. 3.16. This data is scaled with q, and plotted against primary spin magnitude |~χ1| along the horizontal axis and mass scaled frequency m1Ωφ along the vertical axis. Different colours represent the values of the SF coefficients at each point in (χ, m1Ωφ) phase space, correlating with the colourbar at the bottom of each panel.

59 Chapter 3. Fundamental frequency analysis for precessing systems 60

θφ θφ aligned spin KSF(q),1 to linearly increase, while the opposite is true for KSF(q),2. A more robust investigation into these dependencies and the structure of the SF coeﬃcients is left to future work.

θφ θφ θφ In the non-linear least squares fitting to find KSF(q),1, KSF(q),2, the windowing errors in KNR, showcased in Fig. 3.7, are appropriately propagated and included as input errors. This in part affects the 1σ variance seen in the bottom row of Fig. 3.15. In addition to the propagated windowing errors that lead to this overall fitting error in the SF coefficients, there is also deviation between resolutions for the same points in (χ, m1Ωφ) phase space. We consider the resolution error here as the relative difference between identically calculated values across successive resolution Levs. We compute the resolution error between Lev1 and Lev2, and Lev2 and Lev3 for each simulation, and consider these errors to be convergent if the resolution error between the former is larger than the latter on average. If the errors are convergent, the resolution error is assumed to be that between Lev2 and Lev3. If not, the resolution error is the squared sum of the errors from both comparisons. If only two Levs are available, then their relative difference is the reported resolution error. For the systems presented here, all calculations of the SF coefficients are convergent in resolution except for those with |~χ1| = 0.9. The combined errors for the SF coefficients are shown in Fig. 3.17. The top panel shows the resolution error, θφ θφ fitting error, and total combined error of KSF(q),1, and the bottom panel shows the same for KSF(q),2. In both panels, the θφ fitting errors and resolutions errors are similar in magnitude, with the total error averaging around 1-5% for KSF(q),1 θφ and around 10-15% for KSF(q),2. The obvious exception in both cases is the |~χ1| = 0.5, m1Ωφ ∈ [0.030, 0.040] where the total error is dominated by the resolution error and inflates the error total value by an order of magnitude. It can also be seen in Fig. 3.15 that the |~χ1| = 0.5 simulations suffer from large fitting variances as well. It is currently unclear why this is the case, as the dynamics of these systems do not appear to deviate from expectations. This set of simulations were run significantly earlier than the other simulations presented here, and thus used an earlier version of SpEC, as well as being run on a different computing cluster. Additionally, the only major difference found in the dynamics of these simulations was that the overall magnitude of the spin vectors increased with time, unlike all the other systems considered here, which decreased with time. This does not necessarily explain the significantly larger errors that accompany this spin grouping of simulations, but it does indicate some differences between this subset and the other simulations.

The fitting procedure for the ν scaled case follow the same procedure as that for the q scaled case, with the appropriate switch in parameters 1/q → ν and m1Ωφ → MΩφ. Note that the SF estimates found for the ν data are not quite the same as that reported for the q scaled data. Due to the ν resummation, the first order term is technically θφ θφ pushed to second order, and therefore a direct comparison between KSF,1, KSF,2 between the two scalings cannot be made directly. The fits for the ν scaling are presented here for completeness, as opposed to for direct comparisons with the results from the q scaling.

The linear ν extractions are shown in Fig. 3.18 as a discrete contour plot and in Fig. 3.19 against mass scaled frequency, with an error comparison of the linear ﬁt shown in Fig. 3.20. The results for the ν scaling are considerably less robust than those found with the q scaling, with higher error overall, as shown in Fig. 3.20. Such a large disagreement could indicate issues with our ﬁtting procedure, but may also imply that the q scaled equations are

