The Pennsylvania State University

The Graduate School

Eberly College of Science

RINGING IN UNISON: EXPLORING BLACK HOLE

COALESCENCE WITH QUASINORMAL MODES

A Dissertation in Physics by Eloisa Bentivegna

c 2008 Eloisa Bentivegna

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

May 2008 The dissertation of Eloisa Bentivegna was reviewed and approved* by the following:

Deirdre Shoemaker Assistant Professor of Physics Dissertation Adviser Chair of Committee

Pablo Laguna Professor of Astronomy & Astrophysics Professor of Physics

Lee Samuel Finn Professor of Physics Professor of Astronomy & Astrophysics

Yousry Azmy Professor of Nuclear Engineering

Jayanth Banavar Professor of Physics Head of the Department of Physics

*Signatures are on file in the Graduate School.

ii Abstract

The computational modeling of systems in the strong-gravity regime of General Relativity and the extraction of a coherent physical picture from the numerical data is a crucial step in the process of detecting and recognizing the theory’s imprint on our universe. Obtained and consolidated over the past two years, full 3D simulations of binary black hole systems in vacuum constitute one of the first successful steps in this field, based on the synergy of theoretical modeling, numerical analysis and computer science efforts. This dissertation seeks to model the merger of two coalescing black holes, employing a novel technique consisting of the propagation of a massless scalar field on the spacetime where the coalescence is taking place: the field is evolved on a set of fixed backgrounds, each provided by a spatial hypersurface generated numerically during a binary black hole merger. The scalar field scattered from the merger region exhibits quasinormal ringing once a common apparent horizon surrounds the two black holes. This occurs earlier than the onset of the perturbative regime as measured by the start of the quasinormal ringing in the gravitational waveforms, indicating that previous semianalytical evidence on the early validity of perturbative methods during a black hole merger is indeed correct. The scalar quasinormal frequencies are also used to associate a mass and a spin with each hypersurface: this measure is, within our error bars, compatible with the horizon mass and spin computed from the dynamical horizon framework. The emerging physical picture indicates that the behavior of a scalar field propagating on the spacetime of two merged (i.e., surrounded by a common apparent horizon) black holes is very close to that expected on a Kerr spacetime with mass and spin parameters equal to Mf and jf , the mass and spin of the common apparent horizon at the end of the coalescence. Some of the results discussed in this dissertation have been published as: E. Bentivegna, P. Laguna, and D. Shoemaker, The effect of gauge conditions on waveforms from binary black hole coalescence, AIP Conf. Proc., 873:9498 (2006). E. Bentivegna, D. Shoemaker, I. Hinder, and F. Herrmann, Probing the binary black hole merger regime with scalar perturbations, arXiv:0801.3478 (2008).

iii Contents

List of Tables vii

List of Figures viii

Acknowledgements xv

1 Introduction 1 1.1 The black hole concept in mathematical, numerical, astrophysical and quan- tum Relativity ...... 3 1.2 Two-black-hole solutions ...... 5

2 General Relativity and the three-plus-one decomposition 7 2.1 Gravitation according to General Relativity ...... 8 2.1.1 Exact solutions in vacuum: static and stationary black holes ..... 9 2.2 The dynamics of General Relativity ...... 10 2.3 The ADM formulation ...... 11 2.3.1 The 3 + 1 decomposition ...... 12 2.3.2 Projections of the Riemann tensor ...... 14 2.3.3 Projections of Einstein’s equation ...... 15 2.4 Decomposition of the scalar wave equation ...... 18

iv 3 Scalar field evolution on a black hole background 19 3.1 Quasinormal ringing: damped oscillatory modes in open systems ...... 21 3.2 The scalar wave equation on a static, spherically symmetric spacetime .... 22 3.2.1 Green’s function analysis ...... 25 3.3 The scalar wave equation on a stationary, axisymmetric spacetime ...... 28 3.3.1 Green’s function analysis ...... 31

4 General Relativity and numerical methods 33 4.1 Scientific computing in General Relativity ...... 34 4.2 Modelling ...... 34 4.2.1 Evolution schemes and constraint preservation ...... 35 4.2.2 Coordinate conditions ...... 39 4.2.3 Initial data ...... 43 4.2.4 Boundary conditions ...... 45 4.3 Numerical analysis ...... 46 4.3.1 Integration of partial differential equations ...... 46 4.3.1.1 Finite differencing schemes ...... 46 4.3.1.2 Convergence to the continuum solution ...... 48 4.3.2 Function minimization ...... 50 4.4 Computer science ...... 52

5 Gravitational and scalar field evolution codes 55 5.1 BSSN evolution code ...... 56 5.1.1 Scope ...... 56 5.1.2 Structure ...... 56 5.2 Scalar field evolution code ...... 59

v 5.2.1 Scope ...... 59 5.2.2 Structure ...... 59 5.2.3 Code verification ...... 61 5.2.3.1 Flat space tests ...... 62 5.2.3.2 Single black hole tests ...... 64

6 Probing the binary black bole merger regime with scalar perturbations 66 6.1 Binary black hole background ...... 68 6.1.1 Energy and angular momentum ...... 72 6.2 Scalar field evolution ...... 72 6.2.1 Initial data ...... 78 6.2.2 Frequency extraction ...... 78 6.2.3 Mass and spin extraction ...... 100 6.2.4 Scalar field evolution before the first common apparent horizon . . . 109 6.3 Error analysis ...... 109 6.4 Discussion and conclusions ...... 115

References 118

A Gauge conditions and physical observables 130 A.1 Zerilli waveforms ...... 131 A.2 Error budget ...... 131

B Extraction of quasinormal frequencies via an evolutionary strategy 134 B.1 Algorithm ...... 135 B.2 Application to quasinormal ringing waveforms ...... 135

vi List of Tables

6.1 The residual q and the best-fit values of ω20, α20, A20 and φ20, along with the corresponding mass and dimensionless spin estimates...... 107

6.2 The residual q and the best-fit values of ω40, α40, A40 and φ40, along with the corresponding mass and dimensionless spin estimates...... 107

6.3 The residual q and the best-fit values of ω60, α60, A60 and φ60, along with the corresponding mass and dimensionless spin estimates...... 108

vii List of Figures

3.1 The contour integral used to calculate G(x, y; t). The contribution from the poles of G(x, y; ω) in the lower half plane are responsible for the quasinormal mode components of the field, while the half circle as ω -if present at | |→∞ all- generates the prompt response to the initial data and the portion of the contour around the branch cut results in the late-time power-law tails (picture reproduced from [1])...... 27 3.2 The spectrum of quasinormal modes for a Schwarzschild black hole, for the modes ℓ = 2 (diamonds) and ℓ = 3 (crosses). The absolute value of the imaginary part grows in a roughly linear fashion with n, so that higher and higher overtones tend to be more and more suppressed as the field evolves in time (plot reproduced from [2])...... 29

4.1 Evolution of the spatial grid (here collapsed to a 2D plane) during the numer- ical evolution. At each time coordinate, the lapse and the shift are calculated to determine where each grid point is going to be positioned in the next it- eration. In binary black hole simulations, the gauge choices proposed, for instance, in [3] guarantee that the grid points remain far enough from the puncture singularities even when the puncture locations are advected through the grid...... 40

viii 4.2 Lapse function for four gauge types, on the x-y plane, for a single boosted puncture. The λ and γ values refer to the constants in equations (4.2.36) and (4.2.37), while µ = 0 and η = 4. In the case when λ = 1 and γ = 0, the collapsed-lapse region does not track the puncture location efficiently, lagging behind in the evolution and generating the stretched contour shown in the top right panel...... 44

5.1 Structure and interdependencies of the BSSN evolution code used in the runs of chapter 6 ...... 58 5.2 Structure and interdependencies of the scalar evolution code used in the runs of chapter 6 ...... 60 5.3 Convergence factor for the plane wave. After a rapid initial transient, the value of Q settles, as expected, to 15...... 62 5.4 Pointwise convergence for the plane wave, at t/λ = 0, 0.8, 1.6, 2.4, 3.2, 4.0. The difference between the two highest resolution runs is multiplied by the appropriate factor for a fourth order code...... 63 5.5 The angular mode corresponding ℓ = 2, m = 0 on a single Schwarzschild black hole spacetime, using the ”frozen background” approach described in chapter 6 (the time variable is denoted by τ to highlight its distinction from t). The expected value for the real and imaginary part of the frequency is ω = 1 1 0.4836M − (which corresponds to a period of about 13 ) and α =0.0967M − . 65 M 5.6 The real and imaginary part of the extracted complex frequency...... 65

6.1 The trajectory followed by the two black holes in the x-y plane (top). The trajectories of three different resolutions h = /44.8, /51.2 and /57.6 M M M are indistinguishable. The bottom panel shows the coordinate separation of the two black holes as a function of time...... 69 6.2 The real (top) and imaginary (bottom) part of the ℓ = 2, m = 2 mode of the Newman-Penrose scalar Ψ , extracted at r = 50 ...... 70 4 M

ix 6.3 The amplitude (top) and phase (bottom) of the ℓ = 2, m = 2 mode of the Newman-Penrose scalar Ψ , extracted at r = 50 . Note the quasinormal 4 M ringdown exhibited by the linear decay in the amplitude (on this log-linear plot) and the constant frequency...... 71 6.4 The ADM energy (top) and angular momentum (bottom) for an R1 run, as a function of the extraction sphere’s radius ρ. The data points are extrapolated to infinity to yield E =0.9957 and J =0.862 2...... 73 ADM M ADM M 6.5 The radiated energy (top) and angular momentum (bottom) extracted from

the Newman-Penrose scalar Ψ4, as a function of the extraction sphere’s radius ρ, at t 300 . The extrapolation to spatial infinity yields E = 0.032 ∼ M rad M and J = 0.2241 2...... 74 rad − M 6.6 The radiated energy (top) and angular momentum (bottom) extracted from

the Newman-Penrose scalar Ψ4, for five different values of the extraction radius ρ, as a function of t. The detector at 10 and those at 200 and 300 M M M are either too close to the source (and, therefore, not in the “wave zone”) or too far away (and, therefore, causally disconnected from the source), and have

been excluded from the extrapolation procedure for Erad and Jrad...... 75 6.7 The apparent horizons at different stages of a binary black hole merger (in isometric view from the positive-z semiaxis). The two black holes inspiral around each other counterclockwise on the x-y plane, until they become sur- rounded by a distorted common apparent horizon, which then radiates away its higher multipoles reaching an axisymmetric configuration...... 77 6.8 Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 2. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data...... 79 6.9 Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 4. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data...... 80

x 6.10 Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 6. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data...... 81 6.11 Isosurfaces of the scalar field at select values of τ during the evolution, for ℓ = 2 and t = 160 . The white surface represents the apparent horizon. .. 83 M 6.12 The evolution of the ℓ = 2-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The mode { }M was obtained on a sphere of constant coordinate radiusr ˜ =5 ...... 84 M 6.13 The evolution of the ℓ = 4-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The mode { }M was obtained on a sphere of constant coordinate radiusr ˜ =5 ...... 85 M 6.14 The evolution of the ℓ = 6-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The mode { }M was obtained on a sphere of constant coordinate radiusr ˜ =5 ...... 86 M 6.15 The minimum of the fitting residual q in equation (6.2.5) with respect to A,

M, j and φ, as a function of τ0, for the ℓ = 2 mode and for τf fixed and equal to 69 ...... 88 ∼ M 6.16 The values of ω20 (top) and α20 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 89 0 f ∼ M 6.17 The values of A20 (top) and φ20 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 90 0 f ∼ M 6.18 The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 91 0 f ∼ M 6.19 The minimum of the fitting residual q in equation (6.2.5) with respect to A,

M, j and φ, as a function of τ0, for the ℓ = 4 mode and for τf fixed and equal to 69 ...... 92 ∼ M 6.20 The values of ω40 (top) and α40 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 93 0 f ∼ M

xi 6.21 The values of A40 (top) and φ40 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 94 0 f ∼ M 6.22 The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 95 0 f ∼ M 6.23 The minimum of the fitting residual q in equation (6.2.5) with respect to A,

M, j and φ, as a function of τ0, for the ℓ = 6 mode and for τf fixed and equal to 69 ...... 96 ∼ M 6.24 The values of ω60 (top) and α60 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 97 0 f ∼ M 6.25 The values of A60 (top) and φ60 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 98 0 f ∼ M 6.26 The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 ...... 99 0 f ∼ M 6.27 The minimum of the fitting residual q in equation (6.2.5) as a function of t for the ℓ = 2, 4, 6 mode of the scalar field. At early times t the black hole { } has not yet settled down and hence the fit residual q is larger as the scalar field evolved on the black hole background does not exhibit similarly clean ringdown as it does for later times...... 101 6.28 The mass and spin parameters extracted from the fundamental mode fre- quency, for the three angular modes. The error bars for the horizon mass and

spin is included in the curve width. At early times, different choices of τ0 and

τf lead to significant modifications in the behavior of M and j as a function of t. The data error bars in this regime should therefore be considered as a mere constraint of the mass and spin range on each hypersurface...... 102

6.29 Evolution of the complex quasinormal mode frequencies Ω20 = ω20 + iα20 (the point with largest ω corresponds to the hypersurface at t = 160 ), 20 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the intervals M between neighboring lines being equal to 0.01 ), whereas those with positive M slope are the constant-j lines (the thicker being j =0.7, with 0.1 intervals). 104

xii 6.30 Evolution of the complex quasinormal mode frequencies Ω40 = ω40 + iα40 (the point with largest ω corresponds to the hypersurface at t = 160 ), 40 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the intervals M between neighboring lines being equal to 0.01 ), whereas those with positive M slope are the constant-j lines (the thicker being j =0.7, with 0.1 intervals). 105

6.31 Evolution of the complex quasinormal mode frequencies Ω60 = ω60 + iα60 (the point with largest ω corresponds to the hypersurface at t = 160 ), 60 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the intervals M between neighboring lines being equal to 0.01 ), whereas those with positive M slope are the constant-j lines (the thicker being j =0.7, with 0.1 intervals). 106 6.32 The behavior of the scalar field Φ’s mode ℓ = 2 on three different hypersur- faces, corresponding to the first common apparent horizon formation and to 10 before and after it. The results are shown for six choices of the extraction M sphere...... 110 6.33 The difference between the two coarsest resolutions, along with the difference between the two finest resolutions multiplied the expected factor for a first-, second-, third- and fourth-order scheme, for the mode ℓ = 2, extracted at a radius ρ =5 at time t = 200 ...... 112 M M 6.34 Contours of the variation of the mass estimate ∆M(τ , τ ) = M(τ , τ ) 0 f | 0 f − M(25 , 55 ) for ℓ = 2, 4, 6 at t = 160 and t = 260 , as a function of M M | M M τ0 and τf ...... 113 6.35 Contours of the variation of the spin estimate ∆j(τ , τ )= j(τ , τ ) j(25 , 55 ) 0 f | 0 f − M M | for ℓ =2, 4, 6 at t = 160 and t = 260 , as a function of τ and τ . .... 114 M M 0 f

A.1 Real part of the Moncrief function Q (mode ℓ = 2, m = 0)...... 132 A.2 Norm of the difference in waveform between gauges (a) and (b) (top), (a) and (c) (center) and (b) and (c) (bottom)...... 133

3 B.1 A realization of the waveform in (B.2.1), with σ = 10− ...... 136

xiii B.2 Biases and standard deviations for the real and imaginary part of the quasinor- mal frequency and for the quality factor , for the matrix pencil, Kumaresan- Q Tuft, Levenberg-Marquardt and CMA-ES methods. The data relative to the first three algorithms has been extracted from figure 4 in [4]...... 138 B.3 The initial population and generations 25, 40, and 55 for a CMA-ES run on the trial waveform (B.2.1). The contour lines represent the constant value lines for the fitting residual Ψ˜ Ψ(t; A,φ,ω,α) := N Ψ(˜ t ) Ψ(t ; A,φ,ω,α) , || − || n=0 | n − n | where t is the discrete grid on which the waveform is represented. .... 139 { n} P

xiv Acknowledgements

This thesis is dedicated, first and foremost, to all those who have supported me with their love and friendship over the past five years, putting up with the long distances and the long working hours that completing this dissertation has required. This includes my parents, Salvatore and Patrizia, my brother Carlo and countless others. You have made all the difference. I would also like to express my deepest gratitude to the people who make Penn State, the Institute for Gravitation and the Cosmos and the Numerical Relativity group such an exciting work environment. In particular, I thank my adviser Deirdre Shoemaker for giving me so much support and yet so much independence while working on this dissertation; I also thank Bernd Br¨ugmann for introducing me to Numerical Relativity, Pablo Laguna for guiding me through my first binary black hole project and Abhay Ashtekar, Sam Finn and Yousry Azmy for their advice on this work. A special mention goes to all the students and postdocs who have shared our group life and taught me some Numerical Relativity during the past five years, especially Carlos Sopuerta, Uli Sperhake, Andrew Knapp, Shaun Wood, Tanja Bode, Frank Herrmann, Ian Hinder and Erik Schnetter. During this time, I have also enjoyed many stimulating conversations with Alfio Bonanno, Daniele Perini, Tomasz Pawlowski and Alberto Sesana. Finally, I wish to thank those who have shared their Penn State experience with mine, especially Radzie, Daithi, Csaba, Claire and all the State College Italians. I acknowledge the support of the Center for Gravitational Wave Physics funded by the National Science Foundation under Cooperative Agreement PHY-0114375. The simulations presented in this paper were carried out under allocation TG-PHY060013N at NCSA.

xv To my family

xvi Chapter 1

Introduction

Among all the breakthroughs achieved by modern physics in the quest to predict and under- stand phenomena beyond the ordinary experience, the absence of an absolute notion of time and the necessity of a four-dimensional formulation of all the equations of physics are ar- guably two of the greatest. General Relativity, Einstein’s theory of gravitation introduced in 1915, has since provided insight into the behavior of massive objects, shedding light -among others- on nothing less than the nature of space and time and the evolution of our universe. Aside from its formal elegance and symmetry-exposing character, General Relativity has also been dispensing detailed predictions regarding phenomena which would find no natural explanation in Newtonian physics: the gravitational collapse of massive stars, the orbital shrinking due to the emission of gravitational radiation, the dragging of frames around com- pact rotating objects and the cosmic expansion, among many others. While some of these predictions have already met accurate observational confirmation, corroborating our confi- dence in the theory’s founding ideas, the vast majority -and perhaps the most interesting- of General Relativity’s conjectures still await a direct verification. Among these, considerable interest revolves around gravitational collapse, and the nature and behavior of its end products: combining Einstein’s equation to the equation of state of ordinary matter, a novel feature emerges that was not present in Newtonian gravity: in General Relativity, all the contributions to the stress-energy tensor (even the one due to a body’s internal pressure) are sources of gravitation; a compression, increasing the internal pressure, also leads to an augmented gravitational pull, and there exist a critical point beyond

1 which the net effect points inward, and the object inevitably collapses on itself generating a spacetime singularity. The critical mass depends on the object’s equation of state, but in most reasonable cases1 it cannot exceed a few times the mass of the Sun; since stars larger than this size are reasonably common in our universe, gravitational collapse should be expected at the end of their thermonuclear lifetime, and collapsed objects (i.e., black holes, due to the one-way membrane that surrounds each of them) should constitute an appreciable component of our cosmological environment, as confirmed by a growing collection of astrophysical evidence [5–9]. These systems are therefore ideal general-relativistic test bed candidates, in addition to observationally interesting objects in their own right. In this dissertation, I aim to address some questions concerning the transition between a system of two inspiraling black holes and a single perturbed black hole. In particular, I investigate how early during the binary black hole evolution the scalar field probe exhibits the quasinormal ringing phase ordinarily expected from a spacetime containing a single black hole, and what features this phase displays when compared to gravitational wave quasinormal ringing of the single stationary black hole that is left behind by the coalescence process. Based on these considerations and on a preliminary study carried out in [10], I present a novel strategy that consists of (i) simulating a binary black hole merger and extracting horizon information from the merged black hole, (ii) taking snapshots of the spatial geometry obtained in the binary black hole merger at a sequence of coordinate times, (iii) evolving a massless scalar field on the corresponding spatial hypersurfaces, and (iv) observing the scalar field undergo a quasinormal ringing phase and extracting its complex frequency, which in turn can be converted into a mass and a spin parameter using the functional dependence of the quasinormal frequencies on these two parameters prescribed by perturbation theory. I then compare these estimates to the mass and spin extracted directly from the horizon. This method has the benefit of effectively separating the evolution time of the binary system from the relaxation time of the scalar perturbations, allowing for a measure of the “instan- taneous” properties of the foliation. This is particularly important since the interesting transition region lasts only about 0.5 to 0.75 gravitational wave cycles [11], so that an anal- ysis of this regime in terms of gravitational radiation alone is not viable. Notice that this

1The most widely used reference points are the Chandrasekhar limit for white dwarfs and the Tolman- Oppenheimer-Volkoff limit for neutron stars, respectively equal to about 1.4 and 3 solar masses.

2 procedure is effectively equivalent to the evolution of a scalar field on the fictitious spacetime one would obtain by selecting each spatial surface and freezing it in time. This of course is not meant to constitute a physical solution to the coupled scalar-gravitational system: as emphasized above the scalar field is used here exclusively as a probing agent for individual hypersurfaces. This thesis is organized as follows: in this chapter, I will introduce the concept of a black hole and briefly describe the connected two-body problem, along with some of history of its numerical solution. In Chapter 2, I introduce Einstein’s equation (to which black holes are exact solutions) and the Einstein-Hilbert action, and address the crucial transition from a four-dimensional spacetime formalism to a space in time (or “3 + 1”) decomposition, along with the extraction of the true, “observable” degrees of freedom of the gravitational field; chapter 3 describes the physics of a scalar field propagating on a single black hole background, with a frequency-domain analysis of both the Schwarzschild and the Kerr case. Chapter 4 introduces the computational aspects of 3D general-relativistic simulations, and constitutes the last background chapter of this work. Chapter 5 describes the two 3D codes (one for evolving the gravitational field and one for evolving the scalar field) used to carry out the study in chapter 6, which describes the results obtained by simulating a binary black hole coalescence and evolving a scalar field on the corresponding numerical background.

1.1 The black hole concept in mathematical, numeri- cal, astrophysical and quantum Relativity

Far from being an offspring of General Relativity alone, the idea of a gravitational system, so massive that not even light is able to escape from it, has been the object of scientific speculations for over two centuries. In the simple Newtonian framework, for instance, all bodies with an escape velocity greater or equal to the speed of light are, literally, “black”; quite interestingly, in the spherically symmetric case this condition is satisfied by all bodies with a mass M and a radius R such that 2GM/(Rc2) 1, the same condition that holds in ≥ the relativistic case. However, the modern notion of a black hole has only emerged after General Relativity’s

3 formulation of the gravitational interaction, along with its description of horizons and mass- energy equivalence which were absent in Newtonian gravity. Since the introduction of the first black hole solution by Schwarzschild in 1917 [12], this notion has evolved and developed into one of the most polyhedric objects of twentieth century physics. First and foremost, black holes come to light as strong-field solutions of Einstein’s equation, packed with many of the effects that set General Relativity apart from Newtonian gravity, among which is the defining characteristic of black holes: the existence of spacetime regions causally disconnected from the distant observer. The mathematical study of black hole solutions [13], their perturbations and stability [14–19], their symmetries and coordinate extensions and the behavior of null 2-surfaces in the region exterior to the horizon [20, 21] have constituted an appreciable fraction of the research effort in mathematical Relativity over the past few decades. In some cases, the study of black hole systems relies entirely on accurate numerical modeling of the black holes and their surroundings, which has generated a whole new perspective on the black hole concept, centered around its 3+1 dimension representation and its evolution along some predefined time direction. An appropriate formulation of General Relativity, suitable for numerical integration, has proven essential for the advancement of these methods. Third, the strong-field properties of these systems make them premium candidates for the detection of General Relativity’s predictions in the universe. The search for black hole candidates in our cosmic surroundings is a ongoing endeavor that employes the most sophis- ticated observational and modeling techniques, from mass and size estimates built on galactic dynamics [6, 22–25] and spectral emission [26–28], to the observation of the effect of event horizons in accretion disks [5], and to potential gravitational wave astronomy studies [29–33]. Finally, the singular character of the known black hole solutions signals that, in certain regimes, General Relativity does not provide an accurate description of the gravitational interaction; in this sense, the singularities mark the confines of the classical theory and provide a stage for the study of its prospective quantum extensions.

