Compact Gravitational Wave Sources

SCUOLADI DOTTORATO “VITO VOLTERRA” DOTTORATO DI RICERCAIN FISICA –XXII CICLO

THESISSUBMITTEDTOOBTAINTHEDEGREEOF DOCTOROF PHILOSOPHY (“DOTTOREDI RICERCA”) IN PHYSICS OCTOBER 2009

Francesco Pannarale Greco

Program Coordinator Thesis Advisor Prof. Enzo Marinari Prof. Valeria Ferrari

A mia Mamma

If you have an apple and I have an apple and we exchange these apples, then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas. G.B. Shaw

Acknowledgments

As a first thing, I would like to thank my supervisor, Valeria Ferrari, for all of the encour- agement and support over the past three years and for always pushing further ahead my understanding of the physics behind problems. I am also deeply indebted to Leonardo Gualtieri for the many countless and endless discussions, the many answers he gave me, the many right and important questions he posed. I want to thank Luciano Rezzolla, for being such an inspiring thesis referee. Ever since the very first time I spoke to him, I have been learning so much physics and asking myself many fascinating questions, which is one of the foundations of this job. I also want to thank him and everybody at the Albert Einstein Institute for the three exciting weeks of research in Golm during my last year of PhD and for the hospitality and friendliness I encountered when visiting the Numerical Relativity group. I would also like to acknowledge Omar Benhar, Kostas Kokkotas and Anna Watts, with whom I have had brief but valuable work-related discussions and very pleasant non-work- related chats. I thank Stefania Marassi and Riccardo Ciolfi, the two other youngsters of the Rome relativity group along with myself, for the support, the numerous useful tips and the entertaining company they guaranteed all along these three years. I am grateful to Corrado Mascia for the help he gave me when I was trying to solve a problem during my second year of PhD and for being the excellent calculus professor I was lucky to have during my undergrad courses. I owe a lot to Emanuele Berti, for all his precious advice and for being the friend that every frightened young scientist would like to have, and to Andrea Passamonti, for the great time I had with him in Pisa and for being so kind to me every time I contacted him. I thank my Mother, of course, but unfortunately words cannot be enough to express how grateful I am for all her help and all she has done for me and given me in these twenty-six years. Dedicating this thesis to her is a way to say that if I accomplished what I did, it was mainly because of her. It would have never been possible without her support. Now I have to walk on my own, I know I will be alright given all the things she has taught me. I will never forget how nice it was to finally be able to host my cousin Maria Antonietta and to get to know her better. She is such a special and wise person, and I wish her all the best in her life. The wonderful family atmosphere and the immense affection I found when I was hosted by Angela, Enzo, Antonio and her for my last conference will never be forgotten either. I am very very grateful for those fantastic days with them! Gianvito Laterza and Barbara Reeckman both deserve special motioning because of two very important conversations I had with them. They were crucial for me to be where I am now, and I owe them a lot for listening to me and for giving me their advice and full support. I thank all the “saletta”-related people for being fantastic friends, for their support, for the dinners we shared, for the parties we held, for all the dumb things and the smart things we have done together so far: Valentina, Valery, Antonio, Antonino, Paolo, Tommaso, Maria, Giancarlo, Andrea, Alessio, Chiara, Sergio, Pierre, Leone, Alessandro, Lisa. The first time I walked into our... ehm... office-playground-house, I could have never imagined what a lucky day that day was for me: I was about to meet all you great human beings by simply opening a door. I will miss our good old days and I hope our roads will cross as often as possible. During my eight years at La Sapienza, I have met special people who are still very present in my life even though we do not go to class together every morning anymore: Claudio and Nicoletta, with whom I have a very important and tight bond and who were great hosts in Copenhaghen, Federica, who is always very sweet and will run to see you any time you ask her to, Raffaello, who had me laugh throughout these years and has now moved to a better work life in my hometown. Luca deserves a place all to himself since he is the only one who is both a high school friend and a “saletta”-member. This summer he has finally switched to Macs: I had to wait thirteen years for this historical moment. I am very happy for him and I am quite sure Bill didn’t even notice, but then again, we Mac-people our better aren’t we? What about listening to a bit of Oasis now? I wanted to thank more friends that gave me all there support in several moments along these three years: Alessia, Angelo, Dario and Orsetta — who I was very lucky and pleased to get back in touch with — Dario, Gabriele, Eleonora and Samrin — my lifelong friends who I know I will never lose — Donatella and Viviana — who were so nice to host me after my first conference, which I will never forget — Domenico — with whom the whole PhD adventure started — and Manuela — with whom I shared many of the topical moments of our PhD. Special thanks go to John Draskovic and Kyle Howe, who I was very lucky to meet at a school and to get to know more during their visit in Rome. I had a great time with them and I hope our roads will cross again soon.