60 Chapter 3. Fundamental frequency analysis for precessing systems 61

101

) 100 % (

r o r r E

1 1 Res., | 1| = 0.9 , 10 ) q (

F Fit, | 1| = 0.9 S K Total, | 1| = 0.9 2 10 Res., | 1| = 0.5

Fit, | 1| = 0.5

Total, | 1| = 0.5

0.020 0.025 0.030 0.035 0.040 Res., | 1| = 0.5 m1 Fit, | 1| = 0.5

Total, | 1| = 0.5 102 Res., | 1| = 0.9

Fit, | 1| = 0.9

) Total, | 1| = 0.9

% 1 ( 10

r o r r E

2 , ) q

( 0 F 10 S K

10 1

0.020 0.025 0.030 0.035 0.040

θφ θφ Figure 3.17: The resolution, fitting, and total errors associated with KSF(q),1, KSF(q),2 for all available data. The θφ θφ top panel shows errors for KSF(q),1, and the bottom panel shows errors for KSF(q),2. The vertical axis is the percent compared to the associated SF coefficient, and the horizontal axis is mass scaled frequency. The different colours represent different spins, with dotted lines showing fitting errors, dashed lines showing resolution errors, and solid lines showing total, squared sum combined errors. For most of the parameter space considered here, the total error θφ θφ in KSF(q),1 is 1-5% and 10-15% for KSF(q),2.

0.07 KSF( ), 1 KSF( ), 2

0.06

0.05

M 0.04

0.03

0.02

0.9 0.5 0.0 0.5 0.9 0.9 0.5 0.0 0.5 0.9

| 1| | 1|

0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.2 0.1 0.0 0.1 0.2 0.3

θφ θφ Figure 3.18: The left panel shows the ﬁrst order SF correction KSF(ν),1, and the second order SF correction KSF(ν),2 is shown in the right panel, calculated using a linear ﬁt in symmetric mass ratio ν. This data is plotted against primary spin magnitude |~χ1| along the horizontal axis, and against mass-scaled frequency MΩφ along the vertical axis.

61 Chapter 3. Fundamental frequency analysis for precessing systems 62

0.6 0.20 KSF( ), 1 KSF( ), 2

0.15 0.4

| 1| = 0.9, Lev3

0.10 | 1| = 0.9, Lev2 0.2 | 1| = 0.9, Lev1

0.05 | 1| = 0.5, Lev3

| 1| = 0.5, Lev2 0.0 0.00 | 1| = 0.5, Lev1

| 1| = 0.5, Lev3

0.05 | 1| = 0.5, Lev2 0.2 | 1| = 0.5, Lev1

0.10 | 1| = 0.9, Lev3

| 1| = 0.9, Lev2 0.4 0.15 | 1| = 0.9, Lev1

102 101

101 100 K 1 (%) K 1 (%) SF( ), 1 100 SF( ), 2 10 1 0.02 0.03 0.04 0.05 0.06 0.07 0.02 0.03 0.04 0.05 0.06 0.07 M M

θφ Figure 3.19: The upper left panel shows the values for the first order SF correction KSF(ν),1, the upper right shows θφ the second order SF correction KSF(ν),2, and the bottom row shows the respective 1σ variances as percentages of their related values. All panels are plotted against mass scaled frequency MΩφ, with different spin alignments and magnitudes shown in different colours and different resolutions shown with different line styles. The variance in θφ θφ KSF(ν),1 is typically on the order of 5%, while the variance in KSF(ν),2 is on the order of 20%. Different resolutions θφ θφ largely agree for KSF(ν),1, but considerably less so for KSF(ν),2. better served with this analysis on the whole. Despite the larger fitting variances and resolution errors associated with the ν fits, there are visible trends seen in Figs.. 3.18, 3.19. Most notably, the first and second order SF coefficients are both positive for spin aligned systems, θφ θφ and both negative for anti-aligned systems. Additionally, KSF(ν),2 is significantly larger in magnitude than KSF(ν),1, which may imply a surprising dependence on SF corrections that are third order for precession of the orbital plane.