4 1.2 Two-black-hole solutions

This thesis focuses on the study, via numerical means, of the evolution of a pair of classical, equal-mass, non-spinning black holes as they orbit around each other, emitting gravitational radiation and inspiralling until a final, rotating black hole forms. Part of the importance of compact object encounters derives from the fact that these events are premium candidates for the emission of substantial gravitational radiation, thereby possessing a direct detection channel through the gravitational wave observatories currently in operation worldwide. Since gravitational wave astronomy has the potential to disclose details of our universe which are invisible to electromagnetic and neutrino-based observations, these events provide an opportunity to observe our cosmic surroundings under a new lens. Unfortunately, exact solutions for gravitationally collapsed objects are only available in a few idealized cases; in order to assist the study of relativistic effects in more realistic scenarios, such as the encounter and coalescence of two black holes, numerical methods are the only viable option. The numerical simulation of two black hole systems in full 3D General Rela- tivity and the extraction of a coherent physical picture from the numerical data is therefore a crucial step in the process of detecting and recognizing General Relativity’s imprint on our universe. Systems of two black holes have been the object of intense numerical work since the mid sixties [34–36] (see table I in [37] for a comprehensive synopsis of the early codes), in the attempt to provide the simplest numerical description of the two-body problem in General Relativity in all those cases (mass ratio close to unity, small separation between the sources) where analytical techniques failed. Since then, growing computational power in addition to improved instability-free numerical techniques have added to the repertoire of successful binary black hole simulations, which now ranges from head-on and grazing collisions, to circular and eccentric orbits [38–53]; from equal-mass systems to a wide spectrum of mass ratios [54–56]; and from initially non-spinning black holes to a number of spinning configu- rations [57–74]. Recently, a number of astrophysical and cosmological consequences of the new results have been discussed and a few empirical fitting formulæ have been derived as convenient tools to calculate the properties of binary black hole mergers without an explicit evolution [75–78].

5 The field of Numerical Relativity embodies the paradigm of a scientific endeavour where technological and algorithmic advances have assisted each other in the identification of a path to solution: faster components and more efficient parallel architectures have spurred the development of accurate problem formulations and algorithms, which in turn are nec- essary to handle the emerging supercomputer power in a way that is efficient and free of code instabilities. The successful completion of the first simulated merger events in vacuum is the expression of this synergy and an encouraging piece of evidence that innovative com- putational techniques can play a decisive role in the study of relativistic systems and their potential for observation. Moving along this line of research, the interpretation of the new data constitutes an ongoing process that involves a constant dialectic between the mathematical black hole paradigm and its numerical counterpart, necessary -in one direction- to guide the numerical evolutions and -in the other- to decode the multitude of numerical results into a comprehensive, coherent physical picture. In chapters 2 to 4, which enclose all the background material on which my work is founded, I will attempt a description of this iterative process, which has determined the flavor of Numerical Relativity over the past two years.

6 Chapter 2

General Relativity and the three-plus-one decomposition

Relativity has modified our concept of gravitation and encoded fundamental principles such as Lorentz invariance and the equivalence principle into a consistent framework. However, in spite of the profound physical motivation coded in the diffeomorphism invariance of General Relativity, the four-dimensional, coordinate-independent formalism turns out to be fairly obstructive to the extraction of the true degrees of freedom of the gravitational field (which, for a spin-2 field, are expected to be two) from the ten components of the metric tensor, and to the study of its physical modes. In 1962 Arnowitt, Deser and Misner (ADM) tackled the problem by eliminating the parametriza- tion invariance of the Einstein-Hilbert action and recasting it in canonical form [79]; this was accomplished by a 3 + 1 decomposition of the theory, along with the choice of a time direction. The selection of the actual dynamical information, along with a choice of a time coordinate, proves to be extremely convenient when trying to construct solutions to Einstein’s equation by numerical means. The task now consists of the solution of the Cauchy problem repre- sented by the partial differential equations which determine the time evolution. Given an appropriate choice of initial data and boundary conditions, conventional numerical methods can then be applied to generate the solution over the desired time interval (it is worth noting, however, that this is the most diffuse but not the unique path to the numerical integration

7 of Einstein’s equation; for a different approach to the problem, see, for instance, the binary black hole evolutions in [39]). In section 2.1, I will briefly introduce the theory of General Relativity and two of its exact solutions that are relevant to this thesis. I will then discuss the ADM decomposition in section 2.2 and present the corresponding Cauchy problem (in a form that slightly differs from the original work [79], but that is closer to the modern formulation motivated by Numerical Relativity) in the section 2.3, deferring the issue of numerical evolutions till chapter 4.

2.1 Gravitation according to General Relativity

General Relativity is a metric theory of gravity, meaning that the gravitational field is

represented by a four-dimensional tensorial field gab called the metric tensor, which satisfies Einstein’s equation: 1 8πG G := R g R = T (2.1.1) ab ab − 2 ab c4 ab where Rab is the Ricci tensor given by:

a Rab = R bca (2.1.2)

a and R bcd is the Riemann tensor defined by:

Ra v :=( )v =(∂ Γa ∂ Γa +Γa Γe Γa Γe )v (2.1.3) bcd a ∇b∇c − ∇c∇b d c bd − d bc ec bd − ed bc a

a The Ricci scalar R is defined by R = R a. Indices are raised and lowered using the metric ab tensor gab and its inverse g , and the Einstein summation convention:

3 a ab ab R = R a = g Rab := g Rab (2.1.4) a,bX=0 is assumed.

Tab is the stress-energy tensor, representing the properties of the matter contained in the spacetime. Equation (2.1.1) thus realizes the connection between matter and the metric tensor. G and c are Newton’s constant and the speed of light, respectively; as customary, in this thesis the physical units are chosen so that G =1= c.

8 2.1.1 Exact solutions in vacuum: static and stationary black holes

The generic spherically symmetric and static tensor can be written as [80]:

e2Φ 00 0 −  0 e2Λ 0 0  g = ab 2  0 0 r 0   2   0 00 r2 sin θ      where Φ and Λ are arbitrary constants. Solving for these two constants in the vacuum

(Tab = 0) spacetime outside a perfect fluid distribution of radius r˜, one obtains: 1 2M Φ= ln(1 ) (2.1.5) 2 − r 1 2M Λ= ln(1 ) (2.1.6) − 2 − r where M is determined by the boundary conditions at the interface with the fluid:

r˜ M = 4πr2ρ(r)dr (2.1.7) Z0 ρ(r) being the fluid density (nonzero only for r < r˜). This solution is referred to as Schwarzschild geometry [12], and has a line element given by:

2 2M 2 2M 1 2 2 2 2 2 2 ds = (1 )dt + (1 )− dr + r dθ + r sin θdφ (2.1.8) − − r − r

The transformation r R =(r M + √r2 2Mr)/2 casts this line element into a spatially → − − isotropic form:

(1 M/2R)2 M ds2 = − dt2 +(1+ )4(dR2 + R2dθ2 + R2 sin2 θdφ2) (2.1.9) −(1 + M/2R)2 2R

which will prove useful in the construction of initial data for binary black hole systems. An extension of this metric to the axially symmetric and stationary case was worked out by Kerr and Schild while analyzing a metric of the form [81, 82]:

gab = ηab + kakb (2.1.10)

9 where ka needs to be null in order to preserve the signature of gab. Imposing Einstein’s equation leads to:

ka = √Hℓa (2.1.11) with 2Mr3 H = (2.1.12) r4 + a2z2 and 1 1 ℓ = t [r(x x + y y)+ a(x y y x)] z z (2.1.13) a ∇a − r2 + z2 ∇a ∇a ∇a − ∇a − r ∇a where a is a constant. The above metric represents the Kerr solution in Kerr-Schild coor- dinates; the same solution can be represented in Boyer-Lindquist coordinates, as shown in section 3.3 below. Notice that both of the above spacetimes contain a region that is causally disconnected from the exterior: this can be noticed by examining the outgoing null geodesics of each spacetime, and realizing that they become tangent to the surface r = 2M (for the Schwarzschild solu- tion) and r = M +√M 2 a2 (for the Kerr solution). These two surfaces represent the event − horizon of the black hole, defined precisely as the boundary of the causally disconnected region.

2.2 The dynamics of General Relativity

It is a well known result of classical field theory that the action of a system with N field components: N S = dt d3x π φ˙ H(φ, π) (2.2.1) i i − " i=1 # Z Z X can be freely reparametrized by a time transformation t τ(t); if t is then included as → an additional generalized coordinate, with conjugate momentum equal to H, the system’s − action can be rewritten as:

N N+1 3 3 S = dτ d x π φ′ H(φ, π)t′ = dτ d x π φ′ (2.2.2) i i − i i " i=1 # " i=1 # Z Z X Z Z X

10 where it is intended that the constraint π = H holds. This constraint can be added to N+1 − the action with a Lagrange multiplier:

N+1 3 S = dτ d x π φ′ αC (2.2.3) i i − " i=1 # Z Z X where C is equal to π + H, or any other function that has π = H as a simple N+1 N+1 − root. In this form, the action explicitly exhibits the reparametrization invariance involved in the choice of a time coordinate; this symmetric form does however come at a price: the introduction of an additional degree of freedom t(τ). Einstein’s equation in vacuum can be derived, through a variational principle, from the Einstein-Hilbert action: S = d4x√ gR (2.2.4) − Z In [79], Arnowitt, Deser and Misner showed that this action can be cast into the form (2.2.3), so that unphysical degrees of freedom are present in the formulation to make the theory’s coordinate invariance explicit. Formally, the steps that led from (2.2.1) to (2.2.3) can be repeated backwards, in the hope to write the theory’s action in terms of its physical degrees of freedom only. As expected from the spin of the gravitational field, this procedure eliminates

all but exactly two degrees of freedom. The other eight components of gab are eliminated by application of the four constraint equations and by the choice of four coordinate conditions.

2.3 The ADM formulation

The ADM procedure, consisting of (i) a decomposition into 3 + 1 dimensions, (ii) the im- position of the constraint equations and (iii) a choice of coordinates, has become the main paradigm for casting Einstein’s equation in a form suitable for numerical integration. In this section I will present a detailed account of the ADM 3+1 decomposition following [83]; since constraint preservation schemes and coordinate choices are mainly dictated by numerical re- quirements, I will postpone these two topics (and comments on other 3 + 1 decompositions) till the modeling section in chapter 4.

11 2.3.1 The 3+1 decomposition

Given a spacetime ( ,g ), we can foliate it into a family of non-intersecting, three- M ab dimensional hypersurfaces Σ, which are constrained to be spacelike by the following pro- cedure: let t be a scalar function whose level surfaces are, at least locally, represented by the family Σ. We define the one-form: Ω := dt (2.3.1)

which is closed according to dΩ= d(dt) = 0. We further impose that its norm be negative:

2 ab 2 Ω = g t = α− (2.3.2) | | ∇a∇b − and call α the lapse function. A timelike unit normal vector can be defined as:

na := αgabΩ = αgab t (2.3.3) − b − ∇b Since nan = 1, this vector points towards the direction of increasing t. The four dimen- a − sional metric induces, by pull-back, a spatial metric on each Σ:

γab = gab + nanb (2.3.4)

The corresponding projection operators are given by:

a a a γb = δb + n nb (2.3.5)

for projections onto the hypersurfaces and

N a = nan (2.3.6) b − b for projections along na. We can also define the three-dimensional covariant derivative:

D T b := γdγbγf T e (2.3.7) a c a e c ∇d f and the three-dimensional connection coefficients: 1 Γa = γad(γ + γ γ ) (2.3.8) bc 2 bd,c dc,b − bc,d

12 Da is compatible with γbc in the sense that:

D γ = γdγeγf γ = γdγeγf (g + n n )= γdγeγf (n n + n n )=0 (2.3.9) a bc a b c ∇d ef a b c ∇d ef e f a b c e∇d f f ∇d e a The expression for Γbc follows from this property along similar lines as in the for the four- dimension case. From: 0= D γ = ∂ γ Γd γ Γd γ (2.3.10) a bc a bc − ab dc − ac bd permutating the indices we obtain:

d d Γabγdc +Γacγbd = ∂aγbc (2.3.11) d d Γbcγda +Γbaγcd = ∂bγca (2.3.12) d d Γcaγdb +Γcbγad = ∂cγab (2.3.13)

Summing side-by-side, we obtain: 1 Γc = γcd [∂ γ + ∂ γ ∂ γ ] (2.3.14) ab 2 a bd b ad − d ab The three-dimensional Riemann tensor is defined by the conditions:

(D D D D )w = Rd w (2.3.15) a b − b a c cba d d R cband =0 (2.3.16)

d which implies that R cba can be expressed in terms of the connection coefficients just like in d the four-dimensional case. From R cba, index contraction yields the three-dimensional Ricci

tensor Rab and the three-dimensional scalar curvature R. The information about the embedding of each hypersurface in , not contained in any of M the three-dimensional tensors above, is instead encoded in the extrinsic curvature, defined as (minus one-half) the Lie derivative of the spatial metric along the normal na: 1 K := £ γ (2.3.17) ab −2 n ab In terms of the quantities introduced above, this can be rewritten as: 1 K = [nc γ + γ nc + γ nc]= ab −2 ∇c ab ac∇b cb∇a 1 = [nc (n n )+(g + n n ) nc +(g + n n ) nc]= −2 ∇c a b ac a c ∇b cb c b ∇a = n n a (2.3.18) −∇(a b) − (a b) 13 where a := nc nb, and we used the notation A = (A + A )/2. This can also be b ∇c (ab) ab ba rewritten as:

K = n + n a = ab − ∇(a b) (a b) 1 = [ n + n + n a + n a ]= −2 ∇a b ∇b a a b b a 1 = [( α)( t)+ α t + α t + α t + n a + n a ]= −2 ∇a ∇b ∇a∇b ∇b ∇a ∇a∇b a b b a 1 = [ n a + (α t)+( α)( t)+ n a + n a ]= 2 − a b ∇b ∇a ∇b ∇a a b b a = n a n (2.3.19) − a b − ∇a b or simply:

K = n + n a = ab − ∇(a b) (a b) 1 =  n + n nc  n + n + ndn n = −2 ∇a b a ∇c b ∇b a b∇d a = γcγd n  (2.3.20) − a b ∇(c d)

2.3.2 Projections of the Riemann tensor

(4) a Due to the symmetries of the four-dimensional Riemann tensor R bcd, there are only three non-zero projections. First, one can project all of its indices onto the spatial hypersurfaces:

r p q s (4) d r p q ds (4) γaγb γc γd Rrpqsω = = γaγb γc γ Rrpqsωd = = γrγpγq (4)Rd ω = γrγpγq[ ]ω = a b c rpq d a b c ∇r∇p − ∇p∇r q = D D ω D D ω K Kdω + K Kdω = a b c − b a c − ac b d bc a d = Rd ω K Kdω + K Kdω (2.3.21) abc d − ac b d bc a d obtaining Gauss’ equation:

R = γrγpγqγs (4)R + K K K K (2.3.22) abcd a b c d rpqs ac bd − bc ad Second, one can project three indices onto the spatial hypersurface and one normal to it:

γrγpγq (4)R nd = γrγpγq[ ]n = a b c rpqd a b c ∇r∇p − ∇p∇r q = γrγpγq[ K + K ]= a b c −∇r pq ∇p rq = D K + D K (2.3.23) − a bc b ac 14 obtaining Codazzi’s equation:

γrγpγqnd (4)R = D K + D K (2.3.24) a b c rpqs − a bc b ac Finally, projecting twice onto the spatial slice and twice along na yields:

γrγq (4)R nbnd = γrγqnb[ nd nd]= a c rbqd a c ∇r∇q − ∇q∇r = γrγqnb[ ( Kd a nd) ( Kd a nd)] = a c ∇r − q − q − ∇q − r − r = γrγqKd nb + D a + nb K a c q ∇r a c ∇b ac − K nb (γrγq)+ a γra = − rq ∇b a c c a a = γrγqKd nb + D a + a a + nb K + a c q ∇r a c a c ∇b ac +K ( Kq nq)+ K ( Kr nr)= aq − c − ∇c cr − a − ∇a q = Daac + aaac + £nKac + KaqKc = 1 = D D α + £ K + K Kq (2.3.25) α a c n ac aq c that is, Ricci’s equation: 1 γrγqnbnd (4)R = D D α + £ K + K Kq (2.3.26) a c rbqd α a c n ac aq c

The above derivations make use of the fact that ab can be rewritten as Db ln α, since:

a := na n =( αgac t) ( α t) b ∇a b − ∇c ∇a − ∇b α2 na nb = ( t t)+ α α = 2 ∇b ∇a ∇a α α∇a 1 − − = [ga α + nan α]= D ln α (2.3.27) α b ∇a b∇a b

2.3.3 Projections of Einstein’s equation

Gauss’ equation can furtherly be contracted to obtain an equation for the three-dimensional Ricci scalar:

R := γacγbdR = γacγbd[γrγpγqγs (4)R + K K K Kbc]= abcd a b c d rpqs ac bd − ad (4) a b (4) 2 ab = R +2n n Rab + K + KabK (4) a b (4) 2 ab = R +2n n Rab + K + KabK (2.3.28)

15 (4) where we have assumed a vacuum spacetime, Rab = 0. This constitutes the first of three possible projections of Einstein’s equation in vacuum:

(4) Gab =0 (2.3.29) and is referred to as the Hamiltonian constraint. We can project (2.3.29) along na as well, obtaining: 1 na (4)G = na[ (4)R (4)Rg ]= ab ab − 2 ab 1 = nagrgpgcd (4)R (4)Rn = a b rcpd − 2 b = nrγpγcd (4)R = DaK D K (2.3.30) b rcpd ab − b which yields the momentum constraint:

DaK D K =0 (2.3.31) ab − b The constraint equations represent the integrability conditions to embed Σ,γ ,K in . ab ab M The time evolution of Kab and gab is obtained by projecting twice along the time direction. First, we define a new timelike vector in terms of the hypersurface normal na, the lapse α and an arbitrary shift vector βa:

ta := αna + βa (2.3.32)

a a This singles out a more natural time direction, since t is dual to Ωa for any choice of β :

a t Ωa =1 (2.3.33)

a b In particular, this new choice implies that if a tensor Tab is purely spatial (i.e., n n Tab = 0),

so is £tTab:

nanb£ T = nanbtc T + nanbT tc + nanbT tc = tcT nanb (2.3.34) t ab ∇c ab ac∇b cb∇a ab∇c

We then obtain two evolution equations for γab and Kab:

£ γ = tc γ + γ tc + γ T c = t ab ∇c ab ac∇b cb∇a = (αnc + βc) γ + γ (αnc + βc)+ γ (αnc + βc) ∇c ab ac∇b cb∇a = α£ γ + £ γ = 2αK + £ γ (2.3.35) n ab β ab − ab β ab 16 Similarly:

£tKab = α£nKab + £βKab = 1 = α[ndncγqγr (4)R D D α KcK ]+ £ K = a b rdqc − α a b − b ac β ab α = αγqr[R + K K K K ] (4)Rγ D D α αKcK + £ K = bqar ba qr − bq ar − 2 ab − a b − b ac β ab = αR + αKK 2K Kc D D α + £ K ab ab − bc a − a b β ab = D D α + α[R 2K Kc + KK ]+ £ K (2.3.36) − a b ab − bc a ab β ab

Let us now simplify the above equations (2.3.35)-(2.3.36) by specializing to a coordinate system adapted to the foliation, where ta has components (1, 0, 0, 0) and the triad vectors a ei (i = 1, 2, 3) orthogonal to Ωa (so that Ωaei = 0) have components (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. In this system, the shift vector has a zero time component (βa = (0, βi)), the normal n has no spatial components (n = ( α, 0, 0, 0)) and the four- a a − metric takes the form: 2 k α− + βkβ βi gab = − βj γij ! so that the line element reads:

ds2 = α2dt2 + γ (dxi + βidt)(dxj + βjdt) (2.3.37) − ij

If we express the system (2.3.35)-(2.3.36) in these coordinates, we obtain:

∂ γ = 2αK + D β + D β t ij − ij i j j i and

∂ K = D D α + α(R 2K Kk + KK )+ βkD K + K D βk + K D βk (2.3.38) t ij − i j ij − ik j ij k ij ik j kj i along with the usual Hamiltonian and momentum constraints:

R + K2 K Kij = 0 − ij D Kj D K =0 (2.3.39) j i − i

Equations (2.3.38), (2.3.38), (2.3.39) and (2.3.39) represent the 3 + 1 decomposition of Ein- stein’s equation. Not dissimilarly from Maxwell’s equations, this system presents a set of

17 evolution equations determining the change in the field variables γij and Kij along the time direction determined by ta and a set of constraint equations, which the field vari- ables must obey on each spatial hypersurface (notice that, as pointed out in [79], the Bianchi identities guarantee that if the constraints hold initially, they will hold at all times). The analogy with Maxwell’s equations extends beyond the constrained evolution character. As in Electrodynamics, also in General Relativity the physical properties of any solution are invariant under a change in gauge, which in General Relativity’s case amounts to the choice of α and β. Furthermore, notice that an arbitrary combination of the constraints can be added to the evolution equations: all the constraint-satisfying solutions of the new equations will then trivially obey the original system. These considerations will play a major role in the numerical integration of the 3 + 1 evolution equation, since the ADM decomposition will prove inadequate and a modified version of this system will be necessary to guarantee an instability-free evolution. We will expand on this topic in section 4.2.1.

2.4 Decomposition of the scalar wave equation

The 3 + 1 decomposition of the scalar wave evolution equation:

Φ := g a bΦ=0 . (2.4.1) ab∇ ∇

ij i in terms of γij, K , α and β can be found, for instance, in [84] and reads:

(∂ βk∂ )Φ = αΨ (2.4.2) τ − k k α ij ij (∂τ β ∂k)Ψ = ∂i(√γγ ∂jΦ) + αKΨ+ γ ∂iα∂jΦ (2.4.3) − √γ where the auxiliary variable Ψ was introduced in order to reduce the system to a first-order- in-time form. Notice that we label the time coordinate in the above two equations as τ, in order to stress the distinction between this and the time coordinate t associated with the binary black hole evolution.

18 Chapter 3

Scalar field evolution on a black hole background

In the development of General Relativity, the role of evolving test fields on a curved back- ground can hardly be overestimated: in 1957, Regge and Wheeler provided the first descrip- tion of the axial perturbations of a spherically symmetric and static black hole, described as a “small” spin-2 field on the curved Schwarzschild background. The test field is shown to obey the scalar wave equation with a positive definite potential, and the behavior of perturbed-Schwarzchild systems (such as a test particle falling into a black hole) can be as- similated to the familiar flat-space behavior of radiation in the presence of a potential (with phenomena such as scattering [85, 86], resonances [87], barrier reflection and transmission [13], and superradiance [88]), so that well studied concepts and techniques from other areas can be applied to these problems as well. In 1973, Teukolsky extended this treatment to axisymmetric, stationary black holes [19]: in this case, the absence of spherical symmetry complicates the description, causing the angular variables to be fully separable from the radial and time coordinates only in Fourier space. Whilst no single 1D scalar wave equation can describe the evolution of generic initial data in the time domain, a spectral study of the problem’s features is still analytically possible. The investigation of the evolution of small deviations from a black hole grants several advan- tages: in the first place, it sheds light on the stability properties of the background solution, a topic of utmost interest in the early exploration of the nature of black holes and their role

19 in astrophysical scenarios; furthermore, the search for a coordinate independent formula- tion of perturbations (see the work by Moncrief in the spherically symmetric case [18]) has contributed a tremendous help towards an understanding of the nature of these solutions and the invariants that characterize them; finally, perturbative methods provide a powerful framework for the description of all those systems which depart infinitesimally from a black hole, such as extreme mass ratio binary systems and distorted black holes. The study of gravitational test fields on a curved background is also an essential ingredient to describe the propagation of gravitational radiation through the “near zone” that separates the emission region from the asymptotically flat end where the observers are located. This process has been likened to electromagnetic emission (e.g., by a molecular system) inside an optical cavity: the waves that emerge from the cavity and reach a distant observer not only carry the imprint of the source that emitted them, but also contain information on the cavity that affected their propagation towards the observer [1, 87]. Analytically and numerically modelling the wave propagation can therefore help characterize the astrophysical environ- ment that surrounds the sources of gravitational radiation or affects the wave propagation at any point on its path to the detector. Notice that this remark is not limited to gravitational radiation, but is valid for all the radiative fields that are expected to be astrophysically relevant, such as electromagnetic and neutrino fields. Also notice that several “linear optics” counterparts to the curvature scattering of waves (such as gravitational lensing [89]) are already widely used tools to infer certain properties of the wave path between source and observer and the intervening matter sources in between. The first step towards the simulation of the above scenarios, which is free from the complica- cies associated with non-zero spin fields, consists of the evolution of the scalar wave equation on a curved background. Scalar fields have indeed been a popular tool to investigate com- plex dynamics (such as that arising in cosmology and gravitational collapse) in a simplified setting; here they constitute an ideal curvature-tracing tool for the problem under study. In this chapter, we provide some background on the behavior of waves in open systems (with a focus on quasinormal ringing), and review the existing analytical results on wave scattering on stationary black hole spacetimes. A good introduction to the subject can be found in [13] and in Nollert’s [90] and Kokkotas & Schmidt’s review works [2]. This overview will provide

20 the context for our own wave evolution experiments described in chapter 6.