viii Contents

Introduction1

1 An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs7 1.1 Milestones in Gravitational Wave History...... 7 1.2 Gravitational Wave Emission...... 8 1.3 Gravitational Wave Detection...... 11 1.4 Compact Objects...... 12 1.5 Neutron Stars...... 13 1.5.1 Neutron Star Observed Properties...... 14 1.5.2 Neutron Star Anatomy...... 18 1.5.3 Neutron Stars as Continuous Sources...... 21 1.6 Compact Binaries...... 22 1.6.1 Compact Binary Coalescences...... 25 1.7 Gamma-Ray Bursts...... 28

I Analytic Models of Mixed Binaries 33

2 BH-NS and NS-NS Binaries 35 2.1 Quasi-Equilibrium and Pre-Merger Simulations...... 35 2.2 Inspiral and Tidal Disruption...... 36 2.3 Dynamical Calculations of Mergers...... 37 2.3.1 NS-NS Mergers...... 38 2.3.2 BH-NS Mergers...... 39

3 The Affine Model 41 3.1 Kerr Geometry: Parallel Transport and Tidal Tensor...... 42 3.1.1 Newtonian Tidal Tensor...... 42 3.1.2 Relativistic Tidal Tensor...... 43 3.1.3 The Kerr Metric and the Symmetric Tetrad...... 44 3.1.4 The Parallel Propagated Tetrad...... 46 3.1.5 The Tidal Tensor Field for Equatorial Geodesics...... 47 3.2 The Affine Model Equations...... 49 3.2.1 The Affine Constraint...... 49 3.2.2 The Principal Frame...... 51 3.2.3 The Neutron Star Internal Dynamics...... 53

ix 3.2.4 Polytropic EOS and Newtonian Self-Gravity...... 56 3.3 Orbit Descriptions within the Afﬁne Model...... 57

4 Two Important Improvements of the Afﬁne Model 61 4.1 Pseudo-Relativistic Self-Gravity...... 62 4.2 Using any Barotropic EOS...... 66 4.3 A Note About Solving the EOS-Dependent Integrals Numerically..... 66

5 BH-NS Coalescing Binaries: Quasi-Equilibrium Approach 69 5.1 Formulation...... 69 5.2 Code Calibration and Tests...... 71 5.3 Comparison with Recent Relativistic Results...... 72 5.4 New Applications of the Quasi-Equilibrium Approach...... 77 5.4.1 Effects of Varying the NS EOS and the BH Spin...... 77 5.4.2 Varying the NS Mass and Using Other Equations of State...... 80 5.4.3 Measuring the NS Radius with GWs...... 93

6 BH-NS Coalescing Binaries: Dynamical Approach 103 6.1 Formulation...... 104 6.1.1 The Orbit Hamiltonian...... 105 6.1.2 The Hamiltonian for the NS Fluid...... 106 6.1.3 GW Dissipation...... 107 6.1.4 Equations of Motion...... 108 6.1.5 Initial Conditions...... 108 6.2 Results...... 109 6.3 Future Developments...... 111