3.5 Discussion

In this chapter, a simple method for extracting the inclination frequency and accompanying orbital frequency from θφ numerical data is presented and tested. The resulting KNR = Ωθ/Ωφ were compared to the associated test mass Kerr geodesic and used to extract estimates for first and second order SF corrections. θφ It was found that there are oscillations in KNR for some systems, and that there is a stark difference in the shape θφ of the KNR curve for aligned and anti-aligned BBHs (c.f. Figs. 3.8, 3.9, 3.10). As expected, there was a distinct θφ dependence on mass ratio, with the larger the mass ratio the closer the KNR curve goes to the SF test mass limit and clusters. The first and second order SF corrections were extracted using the fundamental frequency ratio from NR and

62 Chapter 3. Fundamental frequency analysis for precessing systems 63

) 10 % (

r o r r E

1 Res., | 1| = 0.9 , ) 0 ( 10 F Fit, | 1| = 0.9 S K Total, | 1| = 0.9

Res., | 1| = 0.5

Fit, | 1| = 0.5 10 1 Total, | 1| = 0.5

0.02 0.03 0.04 0.05 0.06 0.07 Res., | 1| = 0.5 M Fit, | 1| = 0.5

Total, | 1| = 0.5 2 10 Res., | 1| = 0.9

Fit, | 1| = 0.9

) Total, | 1| = 0.9 % ( 1 r 10 o r r E

2 , ) ( F S

K 100

10 1 0.02 0.03 0.04 0.05 0.06 0.07 M

θφ θφ Figure 3.20: The resolution, fitting, and total errors associated with KSF(ν),1, KSF(ν),2 for all available data. The θφ θφ top panel shows errors for KSF(ν),1, and the bottom panel shows errors for KSF(ν),2. The vertical axis is the percent compared to the associated SF coefficient, and the horizontal axis is mass scaled frequency. The different colours represent different spins, with dotted lines showing fitting errors, dashed lines showing resolution errors, and solid lines showing total, squared sum combined errors. For most of the parameter space considered here, the total error θφ θφ in KSF(ν),1 is 5-15% and 10-50% for KSF(ν),2.

Kerr as functions of spin and mass scaled frequency, using linear fits in mass ratio 1/q and symmetric mass ratio ν assuming Eqs. 3.10, 3.16 hold true. In this analysis, the 1/q fits were more reliable, and in general the estimates for the second order SF correction were less robust than for the first order correction. From our analysis, it appears that Eq. 3.10 captures most of the physics for the precession of the orbital plane. θφ Future work will include confirming a model for KPN for direct comparison with the NR counterpart, and further θφ θφ θφ investigations into the dynamics of the systems, frames used, and fitting procedures for both KNR and KSF,1, KSF,2. θφ θφ Additionally, a global fit in q, |~χ1|, m1Ωφ is being pursued for KSF,1, KSF,2 to better describe the SF coefficients’ dependence on frequency, spin magnitude, and spin direction.

63 Chapter 4

Conclusion

The centre of mass (c.m.) project, detailed in Ch. 2, has shown it is possible to remove gauge effects from gravitational waveforms in post-processing using BMS transformations that are allowed by the Bondi gauge. It was further shown that these transformations reliably produced more accurate waveforms, and are effectively random and do not correlate with system parameters. It was found that changing the definition of the c.m. to include PN terms did not improve the waveforms, and that the c.m. drift in the raw data is much larger than that expected from linear momentum recoil. Changing the c.m. correction method from a BMS supertranslation to try and characterize the epicycle motion of the c.m. also proved ineffective and did not further improve the c.m. corrections to the waveform. Investigating the ratios of the fundamental frequencies of BBH systems provides access to gauge-invariant quantities that can be compared with perturbation methods, analytic results, and observations. By focusing on the precession of the orbital plane, or the ratio of the inclination frequency to the orbital frequency, it was found in Ch. 3 that there is not only a dependence on mass ratio, but on magnitude and alignment of the spin of the primary black hole. By comparing the precession of the orbital plane extracted from NR with that calculated for Kerr geodesics, the first and second order SF corrections are available. In particular, it is clear that the second order SF correction is needed to describe precession of the orbital plane in a SF expansion. Future work entails further investigating allowed gauge transformations that may further improve waveforms from SpEC and extending investigations of the fundamental frequencies in terms of parameter space and the ratios themselves. Regarding further mitigating gauge effects and the work presented on c.m. motion, there are a number of loose ends that need to be further investigated. It is still unknown what truly causes the c.m. motion to be so exaggerated in linear drift, acceleration, and epicyclic motion. While the junk radiation phase is the most likely candidate in terms of when this initial displacement and kick of the c.m. occurs, it is unknown how this could impart an acceleration or force the epicycles to be orders of magnitude larger than what can be explained physically.