3.1 Quasinormal ringing: damped oscillatory modes in open systems

Physicists are accustomed to the description of conservative systems, i.e. systems in which certain quantities (usually, some definition of energy) are constant in time. These systems are characterized by Hermitian operators: their eigenfunctions form a complete basis and any state that the system can be found in can be expressed as a superposition of them. The eigenfunctions take the name of normal modes; normal mode expansions are a standard workhorse of classical and quantum physics, and stand behind countless physical ideas and applications. More often than not, however, one has to deal with systems which do dissipate energy (for instance through leakage to infinity, in which case it is customary to speak of open systems). The eigenfunctions of these systems are characterized by an oscillatory behavior with a complex frequency (with the imaginary part responsible for the energy dissipation), and do not usually form a complete set. The lack of completeness makes it impossible to use much of the machinery available for conservative systems (however, see [87] for a unifying treatment close to the Hermitian paradigm), rendering the study of open systems particularly involved. Perturbed black holes (or perturbative fields propagating on top of black hole spacetimes) are an archetypal example of open system: the wave equation that describes the evolution of arbitrary-spin perturbation is solved along with a set of outgoing wave boundary conditions both at spatial infinity and at the horizon. In physical terms, after the initial perturbation the field will keep on decaying, its energy steadily lost to infinity or absorbed by the black hole itself. The complex nature of the frequency spectrum leads to several undesirable consequences, some of which have direct ramifications on our ability to detect and characterize quasinormal ringing in a numerical simulation: first and foremost, the spectrum of quasinormal modes does not constitute a complete set of eigenfunctions [1, 13, 90]: early on, the behavior of

21 the scalar field depends on the process that generated the perturbation, and the field is said to be in the prompt response regime, where it still displays memory of the perturbing mechanism [91]. Additionally, at late times the backscattering off the curvature will overtake the by-now feeble damped oscillations and dominate the waveforms, generating the so-called tails [91]. A Green’s-function analysis of the origin of these three phases, along with a convergence study of the quasinormal mode expansion is presented in [92] for Schwarzschild and in [93] for Kerr black holes. The question then arises about how to identify the onset time of quasinormal ringing, since an accurate determination of this quantity is of the utmost importance while fitting for the quasinormal frequencies. This issue is usually referred to as the time shift problem; for a discussion of its influence on the bias of the fitted parameters, see [94]. Second, concepts such as the fraction of a given waveform that can be identified as each quasinormal mode and the gravitational energy contained in each quasinormal mode cannot be defined in rigorous terms [95]; the absence of such auxiliary notions also poses serious complications to identifying the onset of quasinormal ringing in a numerical waveform, and quantifying its quasinormal content. Berti et al. [96] review this issue from the numerical standpoint. My experimental method to overcome this issues (similar to the method used in [11, 94, 96]) is presented in chapter 6; here I will simply review the general features of black hole scalar perturbations in order to provide a background for that discussion.

3.2 The scalar wave equation on a static, spherically symmetric spacetime

On the background of a Schwarzschild black hole, the scalar wave equation:

Φ := gab Φ=0 (3.2.1) ∇a∇b can be expressed as a flat-space wave equation with a potential [14]. In order to estabilish the conditions under which this is true, let us rewrite the Schwarzschild line element in (2.1.8) as: 2M G(x) ds2 = 1 dt2 dx2 r2(x)dΩ2 (3.2.2) − r(x) − 1 2M −   − r(x)

22 where G(x)=(dr(x)/dx)2 represents the transformation between the areal radius r and the arbitrary coordinate x. In these coordinates, the inverse metric takes the form:

1 2M 0 0 0 1 r(x) − 2M 1 r(x) 1  0 − 0 0  g− = − G(x) 2  0 0 r− (x) 0   −   2 2   0 0 0 r− (x) sin− θ   −    and: √ g = √Gr2 sin θ (3.2.3) − Since ∂ ln √ g = (0, ∂ ln √G +2√G/r, cot θ, 0), in the case of a scalar field Φ we have: a − x 1 Φ = ∂ (√ ggab∂ Φ) = ∂ (gab∂ Φ) + gab∂ ln √ g∂ Φ= √ g a − b a b a − b tt− xx xx θθ φφ = g ∂ttΦ+ g ∂xxΦ+ ∂xg ∂xΦ+ g ∂θθΦ+ g ∂φφΦ+ 2√G +gxx(∂ ln √G + )∂ Φ+ gθθ cot θ ∂ Φ= x r x θ 2√G = gtt∂ Φ+ gxx∂ Φ+ ∂ gxx∂ Φ+ gxx(∂ ln √G + )∂ Φ tt xx x x x r x − 1 (sin2 θ∂ Φ+ ∂ Φ + sin θ cos θ ∂ Φ) (3.2.4) −r2 sin2 θ θθ φφ θ

fℓm Without loss of generality, we can set Φ = ℓm RℓmYℓm = ℓm r Yℓm. We then have:

P ∂ f P √Gf ∂ Φ= Y ∂ R = Y ( x ℓm ℓm ) (3.2.5) x ℓm x ℓm ℓm r − r2 Xℓm Xℓm ∂ f 2√G ∂ Φ= Y ∂ R = Y [ xx ℓm ∂ f xx ℓm xx ℓm ℓm r − r2 x ℓm Xℓm Xℓm 2 G ∂ √G +( | | x )f ] (3.2.6) r3 − r2 ℓm

23 Inserting these into the wave equation, we find:

1 f Φ= Y gtt∂ f ℓm ( ℓ(ℓ + 1) sin2 θ) ℓm r tt ℓm− r3 sin2 θ − Xℓm  xx ∂xxfℓm 2√G 2 G ∂x√G +g 2 ∂xfℓm +( | 3 | 2 )fℓm + " r − r r − r #

xx xx 2√G + ∂xg + g (∂x ln √G + ) " r # ·

∂xfℓm √Gfℓm 2 (3.2.7) · " r − r #)

where we have used the fact that:

sin2 θ∂ Y + ∂ Y + sin θ cos θ ∂ Y = ℓ(ℓ + 1) sin2 θY (3.2.8) θθ ℓm φφ ℓm θ ℓm − ℓm

Setting all the ℓm coefficients equal to zero and dropping the indices on fℓm finally leads to:

gtt∂ f + gxx∂ f + gxx∂ ln( gxx√G)∂ f tt xx x − x − xx xx 2g ∂x√G √G∂xg ℓ(ℓ + 1) + 2 f =0 (3.2.9) − r r − r !

which can be cast in the form:

∂ f F (x)∂ f + H(x)∂ f + V (x)f =0 (3.2.10) tt − xx x with xx F (x)= g − gtt gxx xx√  H(x)= gtt ∂x ln( g G) − xx xx  1 ℓ(ℓ+1) 2g ∂x√G √G∂xg  V (x)= tt 2 g r − r − r   The scalar wave equation turns into a flat-space wave equation provided that F (x) = 1  − and H(x) = 0, which both imply gtt = gxx or G(x)= 1/gxx =1 2M/r. In this case, − − − p

24 the potential becomes:

2 1 ∂ √G √G∂ 1 1 ℓ(ℓ + 1) √G x x √G V (x) = tt 2 + + = g r r r ! 1 ∂ √G 1 ℓ(ℓ + 1) √G x = tt 2 + = g r r !

1 ℓ(ℓ + 1) ∂x ln √G = tt 2 + = g r r ! 2M ℓ(ℓ + 1) 2M = 1 + (3.2.11) − r r2 r3     which is the Regge-Wheeler potential for a spin-zero field. The x coordinate, defined by:

1 2M − dx/dr = 1 (3.2.12) − r   or r x = r +2M log 1 (3.2.13) − 2M

is usually referred to as the tortoise coordinate. Notice that in this coordinate the half-line

problem r 2M gets mapped into the full-line problem x + , so that in these ≥ −∞ ≤ ≤ ∞ coordinates the horizon is at an infinite distance from, say, the peak of V(r(x)).

3.2.1 Green’s function analysis

The resulting equation:

Df(t, x) := ∂ f(t, x) ∂ f(t, x)+ V (x)f(t, x)=0 (3.2.14) tt − xx is formally integrated by:

+ ∞ f(t, x)= dy [G(x, y; t)∂tf(y, 0)+ ∂tG(x, y; t)f(y, 0)] dy (3.2.15) Z−∞ where G(x, y; t) is the associated Green’s function, defined by:

DG(x, y; t)= δ(x y)δ(t), G(x, y, t)=0 for x =0 or y =0 or t 0 (3.2.16) − ≤

25 A Fourier-space derivation of the Green’s function appears to be more convenient; setting

+ ∞ G˜(x, y; ω) := dtG(x, y; t)eiωt (3.2.17) Z−∞ we are left with an ordinary differential equation for G˜(x, y; ω):

D˜G˜ := ω2 d + V (x) G˜(x, y; ω)= δ(x y) (3.2.18) − − xx − If two independent solutions g(x, ω) and h(x, ω) to the homogeneous equation:

Dg˜ (x, ω) = 0 Dh˜ (x, ω)=0 (3.2.19)

with boundary conditions:

g(x) eiωx x ∼ →∞ iωx h(x) e− x (3.2.20) ∼ → −∞ are known, then the Green function can simply be expressed as [97]:

g(x, ω)h(y,ω)/W (ω) xy where W (ω)= g(x, ω)∂ h(x, ω) ∂ g(x, ω)h(x, ω) (3.2.21) x − x is the Wronskian of g and h. Once G˜(x, y; ω) has been worked out, the time-domain Green’s function is given by inverting the Fourier transform in equation (3.2.17), or:

+ 1 ∞ iωt G(x, y; t) = dωG(x, y; ω)e− 2π Z−∞ 1 iωt = dωG(x, y; ω)e− 2π IΓ 1 iωt dωG(x, y; ω)e− − 2π ZΓ1 1 iωt dωG(x, y; ω)e− − 2π ZΓ2 = GQNM(x, y; t)+ GID(x, y; t)+ Gtails(x, y; t) (3.2.22)

26 Figure 3.1: The contour integral used to calculate G(x,y; t). The contribution from the poles of G(x,y; ω) in the lower half plane are responsible for the quasinormal mode components of the field, while the half circle as ω -if present | |→∞ at all- generates the prompt response to the initial data and the portion of the contour around the branch cut results in the late-time power-law tails (picture reproduced from [1]). where: Γ = ω : Im[ω] < 0, ω 2 (3.2.23) 1 | | →∞

and Γ2 is any curve potentially needed to close the contour in the lower half plane and leave

any essential singularity of G(x, y; ω) on the outside (see Figure 3.1); Γ is the union of Γ1,

Γ2 and the real axis. There are thus three possible contributions to G(x, y; t):

1. Prompt response: The contribution from GID(x, y; t) originates in the large fre- quency behavior of G(x, y; ω): if G(x, y; ω) falls off sufficiently rapidly for ω , | |→∞ at least on the lower half plane. Studies indicate that this happens, for instance, for

27 potentials that are not C∞. Finite-differencing evolutions of the scalar wave equation always fall in this category;

2. Quasinormal modes: The contribution from GQNM(x, y; t), due to the residues of the polar singularities of G(x, y; ω). These can derive both from polar singularities in g or h or from the zeros of their Wronskian. Notice that in the latter case, g and h are no longer independent solutions, but are proportional to each other and satisfy both outgoing boundary conditions. In other words, they are the quasinormal modes we have been looking for. For the potential in (3.2.11), the spectrum is discrete, with upper-bounded real part and unbounded imaginary part (see Figure 3.2);

3. Tails: The contribution from Gtails(x, y; t): if G(x, y; ω) has essential singularities, branch cuts, et cetera in the lower half plane, the integration contour will have to be distorted in order to leave out these features. The existence of this contribution (or, equivalently, the existence of essential singularities in G(x, y; ω)) has been related to the asymptotic behavior of the potential V (x): compact or exponentially-decaying

potentials lead to singularity-free solutions, so that Gtails(x, y; t) = 0, whilst potentials with power-law tails lead to non-zero contributions. For the potential in equation α (3.2.11), G (x, y; t) t− with α > 0, generating the power-law decay observed tails ∼ at late times in many numerical studies of black hole perturbations. Physically, this phenomenon has been associated with the backscattering of the scalar field off the curvature in the far region, whose properties are arguably determined by the asymptotic behavior of V (x).

3.3 The scalar wave equation on a stationary, axisym- metric spacetime

The behavior of a massless scalar field Φ on the background of a single Kerr black hole has also been the subject of extensive analytical treatment. In this case, factoring out the angular variables is a much less trivial problem, solved in 1973 by Teukolsky [98] for scalar, electromagnetic, gravitational and neutrino fields.

28 Figure 3.2: The spectrum of quasinormal modes for a Schwarzschild black hole, for the modes ℓ = 2 (diamonds) and ℓ = 3 (crosses). The absolute value of the imaginary part grows in a roughly linear fashion with n, so that higher and higher overtones tend to be more and more suppressed as the field evolves in time (plot reproduced from [2]).

29 In the Boyer-Linquist coordinates t, r, θ and φ, the metric tensor for Kerr solution reads:

2Mr 1 Σ 00 Θ − Σ  0 ∆ 0 0  gab = −  0 0 Σ 0   − 2   Θ 00 sin θ(r2 + a2 + aΘ)   −    with Σ = r2 + a2 cos2 θ,∆= r2 2Mr + a2 and Θ=2Mar sin2 θ/Σ. − The presence of a nonzero gtφ component entangles the temporal and the azimuthal coordi- nates, precluding a separation of the type encountered in the previous section; Teukolsky’s technique consists of attempting a quasinormal mode expansion and searching for solutions iΩt with a temporal dependence of the form e− , where Ω is a constant. Under this assumption, and furtherly assuming that Φ = R(r)S(θ)eimφ, the radial and polar parts of the scalar wave equation separate into [98]:

∂ ∂R K2(Ω, m) ∆ + λ(Ω, m) R =0 (3.3.1) ∂r ∂r ∆ −     and ∂ ∂ sin θ sin θ S + a2Ω2 cos2 θ + (Ω, m) S =0 (3.3.2) ∂θ ∂θ A   with:   K(Ω, m)=(r2 + a2)Ω am (3.3.3) − and λ(Ω, m)= (Ω, m) a2Ω2 +2amΩ (3.3.4) A − For each m, both the radial equation (3.3.1) and the polar equation (3.3.2) admit a discrete spectrum of eigenvalues, parametrized by the quantum numbers n and ℓ, respectively. The separation constants and the complex frequencies Ω depend on ℓ, m and n; however, for A convenience, we will omit the indices whenever possible. Notice that the two transformations: R R = R√r2 + a2 := (3.3.5) → R F and dx r2 + a2 1 r x : = = (3.3.6) → dr ∆ F 2∆ 30 convert (3.3.1) to a Schr¨odinger-type equation of the form:

d2 R + V (x) =0 (3.3.7) dx2 R with K2 V (r(x)) = 3r2∆F 6 [2r(r M) + ∆]F 4 +( λ)F 2 (3.3.8) − − ∆ −

Notice that the potential is now frequency-dependent, so that each quasinormal mode obeys a separate 1D equation, while the time evolution of generic (multi-frequency) initial data is still governed by the full 2D equation. In the case of scalar perturbation, this equation has been solved numerically by Krivan et al. [99], and the qualitative features of the field evolution on a spherically symmetric background are encountered here as well. As in the spherically symmetric case, the quasinormal frequency spectrum has been worked out using a set of methods including continued fractions [100], phase integral methods [101] and WKB [102, 103]. Notice that, due to their dependence on the mass M and spin pa- rameter a of the background Kerr solution, the quasinormal ringing portion of the scattered waves carries information about M and a away from the scattering center (a property which, for instance, makes gravitational quasinormal ringing a powerful tool for gravitational wave astronomy [104]). In chapter 6, we use scalar perturbations precisely to probe the param- eters describing the black hole that results from the coalescence of a binary system, and its evolution from the first common horizon formation to the final equilibrium state. In order to provide as accurate a depiction of the black hole as possible, we will first have to devise a strategy to tackle the ambiguities associated with quasinormal modes and find a reliable strategy to detect and measure the quasinormal components in a waveform. We will elaborate on these considerations as part of our experimental method in chapter 6.

3.3.1 Green’s function analysis

Equation (3.3.7) can be analyzed using a technique similar to that in section 3.2.1. The results, illustrated in [93], are qualitatively coincident with those on a spherically symmetric black hole, providing a theoretical grounding for the existing numerical evidence. The three

31 field regimes (prompt response, quasinormal modes and tails) seem indeed a robust feature of black hole perturbations in general, rather than an exceptional property of solutions with a high degree of symmetry such as the Schwarzschild and the Kerr black hole. The tests carried out in chapter 6 will provide some support to the correctness of this speculation.

32 Chapter 4

General Relativity and numerical methods

Notwithstanding the hundreds of exact solutions to Einstein’s equation known to date, very few systems with direct observational significance are (to our current knowledge) susceptible to analytic solution. The nonlinear character of the equation, the lack of any prescrip- tion regarding the choice of a coordinate system and the ambiguities associated with local conservation laws and initial data all contribute to the complexity of the problem. The most notable example of the emergence of complications even in the simplest circum- stances is perhaps the two black hole problem, which represents the less involved instance of a two comparable-mass body problem in General Relativity. Whilst Post-Newtonian schemes provide a satisfactory analytical description of the problem when the two black holes are well separated, the nonlinear, strong field effects take over as the black holes move closer together, and the simple Keplerian, point-like mass approximation becomes unable to depict the distorted final black hole and the emitted gravitational radiation. As is often the case with the investigation of nonlinear phenomena, the only practicable tool consists of the direct numerical simulation of the system. In this chapter I will illustrate the application of scientific computing to the solution of Einstein’s equation, and describe some of the challenges connected to this task.

33 4.1 Scientific computing in General Relativity

As a scientific method living between analytical methods and experimental (and observa- tional) analysis, the numerical study of a physical system is a complex edifice composed of three parts:

The modelling aspect: connected to the choice of a mathematical model apt to • describe the physical properties of the system and their evolution in time, up to the desired level of accuracy;

The numerical analysis aspect: connected to the adaptation of the mathematical • model (generally living in a spacetime continuum) to the discrete nature of electronic computation. This aspect involves the discretization of the relevant equations, the quantification of the resulting error and the estabilishment of the conditions under which the discrete solution tends to the continuum one as the discretization scale tends to zero;

The computer science aspect: connected to the realization of the numerical algo- • rithms on a given architecture, including, among others, the implementation of paral- lelism and the management of interprocessor communication.

In this chapter, I will address each of these aspects and describe the combination of techniques which constitutes the current state-of-the-art recipe for the simulation of binary black hole systems.

4.2 Modelling

Given Einstein’s (four-dimensional) equation, a number of choices is required in order to turn it into a system of equations describing the time behavior of a physical system: first, as described in chapter 2, the concept of a time evolution implies a 3 + 1 separation of the equations. Quite understandably, this decomposition is not unique, and other formulations may be preferred (especially for numerical convenience) to the choice presented in chapter 2. Additionally, Einstein’s equation leads to a system of constraint equations whose satisfaction

34 must be guaranteed during the evolution. A further choice revolves around the choice of a coordinate system and its possible evolution during the simulations. Finally, as for any initial-boundary value problem, the solution must be specified at the initial time and on the domain’s spatial boundaries in order to ensure the uniqueness of the solutions to the partial differential scheme. In the following sections, I will briefly survey the common solutions to each of these issues. Before doing so, however, I comment on our choice to consider General Relativity as the only candidate theory to model astrophysical systems: whilst alternative theories of gravity are certainly still object of debate and their experimental falsification (especially through gravitational wave tests) is still an open problem, no full 3D numerical effort to model compact objects and their gravitational wave emission according to these theories is currently under way. For our present computational-driven purposes, I will therefore concentrate on General Relativity’s predictions alone, and neglect other theories of gravity.

4.2.1 Evolution schemes and constraint preservation

As observed in [83], it is an empirical result that code instabilities arise in the numerical evolution of Maxwell’s equation due to the presence of mixed temporal/spatial derivative terms, which compromise the hyperbolic character of the system. It has been argued [105] that this phenomenon is due to zero-speed constraint-violating modes, which tend to grow in place due to the non-linear character of the equations. Transforming into an explicitly wave-like system has proven to relieve this, since the constraint violations (which, we remark, are unavoidable at the numerical level) acquire a non-zero propagation speed and seem to quickly propagate outside of the grid. The mixed derivative terms can be removed in several ways: the two most popular solutions are the reduction to a fully hyperbolic system (by combining suitable time derivatives of the original equations; for a review, see [106]) or the introduction of auxiliary variables. In the following, I concentrate on the latter approach as described in [107–109], since this is one of the elements of the binary black hole evolution recipe devised a few years back [40, 41] (but see also [39] for the first successful simulations, obtained with an alternative four- dimensional formulation, and [110] for black hole orbital evolutions with a spectral code),

35 and implemented in the Penn State code. First off, notice that equations (2.3.38) and (2.3.38) can also be used to infer the evolution of the metric and extrinsic curvature’s traces γ and K, obtaining:

∂γ ij i ∂tγ = ∂tγij = γ γ∂tγij =2γ( αK + D βi) (4.2.1) ∂γij − or ∂ ln γ1/2 = αK + Diβ (4.2.2) t − i and:

ij ij ∂tK = γ ∂tKij + Kij∂tγ = = D D α + α(R 2K Kik + K2)+ βkD K − i i − ik k +2K D βk + K [2αKij Diβj Djβi]= ik i ij − − = D2α + αK Kij + βiD K (4.2.3) − ij i where the hamiltonian constraint was used to express R in terms of the extrinsic curvature in the last step. Additionally, the metric can be written as:

4 γij = ψ γ¯ij (4.2.4) where ψ (which can also be expressed as eφ) is the conformal factor. Additionally, a traceless extrinsic curvature can be defined as: 1 A := K K (4.2.5) ij ij − 3 along with its conformal counterpart:

4 A¯ij = ψ− Aij (4.2.6)

36 In terms of these new variables, the old system (2.3.38)-(2.3.38) turns into: ∂ ln γ 1 ∂ φ = t = αK + D βi = t 12 6 − i 1 1 = αK + γij£ γ =  −6 12 β ij 1 1 1 = αK + βk∂ γij + ∂ βk −6 12 k 6 k 1 1 = αK + βk∂ φ + ∂ βk (4.2.7) −6 k 6 k

4φ 4φ ∂ γ¯ = 4e− γ ∂ φ + e− ∂ γ = t ij − ij t t ij 4φ 1 k 1 k = 4e− γ αK + β ∂ φ + ∂ β = − ij −6 k 6 k   4φ 1 2 k k = 2e− α γ K K γ¯ ∂ β 4¯γ β ∂ φ + −3 ij − ij − 3 ij k − ij k   4φ k k k +e− (β ∂kγij + γik∂jβ + γkj∂iβ )) = 2 = 2αA¯ γ¯ ∂ βk +¯γ ∂ βk +¯γ ∂ βk + βk∂ γ¯ (4.2.8) − ij − 3 ij k ik j kj i k ij

1 1 ∂ K = D2α + α (A + γ K)(A + γ K)γikγjl + βiD K = t − ij 3 ij ij 3 ij i   1 = D2α + α A¯ A¯ij + K2 + βiD K (4.2.9) − ij 3 i  

4φ TF TF k ∂ A¯ = e− (D D α) + αR + α KA¯ 2A¯ A¯ + £ A¯ (4.2.10) t ij i j ij ij − ik j β ij where the TF superscript on a tensor denotes the tensor’s trace-free part. The Ricci scalar will accordingly split into two parts:

¯ φ Rij = Rij + Rij (4.2.11) where

Rφ = 2(D¯ D¯ φ +¯γ γ¯lmD¯ D¯ φ)+4 (D¯ φ)(D¯ φ) γ¯ γ¯lm(D¯ φ)(D¯ φ) (4.2.12) ij − i j ij l m i j − ij l m   and R¯ij is the Ricci tensor of the metricγ ¯ij: 1 R¯ = γ¯kl(∂¯ ∂ γ + ∂¯ ∂ γ + ∂¯ ∂ γ )+¯γkl(Γ¯mΓ¯ Γ¯mΓ¯ ) (4.2.13) ij 2 i l kj k j il k l ij il mkj − ij mkl 37 Introducing the new variables:

Γ¯i := γjkΓ¯i = ∂¯ γij (4.2.14) jk − j we obtain the relations: 1 1 γ¯ ∂ Γ¯k = ( γ¯ ∂¯ ∂ γkm γ¯ ∂¯ ∂ γkm)= γ¯km(∂¯ ∂ γ + ∂ ∂ γ ) k(i j) 2 − ki m j − kj m i 2 m j ki m i kj ¯k ¯ lm ¯k ¯ Γ Γ(ij)k = γ ΓlmΓ(ij)k   (4.2.15)

kl Using the fact thatγ ¯ ∂¯i∂jγkl = 0, we can finally write: 1 R¯ = γ¯lm∂ ∂ γ +¯γ ∂ Γ¯k +¯γlm(2Γ¯k Γ¯ + Γ¯k Γ¯ ) (4.2.16) ij −2 l m ij k(i j) l(i j)km im klj

The new variables Γ¯i satisfy an evolution equation given by: 2 ∂ Γ¯i = ∂ ∂¯ γij = ∂ ∂ γ¯ij = ∂ (2αA¯ij 2¯γm(j)∂ βi) + γ¯ij∂ βl + βl∂ γ¯ij (4.2.17) t − t j − j t − j − m 3 l l

Using the momentum constraint, this equation turns into: 2 ∂ Γ¯i = 2A¯ij∂ α 2α∂ A¯ij +2∂ (¯γm(j)∂ βi) ∂ (¯γij∂ βl) ∂ (βl∂ γ¯ij)= t − j − j j m − 3 j l − j l = 2A¯ij∂ α 2α(4e4φAij∂ φ + e4φ∂ Aij)+ − j − j j 2 +2∂ (¯γm(j)∂ βi) ∂ (¯γij∂ βl) ∂ (βl∂ γ¯ij)= j m − 3 j l − j l 1 = 2A¯ij∂ α 2α(4A¯ij∂ φ + e4φD (Kij γijK)) + − j − j j − 3 2 +2∂ (¯γm(j)∂ βi) ∂ (¯γij∂ βl) ∂ (βl∂ γ¯ij)= j m − 3 j l − j l 2 = 2A¯ij∂ α 2α(4A¯ij∂ φ + e4φDiKij 2Γ¯i A¯ki)+ − j − j 3 − jk 2 +2(∂ γ¯m(j)∂ βi) ∂ (¯γij∂ βl) ∂ (βl∂ γ¯ij) (4.2.18) j m − 3 j l − j l

Equations (4.2.7), (4.2.8), (4.2.9), (4.2.10) and (4.2.18), along with the constrain (4.2.14) due to the introduction of the auxiliary variables, constitute the Baumgarte, Shapiro [109], Shibata and Nakamura [108](BSSN) 3+1 formulation (see also [107]).