II Towards QNMs of Rapidly Rotating Neutron Stars 113

7 Perturbation Theory, Quasi-Normal Modes and Instabilities 115 7.1 Eulerian and Lagrangian Approaches...... 116 7.1.1 Gauge Freedom and Gauge Transformations...... 118 7.2 Black Hole Oscillations...... 118 7.3 Oscillations of Relativistic Stars...... 121 7.3.1 Perturbing a Non-Rotating Star...... 122 7.3.2 Effects of Rotation...... 124 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods... 126 7.4.1 The Boundary Conditions...... 127 7.4.2 Spectral Methods for Stellar Oscillations...... 131

8 The Perturbed Einstein Equations for a Rapidly Rotating Star 135 8.1 Geometry of a Rotating Star...... 135 8.2 Perfect Fluids...... 138 8.3 Boundary Conditions...... 140 8.4 Determining the Perturbation Equations...... 140 8.5 Perturbation Equations in Priou’s Approach...... 141 8.5.1 A Note About Gauge Choice...... 142

x 8.6 Perturbation Equations in Hartle and Sharp’s Formalism...... 144 8.6.1 The Regge-Wheeler Gauge...... 148 8.6.2 The Perturbed Einstein Equations...... 149 8.7 Future Developments...... 166

Conclusions and Final Remarks 169

Appendices 172

A Noise Power Spectral Densities of Interferometric GW Detectors 173

B Relativistic Stellar Structure 175 B.1 The TOV Equations...... 175 B.2 Calculating ΦTOV and VbTOV ...... 176

C Equations of Motion for BH-NS Coalescing Binaries 179

D Symmetries of the Afﬁne Model Lagrangian and Conserved Quantities 183

E Essential Mathematical Toolkit 185 E.1 Multipole Expansion of the Metric Perturbation...... 185 E.1.1 Explicit Expressions of the Tensor Basis...... 189 E.1.2 Multipole Expansion of a Vector...... 191 E.2 Pullback and Pushforward...... 191 E.3 Lie Derivative...... 193

F Quasi-Normal Mode Classiﬁcation 195

Bibliography 199

Introduction

Ever since its formulation in 1915-1916 [1, 2], the theory of General Relativity has been carefully tested to an extremely precise extent [3]; moreover, it has proven to be extremely important in explaining the physical mechanisms behind the greatest astrophysical discoveries and high energy astronomical observations of the last decades, such as pulsars, quasars and active galactic nuclei. The examples just given are, in particular, all classes of phenomena involving so-called compact objects (see Section 1.4), whose properties can be understood only in the context of Einstein’s theory of gravitation. In the astrophysics community, the intent of comprehending the inner structure and dynamics of these objects and of the matter that surrounds them (accretion disks), on one side, and the numerous high energy satellites (Chandra, XMM-Newton, Integral, etc.) now fully operative, on the other side, have both greatly stimulated the study of fluid dynamics in strong gravitational field regimes. An additional reason that turned the hydrodynamics of compact objects into one of the hottest topics in theoretical astrophysics is the quest for gravitational waves. Gravitational waves are fluctuations in the curvature of spacetime which propagate (as waves) at the speed of light; their existence is the most elusive prediction of General Relativity, as it has only been proven indirectly so far [4]. The most promising candidate sources of gravitational waves involve compact objects and therefore, studying their hydrodynamics plays a key role in understanding their gravitational wave emission and other physical processes they take part in. The direct observation of gravitational radiation is extremely complicated because (1) it is exceptionally weak (see Eq. (1.1)) and (2) it hardly interacts with matter. The latter point, however, implies that gravitational waves carry unaltered information on their sources and it is thus believed that their detection will open a completely new window on the observable Universe and start the era of gravitational wave astrophysics [5]. If one adds to this thrilling prospect the fact that the indirect detection of gravitational waves and other experimental verifications of General Relativity [3] make us confident that gravitational waves do exist, it is clear why so many intellectual and technological efforts are invested in the direct detection of gravitational waves, one of the biggest goals in physics today. Theoretical and experimental research proceed side by side in this hunt: on one hand sources are modelled and emission mechanisms are sought for, on the other a tremendous technological work goes into building detectors with high enough sensitivities at the frequencies where the first observable gravitational waves are predicted to be. The detection of gravitational waves seems feasible in the next decade thanks to the international network of Earth-based laser interferometer detectors (GEO600 [6], LIGO [7], TAMA300 [8] and Virgo [9]) and to the Laser Interferometer Space Antenna (LISA) project [10]. This thesis is devoted to two compact sources of gravitational waves: black hole-neutron