64 Chapter 4. Conclusion 65

Additionally, attempting to account for the acceleration in the c.m. and the size of the epicycles would motivate analyses into how much of either effect may be true physics and how much is due to gauge effects. It is currently not expected that these components are physically motivated, but it is also unclear how to correct for them using BMS transformations. If it can be confirmed that long-term acceleration and large epicycle motion is not physically motivated, investigations into broadening the application of BMS transformations would be the next logical step. Some physical motivations for the measured behaviour of the c.m. that were not addressed in this work includes using higher order PN terms to account for c.m. motion, and calculating the general relativistic c.m. for the entirety of each simulation - a cumbersome task, but one that may be necessary for further investigation of the c.m. in SpEC simulations. The analysis on precession of the orbital plane, as presented here, is in progress and hence not complete. The immediate next steps for this project are to extract the necessary information from our analysis in order to compute the first and second order SF perturbation coefficients directly from SF codes and then compare with the results presented here. In addition to comparing with SF calculations, we would like to calculate the full PN approximations for the rate of precession of the orbital plane and compare them with the NR equivalents. It is expected that PN approximations will be more accurate at early times and progressively diverge from the NR predictions closer to merger. Additionally, the PN approximations may be better for smaller spin magnitudes and mass ratios. In terms of the NR predictions for the precession of the orbital plane, revisiting the frequency averaging techniques and fine-tuning to acquire better error estimates is pertinent. Regardless of fine-tuning the methods, extracting the fundamental frequencies from NR trajectory data is now readily available with relatively high accuracy and minimal contamination. As such, any combination of the fundamental frequencies into ratios is open to investigation. The methods and techniques showcased in this work may be applied to other physically relevant fundamental frequency ratios, and likewise compared to gain insight into SF and PN equivalents. Lastly, expanding the parameter space for studies of the fundamental frequencies and their ratios will bolster any analysis, and in particular pushing to higher mass ratios and choosing more spin magnitudes. Since the detection of gravitational waves in 2015, the field of compact objects, strong gravity, and gravitational waves has expanded and is picking up speed. There is a plethora of questions to ask and problems to solve, only one niche of them being NR and BBH. This work shows that much analysis is left to be done regarding extracting, analyzing, and understanding gauge effects and gauge-invariant quantities on and from gravitational waves. However, significant progress has been made in reducing gauge effects on the waveforms produced by SpEC and provided by the SXS Collaboration, which will hopefully garner attention to the devil in the details of NR waveforms and extracted quantities.

65 Appendices

66 Appendix A

Note on SpEC

A detailed discussion on the formalisms, assumptions, data outputs, and specifics on the Spectral Einstein Code (SpEC) can be found in the SXS Collaboration 2019 catalog update [71] and on the SpEC website [69]. Presented here is a short overview of the features that concern this work. In regards to BBHs, SpEC uses a multidomain spectral method for evolution. The two black holes are modelled as excisions in the domain, where there is no information about the domain inside of the apparent horizons. The domain of the simulation is divided into subdomains that is controlled by an Adaptive Mesh Refinement (AMR) algorithm which implements both h-refinement, or refinement in regards to the number of subdomains, and p- refinement, or refinement in regards to the number of grid points in a given subdomain. See Resolutions below for more details on resolutions in regards to AMR. During evolution, SpEC keeps information about the metric (or gravitational radiation) at each point in the domain, the trajectories of the apparent horizons, the dimensionless spin vectors of each black hole, and the masses of each black hole. The spin, mass, and trajectory information is stored together in a HDF5 file for each time step.

The gravitational radiation information data is stored as strain h, and as the Weyl tensor Ψ4. The Weyl tensor is a measure of the curvature of the manifold being considered, and the strain is the perturbation from gravitational radiation on the metric. Before evolution, initial data must be constructed to then be evolved. This is discussed in Initial Data and Junk Radiation below. After evolution, the information about the gravitational waveform must be extrapolated to future null inﬁnity and extracted. This is discussed in Extraction of Gravitational Radiation Information below.