38 4.2.2 Coordinate conditions

In chapter 2 and in the above sections we have discussed how to project Einstein’s equations along a time direction ta and on a set of spatial hypersurfaces Σ, once ta and Σ (or alter- natively, α and βa) have been chosen. In principle, this choice can be completely arbitrary, and the physical observables obtained with any evolution should be identical (see Appendix A for an example). Numerical practice, however, has proven that some choices are more suitable for black hole evolutions than others. This is essentially due to the singular nature of the metric tensor at the center of a black hole, which calls for special procedures (known under the generic name of singularity avoidance) devised to lay the numerical grid only over the well-behaved region of the spacetime and away from the singular points. Singularity avoidance usually involves either the use of an inner boundary, complemented with appropriate boundary conditions, or the choice of appropriate, time-dependent α and βa. In this section we explore the second option in greater detail, illustrating some of the tradi- tional gauge conditions along with the new lapse and shift evolution recipe that is part of the moving puncture paradigm. Ideally, the goal is to choose a set of coordinates that adapt to the problem at hand, but at the same time remain reasonably well-behaved and do not lead to coordinate pathologies. Let us first start with lapse (or slicing) conditions. A traditional choice is the so called maximal slicing condition, obtained by ensuring that the trace of the extrinsic curvature be zero at all times:

K =0= ∂tK (4.2.19) This choice extremizes the volume of the spatial hypersurfaces and, since K = na, leads −∇a to divergenceless hypersurface normals. From (4.2.3), this condition can be recast as:

2 ij D α = αKijK (4.2.20) which is an elliptic equation for the lapse function. Since similar expressions are quite expensive to solve numerically, this condition is usually transformed into a more economic one, with roughly the same properties. Here we analize the alternative called the K-driver

39 Figure 4.1: Evolution of the spatial grid (here collapsed to a 2D plane) during the numer- ical evolution. At each time coordinate, the lapse and the shift are calculated to determine where each grid point is going to be positioned in the next it- eration. In binary black hole simulations, the gauge choices proposed, for instance, in [3] guarantee that the grid points remain far enough from the puncture singularities even when the puncture locations are advected through the grid.

40 condition. It is easy to convince oneself that the equation:

∂ K = cK, c> 0 (4.2.21) t − will drive K towards zero, and therefore constitutes an approximate maximal slicing con- dition. Along similar lines, equation (4.2.20) can be substituted by a parabolic equation like: ∂ α = D2α + αK Kij βiD K cK (4.2.22) λ − ij − i − where we have reinserted some terms due to the fact that K is not guaranteed to be zero, and its time derivative is equal to cK. Notice the role of λ as a relaxation parameter: on − each hypersurface, equation (4.2.22) is evolved in λ until the lapse settles down. A somewhat different approach is that adopted by the harmonic slicing condition, accord- ing to which the coordinate t must satisfy: 1 0= t = ∂ (√ ggtb∂ xa) (4.2.23) √ g b − ν − This equation can be turned into a condition for the lapse: ∂ √ g 1 t = ∂ gtb + gtb b − = ∂ gtb + gtbgcd∂ g = b √ g b 2 b cd 3 −i i = α− 2(∂ β ∂ )α + α∂ β + t − i i 1 3 i 2 ij i 2α− (∂ β ∂ )α + α− γ (∂ β ∂ )γ = 2 − t − i t − i ij 3 i 2 = α− (∂ β ∂ )α + α K  (4.2.24) t − i or:   (∂ βi∂ )α = α2K (4.2.25) t − i − A generalization of this condition is the Bona-Mass´oslicing condition:

(∂ βi∂ )α = α2f(α)K (4.2.26) t − i − 1 a Choosing f(α)= α− and β = 0 identically, this equation has an analytic solution:

α = 1 + ln γ (4.2.27)

and is therefore often referred to as the 1 plus log lapse condition (sometimes this denom- ination is extended to the case when βa = 0). 6 41 Once a slicing has been chosen, the next step is to impose a shift, or spatial gauge, condi- tion. As mentioned above, the ideal gauge prescription is dynamic enough to adapt to the system’s evolution, but not so much that its dynamics dominates and obscures the physical evolution. Following this guideline, a gauge condition named the minimal distortion has been introduced: it entails minimizing ∂tγ¯ij, which is realized as follows: first, the quantity

1/3 1/3 uij = γ− ∂t(γ− γij) (4.2.28) is introduced. Notice that uij is traceless, and can therefore be decomposed as:

T T L uij = uij + uij (4.2.29)

T T j T T where uij is such that D uij and

L 1/3 uij = γ £xγ¯ij (4.2.30)

i represents the change inγ ¯ij due to the coordinate change generated by the vector x . We L j therefore choose a gauge where uij = 0, or alternatively D uij = 0. Notice that since the evolution equation forγ ¯ij implies that: 2 u = 2αA + D β + D β + γ¯ D βk (4.2.31) ij − ij i j j i 3 ij k we have: 1 Dju = 2αDj(K γ K) 2A D α + (∆ β) (4.2.32) ij − ij − 3 ij − ij i L i where 2 (∆ β)i := D (Diβj + Djβi + γ¯ijD βk) (4.2.33) L j 3 k Similar properties are obtained by enforcing the less computationally expensive condition i i Γ¯ = 0, or the related ∂tΓ = 0 known as Γ-freezing condition. From the evolution equation for Γi, we obtain an elliptic condition on the shift vector, which again can be turned into a parabolic equation like: i i ∂tβ = k∂tΓ¯ (4.2.34) or an hyperbolic equation like: ∂2βi = k∂ Γ¯i η∂ βi (4.2.35) t t − t

42 When the latter equation is applied along with the 1+log lapse condition, it is observed that binary black hole simulation in the puncture representation tend to exhibit a memory effect, whereby the collapsed region of the lapse (necessary for proper singularity avoidance) lags behind the black hole trajectory, eventually leading to instabilities (see figure 4.2). In order to reduce this delay, the Γ-freezing condition has recently seen a number of variations, mostly consisting of the inclusion of advection terms [3, 40, 41], and in the reduction of the hyperbolic condition to a first-order form. The generic form of these new shift conditions can be written as: 3 ∂ βi = αλBi + µβj∂ βi (4.2.36) t 4 j ∂ Bi = ∂ Γi ηBi + γβj∂ Γi + µβj∂ Bi (4.2.37) t t − j j where λ, µ, η and γ are constants. These variations have proven extremely successful at maintaining a slicing and spatial gauge that allow for instability-free, long-term black hole evolutions in a variety of initial configu- rations [3].

4.2.3 Initial data

The choice of initial conditions for a pair of black holes (i.e., the choice of the spatial metric and the extrinsic curvature at the initial time) must satisfy two criteria:

1. It must satisfy the constraint equations as introduced above;

2. It must represent the binary system with the desired physical properties, such as energy, angular momentum, individual black hole spins, among others;

Subject to these conditions, there are many possible choices, differing in computational complexity [111]. In terms of the conformal factor and the traceless extrinsic curvature, the constraints can be expressed as:

2 5 2 7 ij 8Dψ¯ ψR¯ ψ K + ψ− A¯ A¯ =0 (4.2.38) − − 3 ij 2 D¯ A¯ij ψ6γ¯ijD¯ K =0 (4.2.39) j − 3 j 43 Figure 4.2: Lapse function for four gauge types, on the x-y plane, for a single boosted puncture. The λ and γ values refer to the constants in equations (4.2.36) and (4.2.37), while µ = 0 and η = 4. In the case when λ = 1 and γ = 0, the collapsed-lapse region does not track the puncture location efficiently, lagging behind in the evolution and generating the stretched contour shown in the top right panel.

44 For binary black hole systems, a customary (yet at times disputed) choice includes conformal

flatness (¯γij = 0) and maximal slicing (K =0) [83], which transforms the constraints into:

7 ij 8∆flatψ + ψ− A¯ijA¯ =0 (4.2.40) ij D¯ jA¯ =0 (4.2.41)

The second of the above equations is solved identically by the Bowen-York formula:

K ¯ij 3 i j j i ij i j l A = 2 P(k)n(k) + P(k)n(k) +(η + n(k)n(k))P(k)ln(k) (4.2.42) 2r(k) Xk=0   where P(k) is an arbitrary vector, r(k) is the location of the k-th black hole center and n = x /r ,y /r , z /r . (k) { (k) (k) (k) (k) (k) (k)} This solution can then be inserted in equation (4.2.38), which turns into an elliptic equation for the conformal factor ψ, which in the Penn State code is solved following [112], thanks to the module TwoPunctures.

4.2.4 Boundary conditions

Boundary conditions have a far deeper role than just providing a grid function value for those points where finite differencing is not applicable. On grids that do not extend to infinity, these conditions are also expected to provide the information relative to the solution from the missing, exterior part of the physical domain. In order for the evolution system to be well-posed, appropriate boundary conditions must constrain the incoming modes, and only them. If the conditions are not sufficient to constrain the incoming modes, the system will be underdetermined; if, viceversa, the conditions extend to the modes evolving within the boundaries, or tangent to them, the system will be overdetermined. The first step towards the construction of appropriate boundary conditions is the determina- tion of the characteristic structure of the system, including which modes are not determined by the partial differential equation and should therefore be constrained by the boundary con- ditions. While such an approach is relatively straightforward for first-order, linear systems, it is unfeasible for the BSSN system, and boundary conditions usually consist of setting the grid function values to a constant and making sure that the boundary is causally disconnected from the inner region, where the physical system is evolved.

45 4.3 Numerical analysis

Once a mathematical model of the black hole systems has been estabilished, we need to turn our attention to the numerical integration of the corresponding partial differential equations, which is the subject of section 4.3.1 below. Aside from the algorithm needed for this task, the extraction of the complex quasinormal frequencies from the scalar field waveforms requires appropriate optimization techniques; these will be the subject of section 4.3.2.

4.3.1 Integration of partial differential equations

The numerical solution of partial differential equations is a long-standing field whose con- tributions have proved crucial in countless scientific areas. When closed-form solutions are unavailable and analytical approximations fail, the only viable means to gain insight into a system consists of the numerical integration of its governing equations. Teaming up with the explosive increase of computer resources over the last few decades, numerical techniques have contributed to the development of nearly every branch of physics: from fluid dynamics to material science, from nuclear and subnuclear matter to cosmological large scale simulations. Numerical analysis includes techniques like finite differencing, finite elements, spectral meth- ods and Monte Carlo integration. Given the vastity of the field, there is obviously no hope to do justice to it all in a single section. I will therefore concentrate only on those aspects that are relevant to the code utilized in this thesis.

4.3.1.1 Finite differencing schemes

Let’s consider a first-order-in-time system of partial differential equations for the fields u1(x,y,z,t),u2(x,y,z,t),...,uq(x,y,z,t) =: u(x,y,z,t):

Lu(x,y,z,t) :=(∂t + S)u(x,y,z,t)=0 (4.3.1) where S is a certain spatial differential operator. Given a set of initial data u(x,y,z,t0) (and possibly some condition for u(x,y,z,t) on the spatial boundaries), a traditional strategy to compute the system’s solution is to resort to finite differencing methods, a class of PDE integration techniques which follows the following generic recipe:

46 The spacetime domain is substituted by a I J K N four-dimensional lattice: • × × ×

xi = x0 + i∆x i =1,...,I

yj = y0 + j∆y j =1,...,J

zk = z0 + k∆z k =1,...,K

tn = t0 + n∆t n =1,...,N (4.3.2)

and the fields u(x,y,z,t) are replaced by the grid functions uijkn := u(xi,yj, zk, tn). Notice that while this coordinate discretization usually implies reducing the infinite spatial domain to a finite extension (discarding those portions that are uninteresting from the physical standpoint), this reduction is by no means necessary. A coordinate transformation like φ,ψ,ζ = arctan( x, y, z ), performed prior to the discretization { } { } step, ensures that a finite lattice (in φ, ψ, ζ) covers a infinite domain (in x, y, z). This type of spatial compactification is all but new to Numerical Relativity: some examples of codes that adopt it are described in [39]. If, however, the lattice does cover a finite spatial region, appropriate conditions at the lattice boundaries must be imposed in order to make up for the missing information that results from ignoring the external regions and ensure the well-posedness of the resulting system.

On the lattice, the derivatives of u are represented as differences between the values of • the grid functions at neighboring sites. Using Taylor expansions, one can for instance write: ∆x2 u = u + ∆x∂ u + ∂ u + O(∆x3) i+1,jkn ijkn x |ijkn 2 xx |ijkn 2 ∆x 3 ui 1,jkn = uijkn ∆x∂xu ijkn + ∂xxu ijkn + O(∆x ) (4.3.3) − − | 2 | and solve for ∂ u : x |ijkn 1 2 ∂xu ijkn = (ui+1,jkn ui 1,jkn)+ O(∆x ) (4.3.4) | 2∆x − − The polynomial order of the remainder can be increased by using additional lattice points. For instance, it can be easily verified that another expression for ∂ u is: x |ijkn 1 4 ∂xu ijkn = ( ui+2,jkn +8ui+1,jkn 8ui 1,jkn + ui 2,jkn)+ O(∆x ) (4.3.5) | 12∆x − − − − 47 Similar expansions hold for higher order derivatives. The scheme used to represent derivatives is usually referred to as the (differencing) stencil.

Once all the derivatives have been transformed into finite differences, the original • system turns into a set of algebraic conditions on the lattice values of u, usually referred to as its finite difference approximation, and represented symbolically as

L(∆)u =0 (4.3.6) |ijkn where ∆ represents the vector of lattice spacings. Depending on the form of L(∆) and the procedure used to solve it, the finite differencing method is classified as explicit (when the variables to solve for can be expressed explicitly in terms of known terms) or implicit (where an implicit condition relates known and unknown variables).

4.3.1.2 Convergence to the continuum solution

Naturally, the question arises as to whether (i.e., under which conditions) and how faithfully the solutions to the finite difference equations represent the solutions to the original system of partial differential equations. In order to address this point, let’s concetrate first on the latter issue: the estimate of the errors introduced by finite differencing. This analysis will turn out to provide an answer to the former question: the conditions for convergence of the discretized solutions to the continuum ones. Let’s introduce a few definitions: first, if u is a solution to the partial differential system and u(∆) is the corresponding (i.e., generated from the same initial conditions) solution to the finite difference scheme obtained with a lattice spacing ∆, we call solution error the difference between the two: e(∆) = u u(∆) (4.3.7) − The truncation error is instead defined as:

τ (∆) := L(∆)u (4.3.8)

Since the continuum solution is, for all cases of interest, unknown, neither of these errors can be calculated. However, given a certain L(∆), it is possible to work out the polynomial

48 dependence of the truncation error on the lattice spacing. Additionally, it has been proven that the ansatz: τ (∆) = O(∆ ) e(∆) = O(∆ ) (4.3.9) p → p holds for a large class of systems [113, 114], so that a polynomial form for the solution error is also available for most reasonable PDEs. This observation is the key to answer our original questions on the convergence of the FDA (∆) p solution to the continuum one: if e = O(∆ ), then given two lattice spacings ∆1 and ∆2, one can write, to leading order:

u(∆1) u + e ∆p ≃ p 1 u(∆2) u + e ∆p (4.3.10) ≃ p 2

where the factor ep does not depend on resolution. If ∆1 and ∆2 are sufficiently small, one

can then solve the above system for the two unknowns u and ep and get an estimate of the continuum solution u. Notice, however, that without an estimate of the prefactors multiplying the next-to-leading order terms, there is no a priori knowledge of the magnitude of the remainders, and thus no guarantee that the continuum solution u derived by solving the above system is of any suitable accuracy. Additionally, real codes that implement FDA algorithms rarely provide the scientist with the opportunity to isolate the errors due to finite differencing from other sources of error (such as those originating in the mesh refinement process), so that in principle nothing guarantees that the error due to finite differencing (even in its untruncated form) is not swamped by other sources of uncertainty. A relatively accessible method to assess that the ∆p term is indeed the dominant source of error and that the extrapolation is carried out according to the correct power law has been proposed almost a century ago by L.F. Richardson [114]: it suffices to compute the FDA

solution for a third spacing ∆3 in order to provide a consistency check that the behavior of the solution error with the spacing ∆ is indeed a power law with the exponent equal to p

(and the same prefactor ep). This is usually referred to as a convergence test. Choptuik [113] provides a simple formulation of this check: if equation (4.3.10) holds for

49 three resolutions, then the quantity:

(∆1) (∆2) uijkn uijkn Qijkn = − (4.3.11) u(∆2) u(∆3) ijkn − ijkn should tend to: ∆p ∆p 1 − 2 (4.3.12) ∆p ∆p 2 − 3 for every set of indices ijkn as ∆1, ∆2, ∆3 tend to zero. The condition (4.3.12) is (in principle) straightforward to test and constitute the standard verification strategy for Nu- merical Relativity codes nowadays. The corresponding extraction of the continuum solution u (or, sometimes, the power-law test and extraction procedure as a whole) are referred to as Richardson extrapolation [114].

4.3.2 Function minimization

As discussed in chapter 3, the evolution of a scalar field on a black hole background exhibits a phase known as quasinormal ringing, where the field decays through a period of damped oscillations. The associated spectrum of complex frequencies depends on the mass and spin of the black hole, so that its extraction can be used to gain information on coordinate- independent properties of the black hole. However, the extraction of complex exponentials from a numerical (i.e., intrinsically noisy) waveform is a particularly involved task, on which standard minimization methods often fail to converge or lead to significant biases in the estimated parameters. Since the extracted mass and spin exhibit a pronounced dependence on the complex frequency, and any uncertainty in the latter is extraordinarily amplified in the mass and spin estimates, it is necessary to reduce as much as possible the frequency extraction errors. In light of the recent advancements in binary black hole simulations and the prospect of studying gravitational mode coupling, this issue has been addressed with the introduction to the community of the Kumaresan-Tuft and matrix pencil methods [4, 96]. The class of algorithms used for frequency extraction includes:

Least-squares methods: the implementation of a least-squares method leads, for • complex exponentials, to a system of non-linear equations that requires numerical

50 integration, usually through gradient descent or Newton’s method. These strategies has been observed to produce misconvergences if the value of the initial guess is not close to the global minimum. Furthermore, most iterative methods involve large matrix inversions, and are thus remarkably expensive;

Prony method: an attempt to transform the minimization problem into a linear • system was realized by Prony: given a sum of complex exponentials, calculated on a lattice t = nT , n =1, 2,...,N: { n } K K (iωk αk)tn+φk n 1 Φ(tn)= Ake − = hkzk − (4.3.13) Xk=1 Xk=1

iφk (iωk αk)T we can use the dataset to solve for the 2K variables hk = Ake and zk = e − , provided that the number of points N is at least as large as 2K. In the original Prony method, the second half of the dataset (K +1 n N) is used to determine ≤ ≤ the variables zk through the construction of the characteristic polynomial A(z) := K (z z ) and a forward linear prediction scheme. Alternatively, one could k=1 − k Quse the first half of the data and a backward prediction scheme. Once the zk are solved for, the hk can be derived and one can easily recover the original unknowns Ak,

φk, ωk and αk. The fit is then reduced to the solution of a linear system, which is reasonably economic even for large datasets.

Kumaresan-Tuft method: unfortunately, the Prony method has been proven to lead • to substantial biases in the estimated parameters when the dataset’s signal-to-noise ratio is not small. In those cases, a variation of this strategy, called the Kumaresan- Tuft method, seems to yield better results. Briefly, increasing the prediction order (i.e., the number of exponentials in the sum) has proven to be extremely effective to reduce the bias in the estimated parameters. This, of course, comes at the price of increasing the number of zeros of the characteristic polynomial, but a combination of forward and backward prediction successfully helps to identify the true frequencies contained in the signal from the spurious roots (this is due to the fact that switching from forward to backward prediction -i.e., performing a time reversal- maps the true roots to their complex conjugates, whereas the spurious roots due to noise are left unchanged).

51 Matrix pencil method: this method is also an extension of the Prony method, based • on the construction of a matrix pencil (i.e., a matrix-valued function of z based on two matrices A and B, defined as A zB) from the dataset, and its use instead of the − predictor methods above.

We have chosen to follow a somewhat different, more heuristic approach to the problem by deploying an optimization algorithm called the Covariance Matrix Adaptation Evolution- ary Strategy (CMA-ES). Evolutionary computation has expanded enormously over the past few decades, establishing itself as one of the premium optimization techniques for problems ranging from engineering applications [115] to gravitational wave data analysis [116]. Evo-

lutionary algorithms approach the minimization of a function f(xi) from the black box optimization standpoint: all the information regarding the function must be constructed entirely from evaluating the function over its domain, with no direct access to properties such as its gradient field. In this sense, the function is nothing more than a black box transform-

ing the vector xi into the (usually, real) value f(xi), with all other details (differentiability, invertibility, et cetera) hidden. This principle constitutes both the benefit and the limitation of these algorithms: on one hand, they can be applied (and usually perform cleverly) on the most general optimization problems, including seriously discontinuous functions and functions of discrete variables. At the same time, however, the only means to search for the minimum is the iterated sampling of the function at selected points; needless to say, a carefully crafted strategy is then required to ensure convergence to the global minimum with the smallest possible number of function evaluations. A detailed description of CMA-ES can be found in [117], while its performance on black hole quasinormal ringing waveforms is discussed in appendix B.