1 2 Introduction star binaries and oscillating rapidly rotating neutron stars. Black hole-neutron star binaries (or mixed binaries), are excellent candidate sources for gravitational wave detection and have also attracted much attention as possible short gamma-ray bursts progenitors (see Section 1.7). Studying the final phases of their existence is a very challenging and complicated task which requires the use of elaborate codes and is highly demanding in terms of time and computer resources; numerical relativity is, in fact, the framework which is expected to eventually model their coalescence correctly: “[...] as we continue to observe SGRBs, it is up to simulations to finally determine which systems are realistic progenitors, and to quantify how various features of the observed emission reflect the underlying physical parameters of the system”[11]. In the meantime, approximate models are desirable to start giving the first answers and to provide information which may also possibly be useful to the numerical relativity community. With all this in mind, in the first part of this thesis we build a semi-analytic model with the intention of providing a quick and reliable tool which allows one

• to determine whether a mixed binary is a possible short gamma-ray burst progenitor or not and

• to access parameter space regions (i.e. black hole spin, mass ratio, equation of state, etc.) which are currently inaccessible with other methods.

The model we work on is the affine model developed by Carter, Luminet and Marck [12, 13, 14, 15, 16]. In this description, the a star orbiting a black hole is viewed as a Newtonian self-gravitating ellipsoid which is subject to the black hole tidal field and whose centre of mass moves along timelike geodesics. The many internal degrees of freedom of the star are therefore described with a few global parameters by imposing the ellipsoidal constraint; moreover, the black hole is not dynamic. We improve two important aspects of the original affine model formulation:

1. the Newtonian self-gravity potential is replaced with a pseudo-relativistic potential which is derived from the relativistic stellar structure equations, and

2. all quantities which were determined in previous literature on the subject by specifying a polytropic equation of state are left in integral form so that they may be calculated numerically for any barotropic equation of state.