Initial Data and Junk Radiation

Astrophysical BBH systems undergo thousands if not millions of orbits and more before merging. Since numerical relativity simulations are already lengthy and expensive with a few to a few tens of orbits, it is essential to create some initial data that mimics the positions and velocities of the black holes and basic information of the BBH as if

67 Appendix A. Note on SpEC 68 it had been orbiting for many cycles. For many BBH simulations, the initial separation between the two black holes D0 is chosen as well as their mass ratio and spins. Using PN approximations, the initial orbital frequency Omega0 and the scaled rate of inspiral adot0 are computed. Unless otherwise stated, BBH simulations are assumed to be circular as any orbital eccentricity should radiate out early on, and thus the orbital parameters are estimated so that the BBH has a low eccentricity (typically ≤ 10−3). In addition, a new method for constructing initial data was introduced in 2015 by Ossokine et al. [111]. This method introduced a more flexible domain decomposition, enhanced root-finding for calculating circularized PN approximations for the BBH, AMR, and eliminating a significant portion of linear drift of the BBH throughout the evolution. In regards to the linear drift specifically, this is caused by some randomized kick early on in the simulation that is directly caused by loose constraints on the linear momentum of the system in the initial data. By controlling this kick, or rather by controlling the linear momentum, a significant amount of mode mixing is avoided and the initial data is more reliable. One thing that cannot be avoided in a numerical relativity simulation is junk radiation. Regardless of how close the initial data is to what would be expected from astrophysical systems, once the simulation begins there is some settling or relaxation of the BBH as residual eccentricity is radiated away. This unastrophysical radiation early on in a simulation, or rather, junk radiation, is an artifact in every BBH simulation. The time it takes for the junk radiation to dissipate and leave the domain is called the relaxation time, and is normally several hundred M but typically less than 10% the total length of the simulation. Any momentum imparted onto the BBH during the junk radiation phase is one possible explanation for the c.m. effects discussed in Ch.2.

Extraction of Gravitational Radiation Information

In the same way many orbits cannot be reasonably simulated using numerical relativity, the entirety of the metric also cannot be simulated. Each simulation is computed on a finite domain where the edges of the domain impose constraint-preserving boundary conditions. The h and Ψ4 information is then extracted (separately) from the outer boundary to future null infinity, I +, where it is assumed the gravitational waves are approximately what would be seen with detectors on Earth. There are currently two methods of extraction available for simulations from SpEC. Both extract the gravitational radiation data to concentric spheres at increasing radii from the simulation domain before computing the waveform at I +. The one most commonly used is polynomial extraction (see section 2.4.1 in Ref. [90]), and the other is Cauchy-Characteristic Extraction (CCE) [112, 162–165]. Extraction happens in post-processing, and so the same simulation data can be used regardless of the extraction technique employed. All data referenced and used in this work was extracted via the polynomial extraction method, which is briefly outlined below. Polynomial extraction first assumes that the waveform modes can each be represented by a grid of coordinate l,m l,m l,m times T and radii of fixed concentric coordinate spheres R, such that h = h (Ti, R j). Extending this logic, h may instead be written as being dependent on retarded time ui, j and the areal radius of each fixed coordinate sphere

68 Appendix A. Note on SpEC 69

l,m ri, j, giving h (ui, j, ri, j). The waveforms are interpolated such that they are deﬁned at common retarded times uk, j l,m and transformed into the co-rotating frame, giving hˆ (uk, j, rk, j). Transforming into the co-rotating frame allows the assumption that the waveform modes are varying slowly in time, and can be approximated by a polynomial in r. The waveforms are ﬁt to a polynomial of order N = 2, 3, 4 of the form

N ˆ l,m X h( j) (uk) hˆ l,m(u , r) ≈ , (A.1) k r j+1 j=0 where the polynomial coefficients are found by minimizing the sum of the squared differences between the numerical + data and the polynomial value at each uk. The waveform is then extrapolated to future null infinity I , with the asymptotic waveform found by inverting the co-rotating frame transformation on the zero degree term, i.e. ˆ l,m l,m h(0) (u) → h(0) (u).