4.4 Computer science

As application development extends to more and more complex projects, the need to main- tain a strong consistent control over the quality of the end product demands for a shift in the way traditional programming is organized: first, in order to avoid code duplication and inconstistency, modularity is introduced. Second, functions and methods are appropriately encapsulated, in line with the object-oriented philosophy.

52 This paradigm of computing has provoked the development of the so-called application frameworks; following the definition in [118]:

An application framework is a set of interconnected objects that provides the basic functionality of a working application, but which can be easily specialized (through subclassing and inheritance) to individual applications. An application framework not only allows the reuse of code as a class library, but also the reuse of design structures, because the dependencies between objects are preimplemented through predefined object composition possibilities, event dispatching mechanisms and message flow control.

In other words, an application framework is a prototype of a software application based on a common design and set of development tools, which provides a ”blueprint“ for the application’s structure and workflow: based on this structure, the user will only need to implement the required functionalities in specific software modules, which interact with the infrastructure and between each other according to a predefined set of rules. Using an application framework for software development carries a number of benefits [119]:

Since user modules are programmed according to a standard, predefined structure, • application frameworks simplify code sharing and collaborative efforts, which have become the rule rather than the exception in contemporary software development;

The encapsulated structure also allows application developers to concentrate on the • high-level details of the functionalities to be implemented, while the infrastructure han- dles most of the low-level tasks such as parallelization, data structures, input/output, et cetera. Cross-platform application frameworks also manage the encapsulation of architectural differences under the infrastructure’s hood, making them invisible to the application developer;

The abstract representation of the application’s components, which lies at the base • of the application framework concept, allows the component’s implementation and technology to be changed at any time, without affecting the high-level functionalities provided by the application modules.

53 Application frameworks (often of the open source type) are a diffuse feature of code devel- opment communities, from cross-platform application development to web interfaces, from e-business processes and database management to scientific codes. Cactus is an open source, parallel, modular application framework for the evolution of 3D partial differential equations. Conceived in 1997 and developed within the Numerical Relativity community ever since, this platform provides a basic design based on the following components:

A backbone, referred to as the flesh, which provides the application’s core structure, • providing instructions for how to bind the different modules together, how to control the program flow and furtherly supplying a set of libraries for common tasks such as message passing and data structure definition;

A set of modules, referred to as thorns and furtherly divided into infrastructure • thorns (deputed to purely computational tasks such as I/O management, domain decomposition or interpolation) and application thorns (which implement scientific tasks such as the initial data evolution, the calculation of physical observables or the enforcement of boundary conditions); the thorns access the functionalities provided by the flesh, and can inherit or share grid functions between one another, forming a single cohesive ensemble. Also for cohesion purposes, several thorns providing affine functionalities may be grouped in the so-called arrangements.

Two examples of codes built with Cactus (one for the evolution of the BSSN system and one for the evolution of the scalar wave equation) will be presented in chapter 5.

54 Chapter 5

Gravitational and scalar field evolution codes

Over the past thirty years, numerical evolutions of binary black hole systems have provided deeper and deeper insight into the properties of the vacuum two-body problem of General Relativity. Today a number of codes [39–41, 54, 56, 61, 67, 110] are able to accurately evolve black hole binaries. In particular, the Penn State code is now capable of evolving binary sys- tems of black holes in a broad interval of mass ratios [54], spins [60, 68] and eccentricities [53]. In this dissertation, the code is used to generate the metric tensor of a quasi-circular, equal- mass, non-spinning black hole binary system, to be used as the background for the scalar field propagation. Before turning to the numerical experiments, in this chapter I will describe the two main tools used in this study: the BSSN code used to evolve the black hole binary system (section 5.1) and the scalar field code used to evolve the field on the black hole geometry (section 5.2).

55 5.1 BSSN evolution code

5.1.1 Scope

The Penn State code carries out the evolution of the gravitational variables γij and Kij according to the BSSN system (see section 4.2.1), as implemented in the PSU code through the Kranc generator [120]. The code integrates partial differential equations with fourth order finite differencing, based on the Cactus framework [121]; it is also endowed with Carpet’s mesh refinement capabilities [122]. The gauge variables α and βi are initialized and evolved according to the moving puncture prescription [40, 41], while the initial data are generated following [112].

5.1.2 Structure

The following is a list of commonly used functionalities, along with their thorn implementa- tion:

Domain specification: the task of specifying a grid is managed by the thorn Cart- • Grid3D, that provides the interface for prescribing the lattice spacing, the grid extent and location, possible periodicities in the coordinate axes, and so on. Auxiliary thorns include CoordBase (provide coordinate labels to the grid points), Boundary (generic interface for the application of conditions to the boundary points), Slab (extraction of grid “hyperslabs”, i.e. 3-dimensional subdomains of the base grid), along with a few thorns deputed to symmetry conditions (such as SymBase, ReflectionSymmetry and RotatingSymmetry180);

Mesh refinement: in order to ensure that the numerical solution is appropriately • resolved, the user may decide to refine one or more regions of the base grid, i.e. to locally reduce the spatial and temporal lattice spacing by adding intermediate points. In the Penn State code, mesh refinement is realized according to the Berger-Oliger algorithm [123], implemented in the arrangement Carpet [124]. Following this pro- cedure, a set of refinement levels (i.e., additional grids with smaller lattice spacing) are superimposed to the base grid in those regions that require an increased resolution

56 (such as the locations of the black holes), and refinement levels can in turn have their own subrefinement. In order to apply finite differencing at each level’s boundary, a set of points (called ghosts) are added to the grid, their number depending of course on the width of the differencing stencil. Additionally, adding a so called buffer zone has proved to enhance the convergence properties of the discretized solutions [122]. Each grid (inclusive of its buffer points and ghost zones) must be entirely contained within its parent (i.e, properly nested) in order for the resulting grid hierarchy to be valid. Interaction between refinement levels (including, among others, filling the buffer and ghost zones) that cover the same coordinate region is carried out through the operations of prolongation and restriction. Carpet includes a set of tools and libraries to carry out the above grid management, the I/O and the checkpointing and recovery procedure. In recent years, Carpet’s functionalities have been extended to allow for moving refinement levels [60], e.g. to track the trajectories of the black holes, thus introducing some degree of mesh adaptivity.

Time integration: one of Cactus most widely used infrastructure thorns is MoL, that • implements the method of lines time integrator. In a nutshell, this method implies discretizing only the spatial differencial operator S in equation obtaining an relation of the type:

∂tuijk(t)= s(uijk(t)) (5.1.1) This is nothing but a coupled ordinary differential equation for the vector of func-

tions uijk(t), and can be integrated with any standard ODE integrator. MoL’s options are Runge-Kutta (of several orders), iterative Crank-Nicholson plus a customizable scheme with user-specificied stencil.

Evolution equations: in the Penn State code, the right-hand-side of equation (5.1.1) • is discretized by the Kranc package, which reads the relevant PDEs from a user- provided Mathematica script (which can also contain information about the desired differencing stencil, the initial conditions and the possible interactions with other Cac- tus modules) and generates a thorn with the right-hand-side terms for integration with MoL. Additionally, since most equations make use of ADM variables, the ADMBase thorn (distributed along with Cactus) has the function of a mediator among the different evo- lution modules, and between those and other parts of the code (such as initial data

57 generators and postprocessing tools).

Initial conditions: whilst the standard Cactus distribution includes a number of • initial data modules for simple black hole configurations, the generation of constraint- satisfying initial data requires (as we have described above) the solution of an elliptic system. This is realized in the Penn State code through Marcus Ansorg’s module TwoPunctures [112].

Postprocessing: after evolving the relevant grid functions, it is customary to post- • process them in order to obtain physical information and observables from the simula- tion. The Penn State code is endowed with tools for the search (AHFinderDirect) and analysis (IsolatedHorizon) of apparent horizons, the computation of the Newman-

Penrose scalar Ψ4 (WeylScal4) and the associated radiative properties (Psi4Analysis), and the computation of ADM energy and linear and angular momenta (ADM_EJP). These thorns are supported by modules for spherical mode extraction, black hole tra- jectory tracking, and several others.

Figure 5.1 shows the inheritance plot for this code.

Figure 5.1: Structure and interdependencies of the BSSN evolution code used in the runs of chapter 6

58 5.2 Scalar field evolution code

5.2.1 Scope

As illustrated in chapter 3, the behavior of a massless scalar field Φ on the background provided by a Kerr black hole of mass M and Kerr spin parameter a = J/M is governed by the source-free scalar wave equation:

Φ := g a bΦ=0 . (5.2.1) ab∇ ∇ ij i which, as discussed in chapter 2, can be decomposed in terms ofγ ¯ij, K¯ ,α ¯ and β¯ as follows:

k (∂τ β¯ ∂k)Φ =α ¯Ψ (5.2.2) − α¯ (∂ β¯k∂ )Ψ = ∂ (√γ¯γ¯ij∂ Φ)+α ¯K¯ Ψ+¯γij∂ α∂¯ Φ (5.2.3) τ − k √γ¯ i j i j We evolve these equations on a numerically generated background obtained from a binary black hole evolution: during the course of the evolution, we store the values of the grid functions corresponding to the gravitational variables at a sequence of coordinate times t , which we denote with a bar: { N } γ¯ (xk) γ (t , xk) (5.2.4) ij ≡ ij N K¯ (xk) K (t , xk) (5.2.5) ij ≡ ij N α¯(xk) α(t , xk) (5.2.6) ≡ N β¯i(xk) βi(t , xk) (5.2.7) ≡ N and evolve a scalar field on the background provided by the corresponding frozen geometry.

5.2.2 Structure

The scalar field code has also been built using the Cactus framework; the main evolution routines are contained in the ScalarWave thorn, which implements a first order in time, second order in space wave equation for a scalar field on a curved background as in equations (2.4.2) and (2.4.3). Figure 5.2 shows the inheritance plot for this code.

59 Figure 5.2: Structure and interdependencies of the scalar evolution code used in the runs of chapter 6

ScalarWave was generated using Kranc [120], and also provides the interface with the BSSN

code and storage for α, β~, γij and K as one-timelevel grid functions for use in the calculation of the right-hand sides. The values of the background metric variables can be set along with the initial data by one of the internal calculations, or can be imported from any Cactus run that makes use of the ADMBase thorn. In detail, the thorn includes the following calculations: Initial data calculations:

Plane wave profile (parameter: wlength, IDShift); • Φ(z, 0) = cos(2πz/λ) (5.2.8) 1 i Ψ(z, 0) = 2π sin(2πz/λ)/λ + SIDα− β ∂iΦ (5.2.9)

1D gaussian profile (parameters: sig, z0, IDShift); • − 2 (z z0) Φ(z, 0) = e− 2σ2 (5.2.10) (z−z )2 z z0 0 1 i Ψ(z, 0) = − e− 2σ2 + S α− β ∂ Φ (5.2.11) σ2 ID i

60 3D gaussian profile in Φ (parameters: sig, r0, IDShift, clm); • 2− 2 r r0 4 2 Φ(~r, 0) = r e− 2σ cℓmYℓm (5.2.12) Xℓ,m 2 1 i Ψ(~r, 0) = 2(r r )Φσ + S α− β ∂ Φ (5.2.13) − − 0 ID i

3D gaussian profile in Ψ (parameters: sig, r0, clm); • Φ(~r, 0) = 0 (5.2.14) 2− 2 r r0 4 2 Ψ(~r, 0) = r e− 2σ cℓ,mYℓm (5.2.15) Xℓ,m Background calculations:

Minkowski spacetime; • Single black hole spacetime: • – Schwarzschild coordinates; – Estabrook coordinates;

Numerical background (imported through ADMBase); •

Evolution calculations: implementing the system (2.4.2)-(2.4.3) either in second order or fourth order accuracy (either in the standard derivatives alone or in the advection ones as well); Boundary calculations: implementing Sommerfeld’s radiative boundary conditions.

5.2.3 Code verification

In the following, I present a few test beds carried out to verify the scalar evolution code in scenarios where the exact answer is known via analytical methods. For a convergence study on the actual problem we seek to solve (a scalar field propagation on the background of a binary black hole merger), see section 6.3.

61 5.2.3.1 Flat space tests

The evolution of a plane wave (5.2.8)-(5.2.9) with periodic boundary conditions leads to a fourth order convergent evolution for timescales equal to a few periods. The initial profile, with wlength=2.5, was evolved on an unrefined grid 0.5

Φ(z, t) = cos(2π(z t)/λ) (5.2.16) − In this case, the convergence factor (4.3.11) expected for a fourth order scheme is:

4 Φ(∆1) Φ(∆2) ∆4 ∆4 ∆ Q = − 1 − 2 = 1 1=15 (5.2.17) Φ(∆2) Φ(cont) ≃ ∆4 ∆ − − 2  2  In figure 5.4, the pointwise difference between the three solutions is shown at a sequence of times during the evolution.

20 15 10 Q 5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 t/λ

Figure 5.3: Convergence factor for the plane wave. After a rapid initial transient, the value of Q settles, as expected, to 15.

62 t/λ = 0 t/λ = 0.8 8e-14 ulow umid 1.4e-08 ulow umid 7e-14 15 u| −u | 15 u| −u | · | mid − cont| 1.2e-08 · | mid − cont| 6e-14 1e-08 5e-14 8e-09

Φ 4e-14 Φ 3e-14 6e-09 2e-14 4e-09 1e-14 2e-09 0 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 z/λ z/λ

t/λ = 1.6 t/λ = 2.4 3e-08 4.5e-08 u u u u | low − mid| 4e-08 | low − mid| 2.5e-08 15 umid ucont 15 umid ucont · | − | 3.5e-08 · | − | 2e-08 3e-08 2.5e-08

Φ 1.5e-08 Φ 2e-08 1e-08 1.5e-08 1e-08 5e-09 5e-09 0 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 z/λ z/λ

t/λ = 3.2 t/λ = 4.0 7e-08 5e-08 ulow umid ulow umid 15 u| −u | 6e-08 15 u| −u | · | mid − cont| · | mid − cont| 4e-08 5e-08

3e-08 4e-08 Φ Φ 3e-08 2e-08 2e-08 1e-08 1e-08 0 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 z/λ z/λ

Figure 5.4: Pointwise convergence for the plane wave, at t/λ = 0, 0.8, 1.6, 2.4, 3.2, 4.0. The difference between the two highest resolution runs is multiplied by the appropriate factor for a fourth order code.

63 A second test involved the evolution of an initial 3D gaussian profile with Sommerfeld’s radiative boundary condition: ∂ ∂ + (rΦ)=0 (5.2.18) ∂r ∂t   or: ∂Φ Φ ∂Φ = (5.2.19) ∂t − r − ∂r The test was carried out on a uniform grid ([0 : 1]3), with resolutions ∆x = ∆y = ∆z = 0.02. The pulse is initially centered at the origin with a spread sig equal to 0.3. The finite differencing stencil is fourth-order accurate in the interior points, whilst the boundary conditions are implemented with second-order accurate one-sided derivatives. The global convergence rate is second order (again we used the original resolution, a resolution two times higher and the continuum solution).

5.2.3.2 Single black hole tests

With the same techniques illustrated in chapter 6 for the binary black hole runs, I evolved a single Schwarzschild black hole in isotropic coordinates (2.1.9):

(1 M/2R)2 M ds2 = − dt2 +(1+ )4(dR2 + R2dθ2 + R2 sin2 θdφ2) (5.2.20) −(1 + M/2R)2 2R

The slicing was carried out according to the standard moving puncture recipe, which leads to coordinate dynamics even though the spacetime is itself static. I use a spatial resolution of M/50 near the puncture location, and utilize several mesh refinement levels to place the outer boundaries at 76.8M. I evolve this initial data set, freeze the gravitational variables at coordinate time t = 40M and evolve the scalar wave equation on the corresponding background. The resulting waveform for the ℓ = 2, m = 0 mode is shown in figure 5.5. The frequency extraction shows that the values of the real and imaginary part of the quasinor- 1 mal frequency coincide with the result from perturbation theory (equal to ω =0.4836M − , 1 1 1 α = 0.0967M − ) up to ∆ω . 0.002M − and ∆α . 0.002M − (see figure 5.6), which corre- sponds to ∆M . 0.01 and ∆j . 0.05 for the ℓ = 2 mode. M

64 100 10 1 )

τ 0.1 ( 0.01 20 0.001 Φ 0.0001 1e-05 1e-06 0 10 20 30 40 50 60 τ/M

Figure 5.5: The angular mode corresponding ℓ = 2, m = 0 on a single Schwarzschild black hole spacetime, using the ”frozen background” approach described in chapter 6 (the time variable is denoted by τ to highlight its distinction from t). The expected value for the real and imaginary part of the frequency is ω = 0.4836M 1 (which corresponds to a period of about 13 ) and α = − M 1 0.0967M − .

0.49 0.1

0.485 0.095 | 20 0.48 20 0.09 ω α |

0.475 0.085

0.47 0.08 0 10 20 30 40 50 60 0 10 20 30 40 50 60

τ0 τ0

Figure 5.6: The real and imaginary part of the extracted complex frequency.

65 Chapter 6

Probing the binary black bole merger regime with scalar perturbations

As mentioned in chapter 5, a number of codes [39–41, 54, 56, 61, 67, 110] are able to accurately evolve black hole binaries; in particular, the merger regime itself is now available for studies. It was shown in reference [11, 96] that the merged system seems to enter the perturbative stage while it is still evolving and radiating energy and angular momentum. This is in agreement with earlier semianalytical studies [125, 126]. In order to study the transition between the nonlinear merger stage and the final ringing black hole, we scatter a test scalar field off the geometry generated in the inspiral and merger of binary system. Test fields and perturbation theory have traditionally been used to study excitations on a single stationary black hole spacetime, both for the Schwarzschild [14, 15, 127] and for the Kerr solutions [98]. Numerical methods were used in several scenarios (see for instance [99] and [94]); more recently, these results have been extended to more general black hole solutions, including non-vacuum spacetimes [128] and models with non-flat asymptotic topology [129, 130]. Additionally, perturbative methods have proven accurate not only for spacetimes which can be rightfully considered a perturbation of a single black hole, but even for cases which, at a first glance, would seem to depart significantly from it. A paradigmatic example is represented by the Close Limit approximation [125]: casting the Misner initial data for a pair of black holes [131] into a perturbative Schwarzschild form (with perturbative parameter

66 ǫ =1/ ln µ , where µ is a parameter quantifying the separation between the black holes), | 0| 0 and evolving the corresponding Zerilli function, Price and Pullin obtain the total radiated power as a function of the expansion parameter, which matches the 3D numerical results out to surprisingly large values of ǫ (possibly even for configurations where the black holes are so far from each other that no common apparent horizon surrounds them). Price and Pullin suggest that this agreement can be explained by recognizing that the gravitational radiation moving to infinity can only be influenced by the portion of the spacetime outside the event horizon. Therefore, as long as the horizon and the exterior geometry are approximately static and spherical, the nonspherical interior dynamics can safely be ignored and the emitted radiation should be expected to be consistent with a perturbed Schwarzschild black hole. Based on these observations, the scattering of a test scalar field off the merged (i.e., sur- rounded by a common apparent horizon) system is expected to result in the behavior pre- dicted by perturbation theory (including the QN ringing phase), disregarding whether the spacetime has completely settled down to a single Kerr solution. The novel contribution of our work lies in the fact that additional information can be extracted from the system during this relaxation stage: in particular, our results show that the mass and spin variation that occur during the relaxation leave an imprint on the scalar QN frequencies. As a side remark, the generalization to a non-stationary, spherically symmetric black hole spacetime has an interesting property [128]: the mass extracted from the complex quasi- normal frequency tracks the time-dependent black hole mass quite closely (modulo a delay effect), equaling the constant quasinormal frequency one would obtain on a static spacetime with the mass parameter equal to the instantaneous mass at each time [128]. This adiabatic picture has an intriguing parallel in black hole thermodynamics: the application of the first law to a spacetime containing an axisymmetric dynamical horizon shows that the mass in-

crease between two cross sections with horizon area A1 and A2 and angular momentum J1 and J is given exactly by M(A ,J ) M(A ,J ), where M(A, J) represents the functional 2 2 2 − 1 1 dependence of the mass M on A and J for a Kerr solution [132]. In other words, as far as the first law of thermodynamics is concerned, this portion of the spacetime can formally be pictured as a stack of spatial hypersurfaces, with each slice belonging to a member of the Kerr family with a different mass and spin parameter. In section 5.2.1, I presented the method we used to evolve the scalar wave equation on a

67 numerically-generated curved background; in this chapter, I will discuss the results of the scalar field evolution, and illustrate the complex frequency extraction, and the corresponding black hole mass and spin estimate. Section 6.1 contains the details of the background spacetime, including their gravitational wave contents and some independent mass and spin estimates. Section 6.2 shows the scalar field initial data and waveform, along with the extraction procedure, and section 6.3 attempts to quantify the errors involved in this analysis. Finally, in our concluding section 6.4, we will discuss the results.

6.1 Binary black hole background

For the binary coalescence runs, we choose an initial configuration usually referred to as R1 [43]: this setup represents two non-spinning black holes with irreducible masses equal to 0.505 (where denotes a mass scale related to the ADM mass via E / =0.9957). M M ADM M The two black holes start at a coordinate separation of 6.514 , with linear momenta perpen- M dicular to the separation vector and equal to 0.133 (from the effective-potential condition M for quasi-circularity [133]). Further details regarding our code and our earlier study of this system can be found in [134]. We used a reference spatial resolution of /51.2, and evolved M some simulations at two extra resolutions ( /44.8 and /57.6) to obtain an estimate for M M the truncation errors (see section 6.3 below). In figure 6.1, I plot the two black hole trajectories, while figures 6.2 and 6.3 shows the gravitational wave from the simulated system: the two black holes inspiral around each other for about two orbits before plunging and forming a single final perturbed black hole whose parameters can be inferred, due to energy and angular momentum conservation, from

the emitted radiation in terms of the Newman-Penrose scalar Ψ4. In both of these plots, the curves corresponding to the three different resolutions are indistinguishable.

68 4

2

M 0 y/

-2

-4 -4 -2 0 2 4 x/ M

7 6 5 M /

| 4 2 r 3 −

1 2 r | 1 0 0 50 100 150 200 250 300 350 t/ M

Figure 6.1: The trajectory followed by the two black holes in the x-y plane (top). The trajectories of three different resolutions h = /44.8, /51.2 and /57.6 M M M are indistinguishable. The bottom panel shows the coordinate separation of the two black holes as a function of time.

69 1 10− 10 2 )] − t ( 3 10− (22) 4 10 4 [Ψ − 5 Re 10

r − 6 M 10− 7 10− 100 150 200 250 300 350 t/ M 1 10− 2

)] 10− t ( 3 10− (22) 4 10 4 [Ψ − 5 Im 10

r − 6 M 10− 7 10− 100 150 200 250 300 350 t/ M

Figure 6.2: The real (top) and imaginary (bottom) part of the ℓ = 2, m = 2 mode of the Newman-Penrose scalar Ψ , extracted at r = 50 . 4 M

70 1 10− 2

| 10− ) t ( 3 10− (22) 4

Ψ 4 | 10− r

M 5 10− 6 10− 100 150 200 250 300 350 t/ M 0 -10 -20 ) )

t -30 ( -40

(22) 4 -50

Ψ -60 -70 arg( -80 -90 -100 100 150 200 250 300 350 t/ M

Figure 6.3: The amplitude (top) and phase (bottom) of the ℓ = 2, m = 2 mode of the Newman-Penrose scalar Ψ , extracted at r = 50 . Note the quasinormal 4 M ringdown exhibited by the linear decay in the amplitude (on this log-linear plot) and the constant frequency.