We test the model in this new formulation and establish that the use of the pseudo-relativistic potential improves the affine model considerably. Moreover, within the approach of this improved affine model, we consider several binaries which differ in mass ratio, black hole spin, neutron star mass and neutron star equation of state and study their tidal disruption with the codes we wrote. This allows us to determine what regions of the parameter space are suitable for the system to be a possible short gamma-ray burst progenitor. Finally, we also observe that by measuring the cut-off frequency of the gravitational radiation emitted by such binaries when the neutron star is disrupted by the black hole tidal forces, it is possible to constrain the neutron star radius and equation of state. In the second part of this thesis, we consider oscillating rapidly rotating neutron stars within a perturbative approach. All neutron stars, in fact, rotate. The theoretical lower limit on the rotation period of a young pulsar, the mass-shedding limit, is of about 0.5 ms to about 2 ms; so far, however, observations of young pulsars have not confirmed this prediction: Introduction 3 the initial period of the best known young pulsar, for example, the Crab (PSR0531+21), is estimated to have been of 19 s [17]. On the other hand, old neutron stars may be recycled by accretion in binary systems (Accretion-powered Millisecond Pulsars, or AMPs) and reach rapid rotation rates of the order of a millisecond: the fastest pulsar detected so far, PSR J1748-2446ad, has a period of 1.39 ms [18] and there is a sub-millisecond pulsar candidate in the X-ray transient XTE J1739-285 with a period of 0.89 ms [19]. Moreover it is likely, that every compact star undergoes, during its life, an oscillatory phase: neutron star oscillations may be excited, for example, after a core collapse, during a starquake, during the final stages of a NS binary merger the outcome of which is a NS. When a neutron star oscillates non-radially, it emits gravitational waves at the characteristic frequencies of its quasi-normal modes. These frequencies are complex because gravitational waves acts as a damping mechanism subtracting mechanical energy to the system. Knowing the oscillation frequencies is crucial in order to favour the detection of the emitted gravitational radiation; moreover, since the modes are influenced by the microphysics in the neutron star interior, the emitted gravitational radiation will carry a signature of the physical processes occurring inside the star. The quasi-normal mode spectrum of a NS may be determined by solving the perturbed Einstein equations with suitable boundary conditions, however in the case of rapidly rotating neutron stars the quasi-normal mode spectrum has not been determined yet. Our goal is thus to start extending the linear perturbative approach developed in [20] for slowly rotating neutron stars to the rapid rotation case. We therefore calculate the perturbation equations governing a rapidly rotating neutron star in a form which is appropriate to be handled with a generalization of the numerical techniques used in [20].

Thesis Plan

The first chapter is a general review about the area of physics to which this thesis belongs: we discuss the emission and detection of gravitational radiation and we report on the astrophysics of neutron stars, compact binaries and gamma-ray bursts. Part I, which is devoted to black hole-neutron star binaries, then starts. Its first chapter (Chapter2) is a summary of the current consensus and work on such binaries as well as binary neutron stars, since these two kinds of binaries have various common features. In Chapter3, we present the starting point of the model we developed for mixed binaries: we therefore discuss the Kerr tidal field, which is used for the black hole, and the affine model, which is used for the neutron star. We then improve the affine model in two ways in Chapter4: by introducing a pseudo-relativistic self-gravity potential in the affine model and by extending its formulation to any barotropic equation of state. In Chapter5 we use our improved affine model in the quasi-equilibrium approximation; the code we developed is calibrated, tested against recent relativistic results and then applied to several sets of binary parameters and neutron star equations of state. This way we determine the regions of the parameter space in which the black hole-neutron star binary system is a possible short gamma-ray burst progenitor and show that the neutron star equation of state may be constrained by gravitational wave measurements in the case of binaries whose evolution ends with the disruption of the neutron star by the black hole tidal forces. Chapter4 and part of Chapter5 are based on [ 21]. In the last chapter of this part of the thesis we formulate the dynamical version of the model taking into account the evolution of the orbit due to gravitational wave emission and other post-Newtonian effects; the model in this form is then 4 Introduction used to study the effect of tidal interactions and of the finite size of the neutron star on the emitted gravitational waveform. In the second part of the thesis we perturb rapidly rotating neutron stars. Once again the first chapter (Chapter7) is introductory and reviews relativistic stellar perturbation theory. In Chapter8 we describe rotating stars in General Relativity and then we determine the perturbed equations for an oscillating rotating neutron star.

Units, Conventions, Notations and Abbreviations

Throughout this thesis, for the sake of simplicity, we will use geometrical units

GN = c = 1 where GN is the universal constant of gravitation and c is the speed of light. This way we have

1 = c = 2.9979 · 1010 cm/s −8 3 −1 −2 1 = GN = 6.6720 · 10 cm g s which may be used, for example, to perform the following unit conversions

1 s = 2.9979 · 1010 cm 1 g = 7.423710−29 cm 1 MeV = 1.6022 · 10−6 erg = 1.1605 · 1010 K = 1.3234 · 10−55 cm.