Resolutions

Resolutions in SpEC are designated as “Levs”, usually ascending with increasing resolution. Most simulations will have three resolutions, named “Lev1”, “Lev2”, and “Lev3”. However, simulations may have more or less resolutions depending on the intended use for the simulation, or if a resolution fails and cannot be considered. Resolution levels loosely translate to the number of grid points included in the domain for the simulation, however they are not explicitly set in SpEC. SpEC uses spectral methods to solve Einstein’s equations, which are exponentially convergent. This means that the spatial truncation errors in a subdomain of fixed dimensions decreases exponentially with the number of collocation points. Since AMR is used in SpEC [90], the number of subdomains and their dimensions change dynamically throughout the simulation via the number of grid points (p-refinement) or the number of subdomains (h-refinement) changing. Therefore, in SpEC, instead of resolution levels dictating a specific number grid points, they specify the minimum and maximum truncation errors that can be tolerated in the subdomains for that resolution. If N is the Lev number, the default maximum truncation parameter is 0.0001 × e−N, where e is Euler’s number and the minimum truncation parameter being 1% of the maximum. Unless explicitly changed by the owner of the simulation, the default values for truncation errors are

Resolution Level Max Trunc. Error Min Trunc. Error Lev1 10−4e−1 10−6e−1 Lev2 10−4e−2 10−6e−2 Lev3 10−4e−3 10−6e−3

Issues with SpEC

SpEC is an incredible tool for simulating compact binaries, however like all codes there are issues and limits to its functionality. In particular, SpEC struggles with high aligned spin |~χa| ' 0.9 and high mass ratio q ' 10 runs,

69 Appendix A. Note on SpEC 70 doubly so if the BBH is precessing. The largest mass ratio successfully computed with SpEC to date is q = 15, and the largest spin is |~χa| = 0.998 [71]. Outside of the limits of the code, SpEC is expensive to run. For simulations like those presented in Table 3.1, one of the shortest simulations took 128 CPU hours on 48 processors over the course of several weeks of real time. Simulations with large mass ratios and spins can take months of real time to complete, as SpEC will crash repeatedly and need to be manually tweaked and restarted. To address these issues, the SpEC development team is continuously implementing improvements to SpEC.

70 Appendix B

Spin-Weighted Spherical Harmonics

Spin-weighted spherical harmonics (SWSHs) are typically used to generalize the well-known standard spherical harmonics. Specifically, SWSHs provide a decomposition of general spin-weighted spherical functions (SWSFs) into a sum of SWSHs. Spin-weighted spherical functions themselves provide a vital way to study waves radiating from bounded regions, and so have an obvious and important application in gravitational-wave astronomy, which is the focus of this work. Spin-weighted spherical functions play two key roles in this field: (1) describing the magnitude of the wave given any direction of emission or observation, and (2) providing polarization information. There are a number of subtleties in defining SWSFs and hence SWSHs, including dependencies on the chosen coordinate system and the explicit definition of SWSFs.

The spin weight of a function is deﬁned by how it transforms under rotation of the spacelike vectors <(mµ), =(mµ) where mµ is a complex null vector tangent to the coordinate sphere S 2. The rotation of these spacelike vectors is given by (mµ)0 = eiΨmµ. (B.1)

A function η is then said to have a spin weight s if it transforms as

η0 = esiΨη. (B.2)

In the case of gravitational waves, the metric perturbation h has a spin weight of −2 [86,103] and this decomposition has been used in numerical relativity extensively.

The classic deﬁnition of SWSHs [86] writes the functions in terms of spherical coordinates for S 2, giving them as explicit formulas using polar and azimuthal angles (θ, φ) and using two integer variables that deﬁne the order of spherical harmonic to be used.