71 6.1.1 Energy and angular momentum

For the /51.2 resolution run, starting with an initial ADM mass and angular momentum M equal to E / =0.9957 and J / 2 =0.862, the system radiates E / =0.032 ADM M ADM M rad M ± 0.005 and J / 2 =0.22 0.01, leaving behind a final black hole with E / =0.963 0.005 rad M ± f M ± and J / 2 = 0.64 0.01, in agreement with the estimates of mass and spin provided by f M ± the analysis of the fundamental quasinormal tone of Ψ22 (E / = 0.957 0.005 and 4 QNM M ± J / 2 =0.62 0.01) and with previous results obtained with other codes [43]. QNM M ± Notice that both the ADM energy and angular momentum integrals and the radiated energy

and angular momentum from Ψ4, which should be calculated at spatial infinity in the former case and at null infinity in the latter, can only be computed at a finite distance from the black holes, and during a finite interval of time. An appropriate extrapolation strategy is therefore required to extract the actual energy and angular momentum from the numerically-computed quantities. Figures 6.4 and 6.5 show the value of these quantities at several extraction radii (as a function of the inverse radius, so that the limit at spatial infinity is represented by the intersection of the different extrapolation lines with the 1/ρ = 0 axis). Notice that extrapolation in time for the radiated E and J is trivial, since after a transient period (ending shortly after the gravitational wave passage through each radius) both quantities reach a plateau (see figure 6.6). The errors in the final estimates are given by the root mean square deviation from the fitting line. In the binary coalescence runs, for all t 160 a common apparent horizon is present, and ≥ M becomes approximately axisymmetric at t 165 (see [135] for a description of the Killing ∼ M vector field finding algorithm we deployed).

6.2 Scalar field evolution

Starting at the time of the first common apparent horizon formation (t = 160 ), and until M after the coalescence is complete (at about t = 260 , as indicated by the horizon parame- M ters), we freeze the gravitational variables at regular intervals and perform an evolution of the scalar field Φ as described by equation (5.2.1) on the corresponding geometry, using the reference resolution of /51.2. M

72 300 50 30 20 10 1.16 1.14 1.12 1.1 M

/ 1.08 1.06 ADM

E 1.04 1.02 1 Fit Data 0.98 0 0.02 0.04 0.06 0.08 0.1 0.12 /ρ M 300 50 30 20 10 0.9 Fit 0.85 Data

0.8 2 M

/ 0.75 ADM

J 0.7

0.65

0.6 0 0.02 0.04 0.06 0.08 0.1 0.12 /ρ M

Figure 6.4: The ADM energy (top) and angular momentum (bottom) for an R1 run, as a function of the extraction sphere’s radius ρ. The data points are extrapolated to infinity to yield E = 0.9957 and J = 0.862 2. ADM M ADM M

73 ρ/ M 100 50 30 20 0.05 Fit 0.045 Data

0.04 M

/ 0.035 rad

E 0.03

0.025

0.02 0 0.01 0.02 0.03 0.04 0.05 0.06 /ρ M ρ/ M 100 50 30 20 -0.16 Fit -0.18 Data

-0.2 2 M

/ -0.22 rad

J -0.24

-0.26

-0.28 0 0.01 0.02 0.03 0.04 0.05 0.06 /ρ M

Figure 6.5: The radiated energy (top) and angular momentum (bottom) extracted from

the Newman-Penrose scalar Ψ4, as a function of the extraction sphere’s radius ρ, at t 300 . The extrapolation to spatial infinity yields E = 0.032 ∼ M rad M and J = 0.2241 2. rad − M

74 0.04 ρ = 20 0.035 ρ = 30M = 40M 0.03 ρ ρ = 50M 0.025 ρ = 100M M M

/ 0.02

rad 0.015 E 0.01 0.005 0 0 50 100 150 200 250 300 t/ M 0.05

0

-0.05 2 M

/ -0.1 rad J -0.15 ρ = 20 ρ = 30M M -0.2 ρ = 40 ρ = 50M ρ = 100M -0.25 M 0 50 100 150 200 250 300 t/ M

Figure 6.6: The radiated energy (top) and angular momentum (bottom) extracted from

the Newman-Penrose scalar Ψ4, for five different values of the extraction radius ρ, as a function of t. The detector at 10 and those at 200 and M M 300 are either too close to the source (and, therefore, not in the “wave M zone”) or too far away (and, therefore, causally disconnected from the source),

and have been excluded from the extrapolation procedure for Erad and Jrad.

75 For a discussion of the scalar field evolution before the formation of the first common apparent horizon, see section 6.2.4.

76 Figure 6.7: The apparent horizons at different stages of a binary black hole merger (in isometric view from the positive-z semiaxis). The two black holes inspiral around each other counterclockwise on the x-y plane, until they become sur- rounded by a distorted common apparent horizon, which then radiates away its higher multipoles reaching an axisymmetric configuration.

77 6.2.1 Initial data

For each of the scalar evolutions, we provide initial data for Φ and Ψ as follows:

− 2 (r r0) 4 2 Φ = r e− σ Re[Yℓ0] (6.2.1) 2(r r ) Ψ = − 0 Φ (6.2.2) − ασ2

Notice that, as observed in [94], the fact that Re[Yℓm]=(Yℓm + Yℓ, m)/2 implies the con- − current presence of the (ℓ, m) and (ℓ, m) modes in the initial data. In order to produce − ringing waveforms with a single dominant frequency, we chose only m = 0 modes. The initial isosurfaces of the scalar field Φ and its gradient Ψ for the three modes ℓ = 2, ℓ = 4 and ℓ =6 are shown in figures 6.8, 6.9 and 6.10.

6.2.2 Frequency extraction

Following Dorband et al. [94], we extract the multipoles of the scalar field at a coordinate radiusr ˜ = 5 in all cases (which is large enough compared to the size of the merged M object, of order 2 or less; the effect of the extraction radius on the quasinormal frequency M 1 is illustrated in section 6.3, and amounts to about 0.005 − ). Figure 6.11 shows some M isosurfaces of the scalar field during the evolution, while figures 6.12, 6.13 and 6.14 show its extracted ℓ = 2, ℓ = 4 and ℓ = 6 modes, defined as:

2 Φℓ0(τ) = Yℓ∗0(θ,φ)Φ(ρ,θ,φ,τ)d Ω (6.2.3) ZSr˜

(where Yℓ0 are the standard (spin zero) spherical harmonics, and the angular modes Φℓ0 are extracted on a sphere Sr˜ of fixed coordinate radiusr ˜). Each mode was evolved on eleven different hypersurfaces labeled by t = 160, 162.5, 165, { 167.5, 170, 180, 190, 200, 220, 240, 260 (as stated above, by t = 260 the system has }M M settled down and we expect to observe the typical evolution of a scalar field scattered from a Kerr black hole). As can be clearly seen from these figures, each waveform undergoes a damped oscillatory phase. We observe the emergence of quasinormal ringing on each spatial slice containing a common apparent horizon, regardless of whether the merged system has already settled

78 Figure 6.8: Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 2. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data.

79 Figure 6.9: Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 4. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data.

80 Figure 6.10: Initial isosurfaces (red corresponds to Φ = 10, while blue corresponds to − Φ = 15) for the scalar field Φ (top) and its gradient Ψ (bottom), for the mode ℓ = 6. The vertical direction corresponds to the z-axis, a symmetry axis for the initial data.

81 down (t = 260 ) or is still evolving very dynamically (t = 160 ). Initially the scalar M M field ringing is virtually identical on each of the three hypersurfaces shown, but significant dephasing is obtained towards the end of the scalar field simulation due to the different values of the fundamental frequency in the three cases. This is not an artifact of resolution, but rather shows the difference between the hypersurfaces. In the discussion above, we have mentioned how scalar perturbation theory on a black hole background predicts that, as soon as the initial impulse driving the perturbation propagates away, the scalar field will relax back to its equilibrium configuration through a period of damped (quasinormal) oscillations; numerical simulations (see, e.g., [94]) confirm this pic- ture. In the quasinormal regime the scalar field can be quite accurately described in terms of a set of quasinormal modes identified by the three numbers ℓ, m (labeling the angular eigenstates) and n (labeling the radial eigenstates). In time, each multipole evolves pertur- batively, independently of the others and is characterized by a specific complex frequency

Ωℓmn, whose dependence on the black hole mass and spin (and charge) is known (notice, however, that the dependence on charge is only known for scalar perturbations [136]). In principle, then, it should be possible to extract information about the black hole parameters

M and j by measuring Ωℓmn for a number of different ℓmn-modes. However, the peculiar nature of quasinormal modes complicates this procedure: since quasi- normal modes do not constitute a complete set of eigenstates, they cannot serve as a basis for the solutions of equation (5.2.1), and there is no guarantee that, at any given instant of time, the scalar field can be described as a superposition of quasinormal modes alone. How- ever, the practical evidence is that the system will follow a time evolution path which, for some transitory period, will live almost entirely on the subspace spanned by the quasinormal modes [92, 93].

We fit each Φℓ0(τ) to a superposition of quasinormal modes:

∞ QN iΩℓ0n(M,j)(τ τ0)+φℓ0n Φℓ0 (τ; A, M, j, φ, τ0)=Re Aℓ0ne− − (6.2.4) "n=0 # X where Aℓ0n and φℓ0n are real constants, and the complex quasinormal frequencies Ωℓ0n(M, j)=

ωℓ0n(M, j)+iαℓ0n(M, j) depend on M and j in a somewhat complicated way, which can how- ever be represented quite conveniently through the use of tabulated data [93], complemented with third order interpolation.

82 Figure 6.11: Isosurfaces of the scalar field at select values of τ during the evolution, for ℓ =2 and t = 160 . The white surface represents the apparent horizon. M

83 100

1

0.01

0.0001 ) τ (

20 t = 160 Φ 1e-06 t = 162.M5 t = 165 M t = 167.M5 1e-08 t = 170 M t = 180M t = 190M M 1e-10 t = 200 t = 220M t = 240M t = 260M 1e-12 M 0 10 20 30 40 50 60 70 80 τ/ M

Figure 6.12: The evolution of the ℓ = 2-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The { }M mode was obtained on a sphere of constant coordinate radiusr ˜ = 5 . M

84 100

1

0.01

0.0001 ) τ (

40 t = 160 Φ 1e-06 t = 162.M5 t = 165 M t = 167.M5 1e-08 t = 170 M t = 180M t = 190M M 1e-10 t = 200 t = 220M t = 240M t = 260M 1e-12 M 0 10 20 30 40 50 60 70 80 τ/ M

Figure 6.13: The evolution of the ℓ = 4-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The { }M mode was obtained on a sphere of constant coordinate radiusr ˜ = 5 . M

85 100

1

0.01

0.0001 ) τ (

60 t = 160 Φ 1e-06 t = 162.M5 t = 165 M t = 167.M5 1e-08 t = 170 M t = 180M t = 190M M 1e-10 t = 200 t = 220M t = 240M t = 260M 1e-12 M 0 10 20 30 40 50 60 70 80 τ/ M

Figure 6.14: The evolution of the ℓ = 6-mode of the scalar field is shown on hypersurfaces extracted at t = 160, ..., 260 from a binary black hole inspiral. The { }M mode was obtained on a sphere of constant coordinate radiusr ˜ = 5 . M

86 In order to extract the black hole parameters we minimize the residual of the fit

τf QN dτ Φℓ0(τ) Φ (τ; A, M, j, φ, τ0) τ0 | − ℓ0 | q(A, M, j, φ, τ0, τf )= τf (6.2.5) dτ Φℓ0(τ)) R τ0 | | with respect to A, M, j and φ (with the M and jR dependence given by the table interpolation method mentioned above).

As mentioned above, there is no a priori prescription for the initial and final times τ0 and τf delimiting the fitting window in which the minimum search is performed. In [94] and [11],

τ0 is included as an additional fitting parameter, along with some heuristic prescription for

τf . In [96], the value of τ0 was obtained by cross-correlating different angular modes, with two different cutoff criteria for τf . In this work, we start by choosing the initial and final

times τ0 and τf so that the fitting window excludes the initial prompt response phase and the final wave portion contaminated by numerical and outer boundary error, or τ 25 0 ∼ M and τ 55 . We then perform a two-dimensional analysis of the variations in the best-fit f ∼ M parameters induced by a change in τ0 and τf . This analysis provides an estimate of the frequency extraction error and ensures that small variations on either of these parameters have no appreciable influence on the fit results. Figure 6.34 shows the corresponding contour plots.

As an illustration of the dependence of the extracted parameters on τ0 over a larger interval, for τ fixed and equal to , figures 6.15 to 6.26 show q, ω , α , A , cos(φ ) , M and j f M ℓ0 | ℓ0| ℓ0 | ℓ0 | for each waveform and each angular mode in ℓ =2, 4, 6.

87 1

0.1

t = 160 t = 162M.5 q 0.01 t = 165 M t = 167M.5 t = 170 M t = 180M 0.001 t = 190M t = 200M t = 220M t = 240M t = 260M 0.0001 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.15: The minimum of the fitting residual q in equation (6.2.5) with respect to

A, M, j and φ, as a function of τ0, for the ℓ = 2 mode and for τf fixed and equal to 69 . ∼ M

88 0.54

0.53

0.52

0.51

0.5 t = 160 20 t = 162M.5 ω t = 165 M

M 0.49 t = 167M.5 t = 170 M 0.48 t = 180M t = 190M 0.47 t = 200M t = 220M 0.46 t = 240M t = 260M 0.45 M 0 10 20 30 40 50 60 τ / 0 M 0.115 t = 160 t = 162M.5 0.11 t = 165 M t = 167M.5 0.105 t = 170 M t = 180M t = 190M 0.1 M | t = 200

20 t = 220M α 0.095 t = 240M

M| t = 260M 0.09 M

0.085

0.08

0.075 0 10 20 30 40 50 60 τ / 0 M

Figure 6.16: The values of ω20 (top) and α20 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

89 100

10 t = 160 | M

20 t = 162.5 M A

| t = 165 t = 167M.5 t = 170 M 1 t = 180M t = 190M t = 200M t = 220M t = 240M t = 260M 0.1 M 0 10 20 30 40 50 60 τ / 0 M 3 t = 160 t = 162M.5 M 2.5 t = 165 t = 167M.5 t = 170 M M 2 t = 180 t = 190M |

) t = 200M 20 t = 220M φ 1.5 t = 240M M cos( t = 260 | M 1

0.5

0 0 10 20 30 40 50 60 τ / 0 M

Figure 6.17: The values of A20 (top) and φ20 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

90 1.05

1

t = 160 M M t = 162.5 t = 165 M

M/ 0.95 t = 167M.5 t = 170 M t = 180M t = 190M 0.9 t = 200M t = 220M t = 240M t = 260M M 0 10 20 30 40 50 60 τ / 0 M 1

0.9

0.8 t = 160 t = 162M.5 j t = 165 M M 0.7 t = 167.5 t = 170 M t = 180M t = 190M 0.6 t = 200M t = 220M t = 240M t = 260M 0.5 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.18: The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

91 0.1

t = 160 0.01 t = 162M.5 q t = 165 M t = 167M.5 t = 170 M t = 180M 0.001 t = 190M t = 200M t = 220M t = 240M t = 260M 0.0001 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.19: The minimum of the fitting residual q in equation (6.2.5) with respect to

A, M, j and φ, as a function of τ0, for the ℓ = 4 mode and for τf fixed and equal to 69 . ∼ M

92 1 t = 160 0.99 t = 162M.5 t = 165 M 0.98 t = 167M.5 t = 170 M 0.97 t = 180M t = 190M 0.96 t = 200M

40 t = 220M ω 0.95 t = 240M M t = 260M 0.94 M 0.93 0.92 0.91 0.9 0 10 20 30 40 50 60 τ / 0 M

t = 160 0.12 t = 162M.5 t = 165 M t = 167M.5 M 0.11 t = 170 t = 180M t = 190M M | t = 200

40 0.1 t = 220M α t = 240M

M| t = 260M 0.09 M

0.08

0.07 0 10 20 30 40 50 60 τ / 0 M

Figure 6.20: The values of ω40 (top) and α40 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

93 1000

100

t = 160 | M

40 t = 162.5 10 M A

| t = 165 t = 167M.5 t = 170 M t = 180M 1 t = 190M t = 200M t = 220M t = 240M t = 260M 0.1 M 0 10 20 30 40 50 60 τ / 0 M 3 t = 160 t = 162M.5 M 2.5 t = 165 t = 167M.5 t = 170 M M 2 t = 180 t = 190M |

) t = 200M 40 t = 220M φ 1.5 t = 240M M cos( t = 260 | M 1

0.5

0 0 10 20 30 40 50 60 τ / 0 M

Figure 6.21: The values of A40 (top) and φ40 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

94 1.05

1

t = 160 M M t = 162.5 t = 165 M

M/ 0.95 t = 167M.5 t = 170 M t = 180M t = 190M 0.9 t = 200M t = 220M t = 240M t = 260M M 0 10 20 30 40 50 60 τ / 0 M 1

0.9

0.8

0.7 t = 160 t = 162M.5 j t = 165 M 0.6 t = 167M.5 t = 170 M M 0.5 t = 180 t = 190M t = 200M 0.4 t = 220M t = 240M t = 260M 0.3 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.22: The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

95 1

0.1 t = 160 t = 162M.5 q t = 165 M t = 167M.5 t = 170 M 0.01 t = 180M t = 190M t = 200M t = 220M t = 240M t = 260M 0.001 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.23: The minimum of the fitting residual q in equation (6.2.5) with respect to

A, M, j and φ, as a function of τ0, for the ℓ = 6 mode and for τf fixed and equal to 69 . ∼ M

96 1.44 t = 160 t = 162M.5 1.42 t = 165 M t = 167M.5 1.4 t = 170 M t = 180M 1.38 t = 190M t = 200M

60 1.36 t = 220M ω t = 240M M 1.34 t = 260M M 1.32 1.3 1.28 1.26 0 10 20 30 40 50 60 τ / 0 M 0.15 t = 160 t = 162M.5 0.14 t = 165 M t = 167M.5 0.13 t = 170 M t = 180M t = 190M 0.12 M | t = 200

60 t = 220M α 0.11 t = 240M

M| t = 260M 0.1 M

0.09

0.08

0.07 0 10 20 30 40 50 60 τ / 0 M

Figure 6.24: The values of ω60 (top) and α60 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

97 10

1 t = 160 | M

60 t = 162.5 M A

| t = 165 t = 167M.5 t = 170 M 0.1 t = 180M t = 190M t = 200M t = 220M t = 240M t = 260M 0.01 M 0 10 20 30 40 50 60 τ / 0 M 3 t = 160 t = 162M.5 M 2.5 t = 165 t = 167M.5 t = 170 M M 2 t = 180 t = 190M |

) t = 200M 60 t = 220M φ 1.5 t = 240M M cos( t = 260 | M 1

0.5

0 0 10 20 30 40 50 60 τ / 0 M

Figure 6.25: The values of A60 (top) and φ60 that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

98 1.1

1.05

1

0.95 t = 160 M M 0.9 t = 162.5 t = 165 M M/ t = 167M.5 0.85 t = 170 M t = 180M 0.8 t = 190M t = 200M M 0.75 t = 220 t = 240M t = 260M 0.7 M 0 10 20 30 40 50 60 τ / 0 M 1

0.8

0.6 t = 160 t = 162M.5 j t = 165 M M 0.4 t = 167.5 t = 170 M t = 180M t = 190M 0.2 t = 200M t = 220M t = 240M t = 260M 0 M 0 10 20 30 40 50 60 τ / 0 M

Figure 6.26: The values of M (top) and j that minimize the fitting residual, as a function of τ , for the ℓ = 2 mode and for τ fixed and equal to 69 . 0 f ∼ M

99 As noted in [4], the extraction of the complex exponentials from a quasinormal ringing waveform is a delicate procedure. In order to minimize equation (6.2.5), we have tested the non-linear least-squares Levenberg-Marquardt algorithm, which proved to be inadequate for our purposes. We then chose a fitting algorithm based on an optimization method known as the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES), whose details can be found in [117] and which is available as a convenient open-source Matlab routine. We have tested CMA-ES on synthetic waveforms such as those discussed in [4], obtaining similar results to the Kumaresan-Tufts and matrix pencil methods presented there (see chapter B). Figure 6.27 shows the fit residual q as a function of t for the ℓ = 2, 4, 6 modes. This { } gives us a rough indicator of how well the scattering can be described by quasinormal mode ringing. The fit residual q decreases with increasing t, indicating that the evolution on earlier hypersurfaces leads to an inferior fitting quality. This loss in fit quality at early times is reasonable because the scalar field is expected to depart from the familiar quasinormal ringing behavior since earlier hypersurfaces are representing a black hole in more and more dynamical phases. Also note that the higher modes are more difficult to resolve and hence a larger residual is observed.

6.2.3 Mass and spin extraction

Figure 6.28 shows the black hole mass M and spin j as obtained from the scalar field evolved on different hypersurfaces labeled by t, for the ℓ = 2, 4, 6 modes. For reference, { } we also show the values obtained directly from the dynamical horizon finder. Both M and j show qualitatively similar behavior, with an initial transition regime immediately after the first common apparent horizon formation (at t = 160 ). About 20 after the common M M apparent horizon is found we can reliably extract the spin and mass of the final black hole from the scattered scalar field. We want to emphasize that, while the scalar field modes on the different hypersurfaces appear almost identical, the extracted mass and spins show much larger differences due to the strong dependence of M, j on the extracted quasinormal frequency Ωℓm (notice that the error bars in figure 6.28 represent not only the uncertainty associated with the best-fit procedure, but also the contribution to the error due to finite numerical resolution, to angular mode extraction

100 ℓ = 2 1 ℓ = 4 10− q ℓ = 6

2 10− Residual

3 10− 160 180 200 220 240 260 t/ M

Figure 6.27: The minimum of the fitting residual q in equation (6.2.5) as a function of t for the ℓ = 2, 4, 6 mode of the scalar field. At early times t the black hole { } has not yet settled down and hence the fit residual q is larger as the scalar field evolved on the black hole background does not exhibit similarly clean ringdown as it does for later times.

101 1.04 ℓ = 2 ℓ = 4 1.02 ℓ = 6 1.00 Horizon

M 0.98

M/ 0.96 0.94 0.92

0.80

0.60 j 0.40 ℓ = 2 ℓ = 4 0.20 ℓ = 6 Horizon 160 180 200 220 240 260 t/ M

Figure 6.28: The mass and spin parameters extracted from the fundamental mode fre- quency, for the three angular modes. The error bars for the horizon mass and spin is included in the curve width. At early times, different choices of

τ0 and τf lead to significant modifications in the behavior of M and j as a function of t. The data error bars in this regime should therefore be consid- ered as a mere constraint of the mass and spin range on each hypersurface.

102 at a finite distance from the black hole and to the choice to neglect any overtones or different angular modes in the frequency extraction; these contributions will be discussed in section 6.3). To given an idea of the mass and spin dependence on the complex frequency, the errorboxes in the Ω plane are shown in figures 6.29-6.31 along with the constant mass and spin curves. In the transition regime between 160 and 180 the fit errors (in particular, those from M M varying the choice of τ0 and τf ) become so large that an accurate extraction of the black hole parameters is not possible anymore. Note that the extraction of M and j at early times is

so sensitive to τ0 and τf that the associated uncertainty dominates the bars shown in figure 6.28 for t . 180 . M We conclude that our scattering procedure constitutes an accurate probe of the final black hole potential (defined in the perturbative sense as summarized in section 5.2.1) at times t 180 , consistent with the results from the isolated horizon parameters. Between 160 ∼ M M and 180 , the black hole parameters M and j cannot be reliably extracted; however, M the scalar wave is still showing quasinormal ringing. In this regime, due to the level of uncertainty associated with our numerical resolution and frequency extraction routine, we can only provide a broad constraint for the extracted parameters. For completeness, tables 6.1, 6.2 and 6.3 show a comprehensive list of the fitting residual, the fitting parameters and the corresponding masses and spins.

103 0.098

0.096

0.094 | 20

α 0.092 M| 0.09

0.088

0.086 0.512 0.517 0.522 0.527 ω M 20

Figure 6.29: Evolution of the complex quasinormal mode frequencies Ω20 = ω20 + iα20 (the point with largest ω corresponds to the hypersurface at t = 160 ), 20 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the M intervals between neighboring lines being equal to 0.01 ), whereas those M with positive slope are the constant-j lines (the thicker being j = 0.7, with 0.1 intervals).

104 0.1

0.095 |

40 0.09 α M| 0.085

0.08

0.92 0.925 0.93 0.935 0.94 0.945 0.95 ω M 40

Figure 6.30: Evolution of the complex quasinormal mode frequencies Ω40 = ω40 + iα40 (the point with largest ω corresponds to the hypersurface at t = 160 ), 40 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the M intervals between neighboring lines being equal to 0.01 ), whereas those M with positive slope are the constant-j lines (the thicker being j = 0.7, with 0.1 intervals).

105 0.1

0.095

0.09

| 0.085 60 α 0.08 M| 0.075

0.07

0.065 1.33 1.34 1.35 1.36 1.37 ω M 60

Figure 6.31: Evolution of the complex quasinormal mode frequencies Ω60 = ω60 + iα60 (the point with largest ω corresponds to the hypersurface at t = 160 ), 60 M with the corresponding error boxes. The grey lines with negative slope are the constant-M lines (the thicker one representing M = , and the M intervals between neighboring lines being equal to 0.01 ), whereas those M with positive slope are the constant-j lines (the thicker being j = 0.7, with 0.1 intervals).

106 Table 6.1: The residual q and the best-fit values of ω20, α20, A20 and φ20, along with the corresponding mass and dimensionless spin estimates.

t/ q ω α A φ M/ j M M 20 M 20 20 20 M 2 160.0 1.54 10− 0.524 0.090 10.773 2.511 0.98 0.9 · − 3 162.5 9.47 10− 0.523 0.091 10.872 2.496 0.97 0.8 · − 3 165.0 8.44 10− 0.521 0.092 10.936 2.486 0.97 0.8 · − 3 167.5 7.59 10− 0.520 0.092 10.971 2.471 0.97 0.8 · − 3 170.0 6.92 10− 0.518 0.093 10.984 2.449 0.98 0.8 · − 3 180.0 6.02 10− 0.517 0.094 10.719 2.458 0.97 0.7 · − 3 190.0 5.55 10− 0.518 0.095 10.509 2.507 0.97 0.7 · − 3 200.0 5.32 10− 0.519 0.095 10.379 2.527 0.97 0.7 · − 3 220.0 5.09 10− 0.519 0.095 10.229 2.555 0.97 0.7 · − 3 240.0 4.96 10− 0.519 0.095 10.137 2.573 0.96 0.7 · − 3 260.0 4.85 10− 0.519 0.095 10.075 2.584 0.96 0.7 · −

Table 6.2: The residual q and the best-fit values of ω40, α40, A40 and φ40, along with the corresponding mass and dimensionless spin estimates.

t/ q ω α A φ M/ j M M 40 M 40 40 40 M 2 160.0 3.74 10− 0.940 0.087 21.152 1.186 0.99 0.9 · − − 2 162.5 2.39 10− 0.939 0.089 22.668 1.931 0.98 0.9 · − − 2 165.0 1.79 10− 0.936 0.091 23.540 1.224 0.98 0.8 · − − 2 167.5 1.45 10− 0.933 0.092 23.780 1.894 0.98 0.8 · − − 2 170.0 1.28 10− 0.929 0.092 24.529 1.285 0.98 0.8 · − − 2 180.0 1.05 10− 0.928 0.094 24.514 1.259 0.97 0.7 · − − 3 190.0 9.81 10− 0.930 0.095 24.116 1.168 0.97 0.7 · − − 3 200.0 9.57 10− 0.930 0.095 23.809 1.130 0.96 0.7 · − − 3 220.0 9.36 10− 0.931 0.096 23.429 2.062 0.96 0.7 · − − 3 240.0 9.23 10− 0.931 0.096 23.191 1.046 0.96 0.7 · − − 3 260.0 9.13 10− 0.931 0.096 23.026 1.025 0.96 0.7 · − −

107 Table 6.3: The residual q and the best-fit values of ω60, α60, A60 and φ60, along with the corresponding mass and dimensionless spin estimates.

t/ q ω α A φ M/ j M M 60 M 60 60 60 M 2 160.0 7.83 10− 1.357 0.081 0.738 1.450 1.00 1.0 · − 2 162.5 4.76 10− 1.354 0.086 0.879 1.418 0.99 0.9 · − 2 165.0 3.29 10− 1.350 0.089 0.966 1.400 0.98 0.9 · − 2 167.5 2.43 10− 1.345 0.091 1.027 1.364 0.98 0.8 · − 2 170.0 1.94 10− 1.340 0.093 1.068 1.313 0.98 0.8 · − 2 180.0 1.54 10− 1.338 0.095 1.104 1.349 0.97 0.7 · − 2 190.0 1.52 10− 1.341 0.096 1.093 1.480 0.96 0.7 · − 2 200.0 1.52 10− 1.342 0.096 1.081 1.535 0.96 0.7 · − 2 220.0 1.54 10− 1.342 0.096 1.063 1.607 0.96 0.6 · − 2 240.0 1.54 10− 1.343 0.096 1.052 1.655 0.96 0.6 · − 2 260.0 1.54 10− 1.343 0.096 1.044 1.686 0.96 0.6 · −

108 6.2.4 Scalar field evolution before the first common apparent hori- zon

In this section, we illustrate the behavior of the scalar field on the hypersurfaces around the first common apparent horizon formation, which happens at t = 160 . Figure 6.32 shows M the waveforms for the ℓ = 2 mode, on the three hypersurfaces corresponding to t = 150 , M t = 160 and t = 170 , for six extraction radii. M M The waveforms at t = 150 show a sensible departure from those at t = 160 and 170 , M M M indicating that the scalar field behaves qualitatively differently on this hypersurface versus the hypersurfaces where a common apparent horizon is found.

6.3 Error analysis

At this point we would like to discuss the different error sources for the final extraction of the black hole parameters M and j from the scalar field waveforms. We discuss how sensitive

the extraction of these parameters is to the fit range [τ0, τf ], to resolution, to wave extraction radius and to fitting the waveform to a superposition of more than one quasinormal mode. It is important to keep in mind that the black hole mass and spin parameters show a strong dependence on the scalar quasinormal frequencies Ω and hence a small fractional error in Ω translates into a much amplified fractional error in M and, especially, j. Choice of the fitting window: in order to provide a description of the fitting error due to the choice of the fitting window [τ , τ ], we select τ = 25 and τ = 55 as reference values 0 f 0 M f M and vary these two parameters over a 10 -wide interval centered around each. The induced M variation in M and j is illustrated in figures 6.34 and 6.35 for a few representative cases: this estimate provides a quantitative range for the uncertainty in the fitting parameters. In the following, we limit ourselves to this pair of reference values and quote the error associated with this choice. Resolution: for the three different resolutions listed above (h = /44.8, /51.2 and M M /57.6), we find that both the real and the imaginary part of the extracted frequencies M exhibit a monotonic trend as h decreases, with the difference between neighboring resolu-

109 ρ = 5 ρ = 10 M M 1000 1000 t = 150 t = 150 100 t = 160M 100 t = 160M t = 170M t = 170M 10 M 10 M 1 1 ) ) τ τ ( 0.1 ( 0.1 20 20

Φ 0.01 Φ 0.01 0.001 0.001 0.0001 0.0001 1e-05 1e-05 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 τ/ τ/ M M ρ = 20 ρ = 30 M M 100 100 t = 150 t = 150 10 t = 160M 10 t = 160M t = 170M t = 170M 1 M 1 M ) )

τ 0.1 τ 0.1 ( (

20 0.01 20 0.01 Φ Φ 0.001 0.001 0.0001 0.0001 1e-05 1e-05 20 30 40 50 60 70 80 90 30 40 50 60 70 80 90 100 τ/ τ/ M M ρ = 40 ρ = 50 M M 100 100 t = 150 t = 150 10 t = 160M 10 t = 160M t = 170M t = 170M 1 M 1 M ) )

τ 0.1 τ 0.1 ( (

20 0.01 20 0.01 Φ Φ 0.001 0.001 0.0001 0.0001 1e-05 1e-05 40 50 60 70 80 90 100 110 50 60 70 80 90 100 110 120 τ/ τ/ M M

Figure 6.32: The behavior of the scalar field Φ’s mode ℓ = 2 on three different hypersur- faces, corresponding to the first common apparent horizon formation and to 10 before and after it. The results are shown for six choices of the M extraction sphere.

110 tions decreasing as h 0 and the difference between the highest and the lowest resolution → 1 remaining always under 0.0003 − . M As an example of the convergence properties of our scheme as a function of the grid resolution, figure 6.33 shows the trend in the difference between the extracted field mode ℓ = 2 at the three resolutions discussed above and for t = 200 . The plot shows the difference between M the two coarsest resolutions, along with the difference between the two finest resolutions multiplied the expected factor for a first-, second-, third- and fourth-order scheme, given by:

hn hn c = | low − mid| (6.3.1) n hn hn | mid − high| Even though the differencing stencil is fourth-order accurate, the waveform does not exhibit a perfect fourth-order scaling, possibly due to lower-order contributions to the truncation error, such as those due to the mesh refinement algorithm. However, the scaling seems polynomial throughout most of the evolution, with an index usually larger than one, and often between two and three. The monotonic decrease in the difference between the truncation errors for 1 increasing resolutions appears like a sufficient guarantee that 0.0003 − is indeed an upper M bound for the solution error. Extraction radius: as for the extraction radius, the six different choices (ρ = 10 , 20 ,..., M M 1 50 in addition to the initial ρ = 5 ) lead to frequency shifts of the order of 0.005 − . M M M The uncertainty is also smaller if the extraction is performed on detectors that are farther from outer and refinement boundaries, such as the ones at 5 , 10 and 30 (the different M M M spatial resolutions characteristic of different extraction radii are not dramatically relevant, indicating that the modes are well resolved in all the cases considered). Eigenfunctions for the angular mode extraction: finally, the actual eigenfunctions of the angular part of the wave operator in equation (5.2.1) on a Kerr spacetime are the

(spin-zero) spheroidal harmonics Sℓm, rather than the spherical harmonics Yℓm used here;

2 2 ′ ′ 4 4 however, as discussed in [137], Yℓm = Sℓm +(a Ω ) ℓ′=l Bℓ mSℓ m + O(a Ω ), so that the | | 6 | | use of spherical harmonics only causes a modest amountP of mode mixing in the extracted waveforms (additionally, Berti et al. [138] find that this expansion is surprisingly accurate out to values of a Ω close to unity). To determine the bias caused by such mode mixing | | on the best-fit value of the fundamental frequency, we extend the fitting function (6.2.4) to a superposition of two and three different angular modes. The change in the complex

111 0.05 (l) (m) Φ20 Φ20 0.04 (m) − (h) c1(Φ20 Φ20 ) (m) − (h) c2(Φ20 Φ20 ) 0.03 (m) − (h) c3(Φ20 Φ20 ) (m) − (h) 0.02 c4(Φ Φ ) 20 − 20 0.01 0 -0.01 -0.02 -0.03 -0.04 0 10 20 30 40 50 60 τ/ M Figure 6.33: The difference between the two coarsest resolutions, along with the differ- ence between the two finest resolutions multiplied the expected factor for a first-, second-, third- and fourth-order scheme, for the mode ℓ = 2, extracted at a radius ρ = 5 at time t = 200 . M M

112 ∆M for ℓ = 2 at t = 160 ∆M for ℓ = 2 at t = 260 M M 59 59 0.0004 0.001 57 0.001 57 0.002 0.002 0.003

f 55 0.0035 f 55 0.003 τ τ 53 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29 τ0 τ0

∆M for ℓ = 4 at t = 160 ∆M for ℓ = 4 at t = 260 M M 59 59 0.007 0.001 57 0.008 57 0.002 0.009 0.003

f 55 0.01 f 55 τ 0.015 τ 53 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29

τ0 τ0

∆M for ℓ = 6 at t = 160 ∆M for ℓ = 6 at t = 260 M M 59 59 0.005 0.003 57 0.008 57 0.004 0.009 0.005

f 55 0.01 f 55 τ 0.015 τ 53 0.03 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29

τ0 τ0

Figure 6.34: Contours of the variation of the mass estimate ∆M(τ ,τ ) = M(τ ,τ ) 0 f | 0 f − M(25 , 55 ) for ℓ = 2, 4, 6 at t = 160 and t = 260 , as a function of M M | M M τ0 and τf .

113 ∆j for ℓ = 2 at t = 160 ∆j for ℓ = 2 at t = 260 M M 59 59 0.002 0.005 57 0.005 57 0.01 0.01 0.02

f 55 f 55 0.02 τ τ 0.03 53 0.05 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29 τ0 τ0

∆j for ℓ = 4 at t = 160 ∆j for ℓ = 4 at t = 260 M M 59 59 0.05 0.005 57 0.075 57 0.0075 0.1 0.01

f 55 f 55 0.02 τ τ 53 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29

τ0 τ0

∆j for ℓ = 6 at t = 160 ∆j for ℓ = 6 at t = 260 M M 59 59 0.02 0.01 57 0.03 57 0.02 0.05 0.03

f 55 0.1 f 55 τ 0.2 τ 53 53 51 51 21 22 23 24 25 26 27 28 29 21 22 23 24 25 26 27 28 29

τ0 τ0

Figure 6.35: Contours of the variation of the spin estimate ∆j(τ ,τ ) = j(τ ,τ ) 0 f | 0 f − j(25 , 55 ) for ℓ = 2, 4, 6 at t = 160 and t = 260 , as a function of M M | M M τ0 and τf .

114 frequency of the fundamental mode due to the inclusion of a different number of modes is 1 under 0.0006 − . M Unlike the error due to the frequency fitting, these last three sources of error have virtually no fluctuations from hypersurface to hypersurface. For all times, the cumulative errors associated with the determination of the complex frequency (including the last three sources of error but not the uncertainty introduced by the fitting procedure) then amount to ∼ 1 0.005 − . This number is propagated to M and j via ∆M = M(Ω + ∆Ω) M(Ω) and M − ∆j = j(Ω + ∆Ω) j(Ω) which yields: − ∆M =0.02 , ∆j =0.1 for ℓ =2 M ∆M =0.02 , ∆j =0.15 for ℓ =4 M ∆M =0.03 , ∆j =0.2 for ℓ = 6 . M Finally, these numbers are summed in quadrature to the errors in M and j due to the fitting procedure (shown in figure 6.34), yielding the error bars in figure 6.28.

6.4 Discussion and conclusions

In the past three years, a consistent framework for the full relativistic, three-dimensional numerical evolution of binary black hole systems has emerged, yielding one of the first answers to a thirty-years-old problem. Simulations carried out within this framework have started shedding light on the strongest, most non-linear features of General Relativity, Modelling these new features and sorting them into a coherent ensemble has become the next goal of Numerical Relativity. In this work, we have concentrated on the merger phase, denoted by the formation of a common, highly distorted horizon, which subsequently relaxes to that of a standard Kerr black hole through the emission of gravitational radiation. In particular, we have presented the result of an experiment involving the scattering of a massless scalar field off the curvature of a spacetime corresponding to a black hole merger. This spacetime was generated numerically by a full 3D black hole simulation following the moving puncture paradigm. Each spatial hypersurface obtained served as a fixed background for the propagation of the scalar field and the behavior of the scalar field was analyzed in each case.

115 We observed the emergence of quasinormal ringing on each spatial slice containing a common apparent horizon regardless of whether the merged system had already settled down to a single Kerr black hole. While we could not reliably extract the final black hole parameters immediately after the formation of a common apparent horizon (t = 160 ), we could M still clearly identify quasinormal ringing even in this early phase where the merged black hole is strongly excited and the scattering potential deviates from that of Kerr. This finding agrees with earlier results where scalar perturbations on Vaidya spacetimes were studied and quasinormal ringing could be identified even on time-dependent black hole backgrounds [128]. Our results and interpretation are also consistent with the evidence from the Close Limit approximation [125], where a pair of black holes in a head-on collision was shown to enter the perturbative regime as soon as a common apparent horizon surrounded them. Based on this observation, the scattering of a test scalar field from the merged black hole could provide information about the final black hole state long before the spacetime has settled down to its final state. Soon after horizon formation (i.e. for times t & 180 ) we were able to obtain estimates for M the mass and spin of the final black hole from the scalar field probe in good agreement with direct isolated horizon measures. This indicates that, at this time, the dynamics surrounding the black hole has settled down sufficiently that the scattering potential for the scalar field is essentially that of the final black hole, even though there is still non-trivial dynamics in the vicinity of the horizon. We also observed a qualitative change between the scalar waveforms obtained on the hyper- surfaces that contain a common apparent horizon with respect to those who do not; this change can be attributed to a different response of the scalar field to the background geome- try or to the failure of our simple frozen-background approach at such early times. In order to estabilish which is the source of this effect, it is necessary to carry out alternative evo- lutions (based, for instance, on a concurrently evolving background) and mode extractions (for instance, for different choices of the extraction surfaces). Other intriguing follow-ups include the extension of the present work to spinning black hole binaries: in this case, it would be interesting to explore the different merger dynamics and verify whether the quasinormal frequencies are able to track the binary’s spin evolution. Finally, a comparison of the system’s properties (measured through the quasinormal modes)

116 to the horizon multipole evolution could yield a complementary view on the highly relativistic merger phase.

117 References

[1] Ching, E. S. C., Leung, P. T., Suen, W. M., & Young, K. Wave propagation in gravitational systems: Completeness of quasinormal modes. Phys. Rev., D54:3778– 3791, 1996.

[2] Kokkotas, Kostas D., & Schmidt, Bernd G. Quasi-normal modes of stars and black holes. Living Rev. Rel., 2:2, 1999.

[3] van Meter, James R., Baker, John G., Koppitz, Michael, & Choi, Dae- Il. How to move a black hole without excision: gauge conditions for the numerical evolution of a moving puncture. Phys. Rev., D73:124011, 2006.

[4] Berti, Emanuele, Cardoso, Vitor, Gonzalez, Jose A., & Sperhake, Ul- rich. Mining information from binary black hole mergers: A comparison of estimation methods for complex exponentials in noise. Phys. Rev., D75:124017, 2007.

[5] Menou, Kristen, Quataert, Eliot, & Narayan, Ramesh. Astrophysical evi- dence for black hole event horizons. 1997.

[6] Magorrian, John, et al. The Demography of Massive Dark Objects in Galaxy Cen- tres. Astron. J., 115:2285, 1998.

[7] Garcia, Michael R., McClintock, Jeffrey E., Narayan, Ramesh, Callanan, Paul, & Murray, Stephen S. New evidence for black hole event horizons from chandra. 2000.

[8] Narayan, Ramesh, Garcia, Michael R., & McClintock, Jeffrey E. X-ray novae and the evidence for black hole event horizons. 2001.

118 [9] Komossa, Stefanie. Observational evidence for supermassive black hole binaries. AIP Conf. Proc., 686:161–174, 2003.

[10] Shoemaker, Deirdre M., Pfeiffer, Harald, Kidder, Larry, Laguna, Pablo, Lindblom, Lee, Scheel, Mark A., & Teukolsky, Saul A. Mining for observables: A new challenge in numerical relativity. AIP Conf. Proc., 848:669– 676, 2005.

[11] Buonanno, Alessandra, Cook, Gregory B., & Pretorius, Frans. Inspiral, merger and ring-down of equal-mass black-hole binaries. 2006.

[12] Schwarzschild, Karl. On the gravitational field of a mass point according to Ein- stein’s theory. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ), 1916:189–196, 1916.

[13] Chandrasekhar, S. The mathematical theory of black holes. Oxford University Press, 1985.

[14] Regge, Tullio, & Wheeler, John A. Stability of a Schwarzschild singularity. Phys. Rev., 108:1063–1069, 1957.

[15] Zerilli, Frank J. Effective potential for even parity regge-wheeler gravitational perturbation equations. Phys. Rev. Lett., 24:737–738, 1970.

[16] Price, Richard H. Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations. Phys. Rev., D5:2419–2438, 1972.

[17] Davis, M., Ruffini, R., & Tiomno, J. Pulses of gravitational radiation of a particle falling radially into a Schwarzschild black hole. Phys. Rev., D5:2932–2935, 1972.

[18] Moncrief, V. Gravitational perturbations of spherically symmetric systems. I. The exterior problem. Ann. Phys., 88:323–342, 1974.

[19] Teukolsky, Saul A. Gravitational perturbations of a rotating black hole. PhD thesis, California Institute of Technology, 1974.

[20] Hayward, S. A. General laws of black hole dynamics. Phys. Rev., D49:6467–6474, 1994.

119 [21] Ashtekar, Abhay, & Krishnan, Badri. Isolated and dynamical horizons and their applications. Living Rev. Rel., 7:10, 2004.

[22] Ferrarese, Laura, & Merritt, David. A Fundamental Relation Between Super- massive Black Holes and Their Host Galaxies. Astrophys. J., 539:L9, 2000.

[23] Gebhardt, Karl, et al. A Relationship Between Nuclear Black Hole Mass and Galaxy Velocity Dispersion. Astrophys. J., 539:L13, 2000.

[24] Genzel, R., et al. The Stellar Cusp Around the Supermassive Black Hole in the Galactic Center. Astrophys. J., 594:812–832, 2003.

[25] Ghez, A. M., et al. Stellar Orbits Around the Galactic Center Black Hole. Astrophys. J., 620:744–757, 2005.

[26] Ferrarese, Laura, & Ford, Holland. Supermassive Black Holes in Galactic Nuclei: Past, Present and Future Research. 2004.

[27] Kewley, Lisa J., Groves, Brent, Kauffmann, Guinevere, & Heckman, Tim. The Host Galaxies and Classification of Active Galactic Nuclei. Mon. Not. Roy. Astron. Soc., 372:961–976, 2006.

[28] Stasinska, Grazyna, Cid Fernandes, Roberto, Mateus, Abilio, Sodre, Laerte, Jr., & Asari, Natalia V. Semi-empirical analysis of Sloan Digital Sky Survey galaxies III. How to distinguish AGN hosts. 2006.

[29] Laser Interferometer Gravitational Wave Observatory (LIGO), http://www.ligo.caltech.edu.

[30] GEO, http://geo600.aei.mpg.de.

[31] VIRGO, http://www.virgo.infn.it.

[32] TAMA, http://tamago.mtk.nao.ac.jp.

[33] Laser Interferometer Space Antenna (LISA), http://lisa.nasa.gov.

[34] Hahn, S. G., & Lindquist, R. W. The two-body problem in geometrodynamics. Annals of the New York Academy of Science, 29:304–331, 1964.

120 [35] May, M. M., & White, R. H. Hydrodynamic Calculations of General-Relativistic Collapse. Phys. Rev., 141:1232–41, 1966.

[36] Smarr, L., Cadez, A., DeWitt, Bryce S., & Eppley, K. Collision of Two Black Holes: Theoretical Framework. Phys. Rev., D14:2443–2452, 1976.

[37] Piran, Tsvi. Problems and solutions in numerical relativity. Annals of the New York Academy of Science, 375:1, 1981.

[38] Bruegmann, Bernd, Tichy, Wolfgang, & Jansen, Nina. Numerical simulation of orbiting black holes. Phys. Rev. Lett., 92:211101, 2004.

[39] Pretorius, Frans. Evolution of binary black hole spacetimes. Phys. Rev. Lett., 95:121101, 2005.

[40] Campanelli, Manuela, Lousto, C. O., Marronetti, P., & Zlochower, Y. Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett., 96:111101, 2006.

[41] Baker, John G., Centrella, Joan, Choi, Dae-Il, Koppitz, Michael, & van Meter, James. Gravitational wave extraction from an inspiraling configuration of merging black holes. Phys. Rev. Lett., 96:111102, 2006.

[42] Campanelli, Manuela, Lousto, C. O., & Zlochower, Y. The last orbit of binary black holes. Phys. Rev., D73:061501, 2006.

[43] Baker, John G., Centrella, Joan, Choi, Dae-Il, Koppitz, Michael, & van Meter, James. Binary black hole merger dynamics and waveforms. Phys. Rev., D73:104002, 2006.

[44] Pretorius, Frans. Simulation of binary black hole spacetimes with a harmonic evolution scheme. Class. Quant. Grav., 23:S529–S552, 2006.

[45] Sperhake, Ulrich. Binary black-hole evolutions of excision and puncture data. Phys. Rev., D76:104015, 2007.

[46] Tichy, Wolfgang. Black hole evolution with the BSSN system by pseudo-spectral methods. Phys. Rev., D74:084005, 2006.

121 [47] Bruegmann, Bernd, et al. Calibration of Moving Puncture Simulations. 2006.

[48] Baker, John G., et al. Binary black hole late inspiral: Simulations for gravitational wave observations. Phys. Rev., D75:124024, 2007.

[49] Baker, John G., Campanelli, Manuela, Pretorius, Frans, & Zlochower, Yosef. Comparisons of binary black hole merger waveforms. Class. Quant. Grav., 24:S25–S31, 2007.

[50] Marronetti, Pedro, et al. Binary black holes on a budget: Simulations using workstations. Class. Quant. Grav., 24:S43–S58, 2007.

[51] Pretorius, Frans, & Khurana, Deepak. Black hole mergers and unstable circular orbits. Class. Quant. Grav., 24:S83–S108, 2007.