71 Appendix B. Spin-Weighted Spherical Harmonics 72

Spin-weighted spherical harmonics are thus classically deﬁned as

 h i1/2  (l−s)! ðsY , 0 ≤ s ≤ l,  (l+s)! l,m sYl,m = (B.3)  h i1/2  s (l+s)! ¯−s (−1) (l−s)! ð Yl,m, −l ≤ s ≤ 0, where ð is effectively a covariant differentiation operator in the surface of the sphere. ð is defined [86] as

( ) ∂ i ∂ ðη = −(sin θ)s + {(sin θ)sη} (B.4) ∂θ sin θ ∂φ when operating on some function η that has a spin weight s. The above classic method inherits an unfortunate dependency on the chosen coordinates. In particular, SWSHs cannot be written as functions on the sphere S 2; at best they can only be written as functions on coordinates of S 2. As such, SWSHs as defined in Eq. (B.3) do not transform among themselves under rotation of the sphere (or, equivalently, rotation of coordinates of the sphere). That is, a SWSH in a given coordinate system cannot generally be expressed as a linear combination of SWSHs in another coordinate system. A more correct method for defining SWSFs that does not inherit these coordinate-system dependencies is to represent them as functions from Spin(3) ≈ SU(2), which in turn represents orthonormal frames on S 2 [166]. By forming a representation of Spin(3), SWSHs defined in this way do transform among themselves and still agree with the classic definition. SWSHs may then be defined as r 2l + 1 Y (R):= (−1)s D(l) (R), (B.5) s l,m 4π m,−s where D is a Wigner matrix, which are representations of the spin group, and R is the Spin(3) argument. Taking R to be in the unit-quaternion representation of Spin(3), the D matrices may be expressed as

s ρ2 0! 0 ! (l) (l + m)!(l − m)! X l + m l − m D 0 (R) = m ,m (l + m0)!(l − m0)! ρ l − ρ − m ρ=ρ1

ρ l+m0−ρ l−ρ−m ρ−m0+m ρ (−1) Rs R¯ s Ra R¯a, (B.6)

0 0 where ρ1 = max(0, m − m), ρ2 = min(l + m , l − m), and Rs and Ra are the geometric projections of the quaternion into symmetric and antisymmetric parts under reﬂection along the z axis, which are essentially complex combinations of components of the quaternion:

Rs B Rw + iRz and Ra B Ry + iRx. (B.7)

The deﬁnition given in Eq. (B.5) is consistent with the deﬁnition of SWSHs typically used within the SXS collaboration and is the assumed formulation for this work. For further information on SWSFs and SWSHs, a comprehensive in-depth discussion of the history, details, and additional formulations of SWSHs can be found in [166].

72 Appendix C

Post-Newtonian Correction to the c.m.

As discussed in Sec. 2.5.1, the 1PN and 2PN corrections to the c.m. used are those outlined in Ref. [132]. The c.m. up to 2PN order is given in Eq. (4.5) of Ref. [132] as

" !# 1 Gm m m v2 Gi m yi yi − a b a a = a a + 2 a + (C.1) c 2rab 2 " ! 2 2 2 2 1 7 7 5G m m 7G mam vi Gm m − n v − n v yi − a b b + 4 a a b ( ab a) ( ab b) + 1 2 + 2 c 4 4 4rab 4rab 4 3mava Gmamb 1 2 1 1 2 + + − (nabva) − (nabva)(nabvb) + (nabvb) 8 rab 8 4 8 !!# 19 7 7 + v2 − (v v ) − v2 + a ⇐⇒ b 8 a 4 a b 8 b where the superscript i designates the vector component being considered; subscripts a, b designate which object is being considered; ~y is the position of the body being considered; rab = |~ya − ~yb| is the distance between body a and b; ~v is the velocity of the body being considered and likewise v is the magnitude of the velocity; and ~nab = ~rab/rab.

Parentheses here represent the scalar product of the interior values, e.g., (nabvb) = ~nab · ~vb. Note that this representation of the c.m. position does not include an overall division by the total mass of the system, and so the calculations used deviate from Eq. (C.1) only by including an overall denominator M = m1 + m2.