[52] Baker, John G., et al. Gravitational waves from black-hole mergers. 2007.

[53] Hinder, Ian, Vaishnav, Birjoo, Herrmann, Frank, Shoemaker, Deirdre, & Laguna, Pablo. Universality and final spin in eccentric binary black hole inspirals. 2007.

[54] Herrmann, Frank, Hinder, Ian, Shoemaker, Deirdre, & Laguna, Pablo. Unequal mass binary black hole plunges and gravitational recoil. Class. Quant. Grav., 24:S33–S42, 2007.

[55] Baker, John G., et al. Getting a kick out of numerical relativity. Astrophys. J., 653:L93–L96, 2006.

[56] Gonzalez, Jose A., Sperhake, Ulrich, Bruegmann, Bernd, Hannam, Mark, & Husa, Sascha. Total recoil: The maximum kick from nonspinning black- hole binary inspiral. Phys. Rev. Lett., 98:091101, 2007.

[57] Campanelli, Manuela, Lousto, C. O., & Zlochower, Y. Gravitational radia- tion from spinning-black-hole binaries: The orbital hang up. Phys. Rev., D74:041501, 2006.

[58] Campanelli, Manuela, Lousto, C. O., & Zlochower, Yosef. Spin-orbit interactions in black-hole binaries. Phys. Rev., D74:084023, 2006.

122 [59] Campanelli, Manuela, Lousto, Carlos O., Zlochower, Yosef, Krishnan, Badri, & Merritt, David. Spin Flips and Precession in Black-Hole-Binary Merg- ers. Phys. Rev., D75:064030, 2007.

[60] Herrmann, Frank, Hinder, Ian, Shoemaker, Deirdre, Laguna, Pablo, & Matzner, Richard A. Gravitational recoil from spinning binary black hole mergers. 2007.

[61] Koppitz, Michael, et al. Getting a kick from equal-mass binary black hole mergers. 2007.

[62] Campanelli, Manuela, Lousto, Carlos O., Zlochower, Yosef, & Mer- ritt, David. Large Merger Recoils and Spin Flips From Generic Black-Hole Binaries. Astrophys. J., 659:L5–L8, 2007.

[63] Choi, Dae-Il, et al. Recoiling from a kick in the head-on collision of spinning black holes. Phys. Rev., D76:104026, 2007.

[64] Gonzalez, J. A., Hannam, M. D., Sperhake, U., Brugmann, B., & Husa, S. Supermassive kicks for spinning black holes. Phys. Rev. Lett., 98:231101, 2007.

[65] Campanelli, Manuela, Lousto, Carlos O., Zlochower, Yosef, & Mer- ritt, David. Maximum gravitational recoil. Phys. Rev. Lett., 98:231102, 2007.

[66] Baker, John G., et al. Modeling kicks from the merger of non-precessing black-hole binaries. Astrophys. J., 668:1140–1144, 2007.

[67] Tichy, Wolfgang, & Marronetti, Pedro. Binary black hole mergers: Large kicks for generic spin orientations. Phys. Rev., D76:061502, 2007.

[68] Herrmann, Frank, Hinder, Ian, Shoemaker, Deirdre M., Laguna, Pablo, & Matzner, Richard A. Binary black holes: Spin dynamics and gravitational recoil. Phys. Rev., D76:084032, 2007.

[69] Rezzolla, Luciano, et al. Spin Diagrams for Equal-Mass Black-Hole Binaries with Aligned Spins. 2007.

123 [70] Lousto, Carlos O., & Zlochower, Yosef. Further insight into gravitational recoil. 2007.

[71] Pollney, Denis, et al. Recoil velocities from equal-mass binary black-hole mergers: a systematic investigation of spin-orbit aligned configurations. Phys. Rev., D76:124002, 2007.

[72] Marronetti, Pedro, Tichy, Wolfgang, Brugmann, Bernd, Gonzalez, Jose, & Sperhake, Ulrich. High-spin binary black hole mergers. 2007.

[73] Rezzolla, Luciano, et al. The final spin from the coalescence of aligned-spin black- hole binaries. 2007.

[74] Baker, John G., et al. Modeling kicks from the merger of generic black-hole binaries. 2008.

[75] Schnittman, Jeremy D., & Buonanno, Alessandra. The Distribution of Recoil Velocities from Merging Black Holes. 2007.

[76] Schnittman, Jeremy D., et al. Anatomy of the binary black hole recoil: A multipolar analysis. 2007.

[77] Boyle, Latham, Kesden, Michael, & Nissanke, Samaya. Binary black hole merger: symmetry and the spin expansion. 2007.

[78] Buonanno, Alessandra, Kidder, Lawrence E., & Lehner, Luis. Estimating the final spin of a binary black hole coalescence. Phys. Rev., D77:026004, 2008.

[79] Arnowitt, R., Deser, S., & Misner, C. W. The dynamics of general relativity. 1962.

[80] Misner, C. W., Thorne, K. S., & Wheeler, J. A. Gravitation. San Francisco 1973, 1279p.

[81] Kerr, Roy P. Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett., 11:237–238, 1963.

[82] Kerr, Roy Patrick. Discovering the Kerr and Kerr-Schild metrics. 2007.

124 [83] Baumgarte, Thomas W., & Shapiro, Stuart. Numerical relativity and compact binaries. 2002.

[84] Kurki-Suonio, Hannu, Laguna, Pablo, & Matzner, Richard A. Inhomoge- neous inflation: Numerical evolution. Phys. Rev., D48:3611–3624, 1993.

[85] Futterman, J. A. H., Handler, F. A., & Matzner, R. A. Scattering from black holes. Cambridge, UK: University Press (1988) 192 P. (Cambrisge Monographs on Mathematical Physics).

[86] Andersson, Nils, & Jensen, Bruce P. Scattering by black holes. 2000.

[87] Ching, E. S. C., Leung, P. T., Maassen van den Brink, A., Suen, W. M., Tong, S. S., & Young, K. Quasinormal-mode expansion for waves in open systems. Rev. Mod. Phys., 70(4):1545–1554, 1998.

[88] Glampedakis, Kostas, & Andersson, Nils. Scattering of scalar waves by rotating black holes. Class. Quant. Grav., 18:1939–1966, 2001.

[89] Peacock, J. A. Cosmological physics. 1999. Cambridge, UK: Univ. Pr.

[90] Nollert, Hans-Peter. Topical review: Quasinormal modes: The characteristic ‘sound’ of black holes and neutron stars. Class. Quant. Grav., 16:R159–R216, 1999.

[91] Ching, E. S. C., Leung, P. T., Suen, W. M., Tong, S. S., & Young, K. Waves in open systems: Eigenfunction expansions. Rev. Mod. Phys., 70:1545, 1998.

[92] Leaver, Edward W. Spectral decomposition of the perturbation response of the Schwarzschild geometry. Phys. Rev., D34:384–408, 1986.

[93] Berti, Emanuele, & Cardoso, Vitor. Quasinormal ringing of kerr black holes. I: The excitation factors. Phys. Rev., D74:104020, 2006.

[94] Dorband, Ernst Nils, Berti, Emanuele, Diener, Peter, Schnetter, Erik, & Tiglio, Manuel. A numerical study of the quasinormal mode excitation of kerr black holes. Phys. Rev., D74:084028, 2006.

125 [95] Nollert, Hans-Peter, & Price, Richard H. Quantifying excitations of quasi- normal mode systems. J. Math. Phys., 40:980–1010, 1999.

[96] Berti, Emanuele, Cardoso, Vitor, Gonzalez, Jose A., Sperhake, Ulrich, Hannam, Mark, Husa, Sascha, & Bruegmann, Bernd. Inspiral, merger and ringdown of unequal mass black hole binaries: A multipolar analysis. 2007.

[97] Barton, Gabriel. Elements of green’s functions and propagation - potentials, diffu- sion and waves. Clarendon Press, Oxford, 1989.

[98] Teukolsky, Saul A. Perturbations of a rotating black hole. 1. Fundamental equa- tions for gravitational electromagnetic, and neutrino field perturbations. Astrophys. J., 185:635–647, 1973.

[99] Krivan, William, Laguna, Pablo, & Papadopoulos, Philippos. Dynamics of scalar fields in the background of rotating black holes. Phys. Rev., D54:4728–4734, 1996.

[100] Leaver, E. W. An analytic representation for the quasi normal modes of kerr black holes. Proc. Roy. Soc. Lond., A402:285–298, 1985.

[101] Froeman, N., Froeman, P. O., Andersson, N., & Hoekback, A. Black hole normal modes: Phase integral treatment. Phys. Rev., D45:2609–2616, 1992.

[102] Mashhoon, B. Quasi-Normal Modes of a Black Hole. Proceeding of the Third Marcel Grossman Meeting on Recent Developments of General Relativity, Shangai, China, 1983.

[103] Schutz, Bernard F., & Will, Clifford M. Black hole normal modes: A semi- analytic approach. Print-85-0063 (WASH.U.,ST.LOUIS), 1985.

[104] Berti, Emanuele, Cardoso, Vitor, & Will, Clifford M. On gravitational- wave spectroscopy of massive black holes with the space interferometer lisa. Phys. Rev., D73:064030, 2006.

[105] Knapp, A. M., Walker, E. J., & Baumgarte, T. W. Illustrating Stability Properties of Numerical Relativity in Electrodynamics. Phys. Rev., D65:064031, 2002.

126 [106] Reula, Oscar A. Hyperbolic methods for Einstein’s equations. Living Rev. Rel., 1:3, 1998.

[107] Nakamura, T., Oohara, K., & Kojima, Y. General relativistic collapse to black holes and gravitational waves from black holes. Prog. Theor. Phys. Suppl., 90:1–218, 1987.

[108] Shibata, M., & Nakamura, T. Evolution of three-dimensional gravitational waves: Harmonic slicing case. Phys. Rev., D52:5428–5444, 1995.

[109] Baumgarte, Thomas W., & Shapiro, Stuart L. On the numerical integration of einstein’s field equations. Phys. Rev., D59:024007, 1999.

[110] Pfeiffer, Harald P., et al. Reducing orbital eccentricity in binary black hole sim- ulations. Class. Quant. Grav., 24:S59–S82, 2007.

[111] Cook, Gregory B. Initial Data for Numerical Relativity. Living Rev. Rel., 3:5, 2000.

[112] Ansorg, Marcus, Brugmann, Bernd, & Tichy, Wolfgang. A single-domain spectral method for black hole puncture data. Phys. Rev., D70:064011, 2004.

[113] Choptuik, Matthew. Numerical analysis with applications in theoretical physics (lectures for taller de verano 1999 de fenomec), 1999.

[114] Richardson, L.F. On the approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Royal Society of London Proceedings Series A, 83:335–336, 1910.

[115] Reed, P. M., Minsker, B. S., & Goldberg, D. E. A multiobjective approach to cost effective long-term groundwater monitoring using an elitist nondominated sorted genetic algorithm with historical data. Journal of Hydroinformatics, (3):71–89, 2001.

[116] Crowder, Jeff, Cornish, Neil J., & Reddinger, Lucas. Darwin meets Ein- stein: LISA data analysis using genetic algorithms. Phys. Rev., D73:063011, 2006.

[117] Hansen, Nikolaus, & Ostermeier, Andreas. Completely derandomized self- adaptation in evolution strategies. Evolutionary Computation, 9(2):159–195, 2001.

127 [118] Eichelberg, Dominik, & Ackermann, Philipp. Integrating interactive 3d- graphics into an object-oriented application framework. In Grechenig, Thomas, & Tscheligi, Manfred, editors, Human Computer Interaction, volume 733 of Lecture Notes in Computer Science. Springer, 1993.

[119] Goodale, Tom, Allen, Gabrielle, Lanfermann, Gerd, Masso’, J., Radke, Thomas, Seidel, Edward, & Shalf, John. The cactus framework and toolkit: Design and applications. 2003.

[120] Husa, Sascha, Hinder, Ian, & Lechner, Christiane. Kranc: A mathematica application to generate numerical codes for tensorial evolution equations. Computer Physics Communications, 174:983, 2006.

[121] Cactus, http://www.cactuscode.org.

[122] Schnetter, Erik, Hawley, Scott H., & Hawke, Ian. Evolutions in 3d numer- ical relativity using fixed mesh refinement. Class. Quant. Grav., 21:1465–1488, 2004.

[123] Berger, M., & Oliger, J. Local adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 1984.

[124] Carpet, www.carpetcode.org.

[125] Price, Richard H., & Pullin, Jorge. Colliding black holes: The close limit. Phys. Rev. Lett., 72:3297–3300, 1994.

[126] Baker, John G., Campanelli, Manuela, & Lousto, Carlos O. The lazarus project: A pragmatic approach to binary black hole evolutions. Phys. Rev., D65:044001, 2002.

[127] Vishveshwara, C. V. Scattering of gravitational radiation by a Schwarzschild black- hole. Nature, 227:936, 1970.

[128] Shao, Cheng-Gang, Wang, Bin, Abdalla, Elcio, & Su, Ru-Keng. Quasinor- mal modes in time-dependent black hole background. Phys. Rev., D71:044003, 2005.

[129] Molina, C., Giugno, D., Abdalla, E., & Saa, A. Field propagation in the reissner-nordstroem-de sitter black hole. Phys. Rev., D69:104013, 2004.

128 [130] Chan, J. S. F., & Mann, Robert B. Scalar wave falloff in asymptotically anti-de sitter backgrounds. Phys. Rev., D55:7546–7562, 1997.

[131] Misner, C. W. Wormhole initial conditions. Phys. Rev., 118(4):1110–1111, 1960.

[132] Ashtekar, Abhay. How black holes grow. 2003.

[133] Baumgarte, Thomas W. The innermost stable circular orbit of binary black holes. Phys. Rev., D62:024018, 2000.

[134] Vaishnav, Birjoo, Hinder, Ian, Herrmann, Frank, & Shoemaker, Deirdre. Matched filtering of numerical relativity templates of spinning binary black holes. Phys. Rev., D76:084020, 2007.

[135] Dreyer, Olaf, Krishnan, Badri, Shoemaker, Deirdre, & Schnetter, Erik. Introduction to isolated horizons in numerical relativity. Phys. Rev., D67:024018, 2003.

[136] Berti, Emanuele, & Kokkotas, Kostas D. Quasinormal modes of kerr-newman black holes: Coupling of electromagnetic and gravitational perturbations. Phys. Rev., D71:124008, 2005.

[137] Press, W. H., & Teukolsky, S. A. Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric. Astrophys. J., 185:649–674, 1973.

[138] Berti, Emanuele, Cardoso, Vitor, & Casals, Marc. Eigenvalues and eigen- functions of spin-weighted spheroidal harmonics in four and higher dimensions. Phys. Rev., D73:024013, 2006.

129 Appendix A

Gauge conditions and physical observables

In this appendix, I study the effects of the gauge choice for a plunging binary black hole system, starting out from the last stable quasi-equilibrium configuration, known as QC0. The sequence of stable orbits is found by imposing a stationariety condition for the binding

energy Eb, defined as the difference between the ADM mass of the spacetime and the sum of the masses of the black holes: ∂E ∂ [M 2M] b := d ADM − = 0 (A.0.1) ∂d ∂d where d is the separation between the holes. The last stable orbit denoted as QC-0 is the

saddle point between the binding energy stationary points that minimize Eb and those that maximize it: ∂2E b = 0 (A.0.2) ∂d2 The two holes start approximately 2.5M apart and merge after less than one orbit. We have used this system to test the impact of three currently reported gauge choices on the physical output of the runs. The simulations involve a binary black hole system starting from a QC-0 quasi-equilibrium configuration, with outer boundaries at 64M, six refinement levels and bitant symmetry.

130 The tests are run for the following gauge choices:

∂ α = 2αK + βi∂ α t − i i 3 i (a)[40]  ∂tβ = 4 B  ∂ Bi = ∂ Γ˜i ηBi t t −  ∂ α = 2αK + βi∂ α t − i (b)[41] i 3 i  ∂tβ = 4 αB  ∂ Bi = ∂ Γ˜i ηBi βj∂ Γ˜i t t − − j  ∂ α = 2αK + βi∂ α t − i i 3 i j i (c)[3]  ∂tβ = 4 B + β ∂jβ  ∂ Bi = ∂ Γ˜i ηBi + βj∂ (Bi Γ˜i) t t − j −  A.1 Zerilli waveforms

We show the real part of the Moncrief Q function (mode ℓ = 2, m = 0) for two different resolutions and two different detector locations. As expected, the agreement between the different predictions improves with increasing resolution and at increasing distance from the merger region.

A.2 Error budget

We discuss two possible sources of error in the comparison:

Resolution: the differences could be induced by a different effect of the grid spacing • h on each gauge. An appropriate comparison should therefore confront the limit of the three gauge conditions for h ! 0, and not the results at finite resolution.

Deviations from geodesicity: the same coordinate label is affixed to different space- • time events in different runs, due precisely to gauge effects. The detectors closer to the merger region are more affected by this deviation, whereas those far away have nearly geodetical worldlines. The plots of the norm of the difference between the waveform

131 Figure A.1: Real part of the Moncrief function Q (mode ℓ = 2, m = 0).

132 predictions of -respectively- gauge (a) and (b), (a) and (c) and (b) and (c) are shown below.

Figure A.2: Norm of the difference in waveform between gauges (a) and (b) (top), (a) and (c) (center) and (b) and (c) (bottom).

133 Appendix B

Extraction of quasinormal frequencies via an evolutionary strategy

The Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) is a optimization al- gorithm based on the evolution of an initial population of sample points, whereby at each iteration the population’s best points’ covariance matrix is constructed and used to generate a new, multivariate normal distribution of points, which will constitute the next iteration’s population. Due to its reiterated adaptative properties, this strategy has proven to be extremely success- ful with multimodal and noisy landscapes, where classical local search methods are attracted to the nearest peak and fail to reach the true optimum. CMA-ES’s performance on a set of benchmark objective functions is presented in [cite] and compared to other algorithms. In the following, we will concentrate on CMA-ES performance when fitting the (superposi- tions of) damped oscillatory functions in white noise analyzed in [4]. The test functions are designed to reproduce the features of quasinormal ringing waveforms from perturbed black holes, while retaining desirable properties such as exactly-known frequency, amplitude and phase parameters and a tunable amount of noise.

134 B.1 Algorithm

Let us illustrate CMA-ES’s structure in greater detail: assume that the optimization problem consists of finding the minimum of a function f : Rn R, where, for the present purposes, → we ignore the possible presence of constraints on the variables in Rn. The algorithm will proceed along the following steps:

1. A random population of points x : x Rn, i =1,...,λ is drawn initially at random; { i i ∈ }

2. The µ λ points x˜j corresponding to the µ lowest values of f are selected, ≤ { }j=1,...,µ and the sample’s covariance matrix is constructed according to: µ 1 C = (x m) (x m) (B.1.1) µ µ j − ⊗ j − j=1 X where m is the mean of the µ best points: µ 1 m = x˜ (B.1.2) µ i i=1 X 3. A new population of λ points is generated from the probability distribution (m, C ) N µ and associated density:

1 T (x m) Cµ(x m) P [ (m, C ), x] := e− 2 − − (B.1.3) N µ 4. The procedure is repeated until the desired stopping criterion is met.

This basic procedure (and especially point 2, the generation of the data’s covariance matrix) has a number of variations meant to enhance the reliability of the adaptation step and increase its speed: these involve the use of weighted sums for the mean and covariance matrix calculations and the use of the information from several past generations into the covariance matrix construction.

B.2 Application to quasinormal ringing waveforms

In order to compare CMA-ES’s performance to that of some of the existing algorithms when extracting parameters from quasinormal ringing waveforms, we perform the test discussed

135 100

1 10−

2 10− )

t 3 10− ˜ Ψ( 4 10−

5 10−

6 10− 0 1 2 3 4 5 t/T

3 Figure B.1: A realization of the waveform in (B.2.1), with σ = 10− . in [4]: we construct a waveform Ψ˜ corresponding to the real part of the fundamental tone of the ℓ = 2, m = 2 mode of Ψ , which has a complex frequency Ω := ω + iα =0.3736716844 4 − 0.0889623157i; to this we superimpose some white noise according to:

αtruet Ψ(˜ t) = cos(ω t)e− + σ 2 ln u cos(2πu ) (B.2.1) true − 1 2

3 p where σ = 10− and, for each t, the quantities u1 and u2 are random numbers uniformly distributed between 0 and 1. A realization of this waveform is shown in figure B.1. We generate N = 100 realizations of this noisy waveform and fit each of them to a functional form αt Ψ(t; A,φ,ω,α)= A cos(ωt + φ)e− (B.2.2) in the interval 0 t 5T , where T =2π/ω, comparing to the results in [4]. ≤ ≤ Figure B.2 shows the relative bias: ∆x x¯ x = − true (B.2.3) xtrue xtrue

136 and standard deviation: σ 1 1 N x = (x x¯)2 (B.2.4) x x vN 1 i − true true u i=1 u − X with t 1 N x¯ = x (B.2.5) N i i=1 X for three quantities ω, α and := ω/(2α). The results illustrate how CMA-ES performs at Q least as well as the other two algorithms (matrix pencil and Kumaresan-Tuft) presented in [4], while the non-linear least-squares Levenberg-Marquardt recipe leads to sensibly higher biases. As a further illustration of the working principles of CMA-ES, figure B.3 shows the population of points in the course of the evolution, along with the contours of the fitting residual Ψ˜ Ψ(t; A,φ,ω,α) , on the ω-α plane alone. The distribution of points progressively || − || adapts to the the shape of the minimum in the residual, eventually converging to it.

137 1 10− ∆ω/ω 2 10− ∆α/α 3 ∆ / 10− Q Q 4 10− 5 10− 6 10− 7 10− MP KT LM ES

1 10− σω/ω σ /α 2 α 10− σ / Q Q 3 10−

4 10−

5 10− MP KT LM ES

Figure B.2: Biases and standard deviations for the real and imaginary part of the quasi- normal frequency and for the quality factor , for the matrix pencil, Q Kumaresan-Tuft, Levenberg-Marquardt and CMA-ES methods. The data relative to the first three algorithms has been extracted from figure 4 in [4].

138 Initial population Generation 25

0.12 0.12 0.11 0.11 0.1 0.1 α 0.09 α 0.09 | | | | 0.08 0.08 0.07 0.07 0.06 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ω ω

Generation 40 Generation 55

0.12 0.12 0.11 0.11 0.1 0.1 α 0.09 α 0.09 | | | | 0.08 0.08 0.07 0.07 0.06 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ω ω

Figure B.3: The initial population and generations 25, 40, and 55 for a CMA-ES run on the trial waveform (B.2.1). The contour lines represent the constant value lines for the fitting residual Ψ˜ Ψ(t; A, φ, ω, α) := N Ψ(˜ t ) || − || n=0 | n − Ψ(t ; A, φ, ω, α) , where t is the discrete grid on which the waveform is n | { n} P represented.

139 Vita

Eloisa Bentivegna was born in Palermo (Italy) on July 28th, 1978. From 1996 to 2002, she attended the Theoretical Physics program at University of Catania, where she earned a M.Sc. cum laude presenting a thesis titled ”Robertson-Walker cosmologies and recent data from type-Ia supernovae“ (advisers: A.M. Anile and A. Bonanno). Upon graduation, she worked as a research assistant at the Catania Astrophysical Obser- vatory until November of 2002, when she was awarded a Ministry of Education three-year doctoral fellowship for the University of Catania Physics Ph.D. program. In 2003, she trans- ferred to Penn State, where she became interested in numerical aspects of classical and quantum gravity. She has since been working on the study of dynamical black hole simula- tions under the supervision of Deirdre Shoemaker, analyzing gauge conditions for black hole mergers first, and then focusing on the dynamics of scalar fields on the curved background generated in a binary black hole coalescence. Over the past year, she has also completed the study of deSitter and anti-deSitter cosmologies in Loop Quantum Gravity, in collaboration with Tomasz Pawlowski and Abhay Ashtekar. As a secondary interest, Eloisa is minoring in High Performance Computing, and has worked on a number of computational projects spanning from evolutionary algorithms to pervasive systems. Eloisa is a student member of the American Physical Society, the European Physical Society and the Graduate Women in Science (Sigma Delta Epsilon, Penn State Nu Chapter), where she has served as a webmaster since 2006. Her latest recognitions include a Braddock fel- lowship (PSU, 2003), four Duncan fellowships (PSU, 2004-2007), and a Wheeler fellowship (University of Texas at Austin, 2006).