73 Appendix D

Linear Momentum Flux from hl,m modes

As mentioned in Sec. 2.5.2, the calculation of the linear momentum flux from the simulations is based on the formalism outlined in Ref. [102], which uses the SWSH structure of the gravitational strain h. Starting from Eq. (2.24), which gives the general form of the linear momentum flux in hl,m modes, the components of the linear momentum flux may be calculated as:

r   2 X 0 l l0 1 R m (2l + 1)(2l + 1)   p˙ x = − (−1)   16π 2   l,l0,m 2 −2 0     " 0 0 # 0  l l 1  0  l l 1 ˙ l,m ˙¯ l ,m−1   ˙¯ l ,m+1   h h   − h   m 1 − m −1 m −1 − m 1 r   2 X 0 l l0 1 R m (2l + 1)(2l + 1)   p˙y = − i (−1)   16π 2   l,l0,m 2 −2 0     " 0 0 # 0  l l 1  0  l l 1 ˙ l,m ˙¯ l ,m−1   ˙¯ l ,m+1   h h   + h   m 1 − m −1 m −1 − m 1     2 0 0 X 0  l l 1 l l 1 R m l,m ¯ l ,m p 0     p˙z = (−1) h˙ h˙ (2l + 1)(2l + 1)     , (D.1) 16π     l,l0,m m −m 0 2 −2 0 where x, y, z refer to the simulation coordinates, with the orbit typically lying in the x − y plane, R is the distance to the observation sphere, and h˙, h˙¯ are the time derivative and its conjugate of the mode amplitudes. The matrices throughout the summations are Wigner 3-j symbols.

74 Appendix E

Fundamental Frequencies from Kerr Geodesics

Following the discussion in Ref. [23], the fundamental frequencies Ωr, Ωθ, Ωφ for BBHs can be calculated from Kerr geodesics by assuming that the radiation reaction time is much larger than the periods associated with any of the fundamental frequencies. This further imposes the assumptions that the orbit is arbitrary, bound, non-plunging, and adiabatic.

In general, the Hamiltonian may be expressed as

1 H(xα, p ) = gµν p p , (E.1) β 2 µ ν

α µν where x is the physical location, pβ are the conjugate momenta, and g is the metric. Transforming into Boyer- Lindquist coordinates [18],

√ x = r2 + a2 sin θ cos φ, (E.2) √ y = r2 + a2 sin θ sin φ, √ z = r2 + a2 cos θ, (r2 + a2)2 − ∆a2 sin2 θ 2aMr → HBL(xα, p ) = − (p )2 − p p (E.3) β 2∆Σ t ∆Σ t φ 2 2 ∆ − a sin θ 2 ∆ 2 1 2 + (pφ) + (pr) + (pθ) , 2∆Σ sin2 θ 2Σ 2Σ where a = |J~|/M is the Kerr parameter, in which J~is the total angular momentum, and ∆ = r2 − 2Mr + a2. r, θ, φ are the usual polar coordinates. From the equations of motion resulting from this Hamiltonian, the conjugate momenta

75 Appendix E. Fundamental Frequencies from Kerr Geodesics 76 are

pt = − E, (E.4)

pφ =Lz,

2 2 ∆ pr =R,

2 pθ =Θ,

h 2 2 2i h 2 2 2 i R = (r + a )E − aLz − ∆ m2r + (Lz − aE) + Q , (E.5) " L2 # Θ =Q − (m2 − E2)a2 + z cos2 θ, 2 sin2 θ where Q is the Carter constant, which describes the separation of the polar and radial equations of motion, E is the total energy of the system, and Lz is the projection of the orbital angular momentum onto the z axis.

Noting that the canonical one-form of Hamiltonian mechanics, Θ = −Edr + prdr + pθdθ + Lzdφ, is also the α 2 one-form corresponding to the relativistic four-momentum p such that hΘ, pi = pα p = −m2, the action variables are then deﬁned as

1 I Jk = Θ, (E.6) 2π −1 Φ √(Ck) 1 I R → J = dr, (E.7) r 2π ∆ 1 I J = Θdθ, (E.8) θ 2π

Jφ =Lz, (E.9)

−1 where Φ (Ck) is a loop on the respective phase space torus. Lastly, the fundamental frequencies can be deﬁned as

δH m2Ωk = . (E.10) δJk

Completing these integrals and shifting the frequencies to be dimensionless brings the calculation to Eq. 33, 34, 35 in Ref. [23], which are the deﬁnitions used in Ch. 3. It should be noted that these deﬁnitions and hence the θφ calculations used to compute Kgeo are presented in the precessing frame, and hence provide additional motivation θφ for the calculations in Ch.3 for KNR to likewise be presented in the precessing frame.

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