We will often use astrophysical quantities such as

33 M = 1.989 · 10 g = 1.4768 · 105 cm 10 R = 6.960 · 10 cm 1 pc = 3.26 light years = 3.09 · 1018 cm.

The Greek alphabet is used for indices that run from 0 to 3, whereas Latin letters run from 1 to 3 with the exceptions of the letters l and m that are reserved to the harmonic indices. Introduction 5

The following is a list of the abbreviations used throughout this thesis; other abbreviations introduced and used in single Sections of the thesis are not listed bellow.

BBH Binary Black Hole BH Black Hole BNS Binary Neutron Star CBC Compact Binary Coalescence EOS Equation of State GR General Relativity GRB Gamma-Ray Burst GW Gravitational Wave ISCO Innermost Stable Circular Orbit KBH Kerr Black Hole MWEG Milky Way Equivalent Galaxy NS Neutron Star ODE Ordinary Differential Equation PDE Partial Differential Equation PN Post-Newtonian QNM Quasi-Normal Mode SBH Schwarzschild Black Hole SGRB Short Gamma-Ray Burst SNR Signal to Noise Ratio TOV Tolman-Oppenheimer-Volkoff

Chapter 1

An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

In this Chapter, we will shortly review the history of Gravitational Waves (GWs) in physics, their emission and their detection. As Eq. (1.2) will make clear, the most promising gravitational wave sources involve compact objects: we therefore also give an overview on neutron stars and compact binaries which are the sources considered in this thesis. More detailed accounts of the current status of the art in compact binary coalescences and of neutron stars as gravitational wave sources are given in the opening chapters of Part I and Part II respectively. Finally we also discuss gamma-ray bursts, especially short ones, since compact binaries containing at least a neutron star are leading candidates as progenitors of these violent events.

1.1 Milestones in Gravitational Wave History

In 1907, Poincaré mentioned the probable existence of some “ondes gravifiques” in any relativistic theory of gravitation, due to the fundamental principle that no interaction may propagate instantaneously between two points in spacetime. As soon as 1916, Einstein understood that his new theory of gravitation, General Relativity (GR), admitted solutions describing undulatory perturbations of the gravitational field propagating at the speed of light c: this was the first mathematical description of gravitational waves [22]. Two years later, he found a formula for the gravitational emissivity of a relativistic fluid ball with weak gravitational field and slow internal motions. This quadrupole formula showed that, in the context of GR, gravitational radiation propagating at the speed of light is emitted by a mass-energy system whose quadrupole moment varies in time [23]. This solution describing gravitational waves was determined in a given coordinate system and linearising the Einstein equations (of General Relativity) and therefore the question of the physical relevance of gravitational waves was naturally lifted. In GR, in fact, all reference systems are equivalent and none of them directly has a physical meaning: it could hence be thought that the waves were just “coordinate waves” lacking any physical content. This issue had to wait until 1956 for the answer to start emerging. That year, Pirani posed himself the question of what would happen to a GW detector if a GW went through it and answered it by proving that

7 8 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs this phenomenon would actually be linked with an energy deposit [24]. Even before the rigorous proof of the existence of GWs in Einstein’s theory [25], the pioneer J. Weber had started to look for them experimentally. Direct detection of gravitational radiation has its roots in 1959 when Weber built the first resonant gravitational wave detectors, a set of solid aluminum cylinders about 2 meters long and 1 meter in diameter and suspended on steel wires. A passing gravitational wave would set one of these cylinders vibrating at its resonant frequency — about 1660 Hz — and piezoelectric crystals firmly attached around the waist of the cylinder would convert (“bar-detector”). Shortly after beginning to collect data, Weber announced that he had successfully detected a signal. This detection was turned down later, but Weber’s work and supposed results deserve invaluable credit for having definitely boosted the race for GW detection. Quite ironically, the first indirect observational proof of the existence of GWs came when most of the earliest bar experiments were being left. The proof was based on the 1975 discovery by Hulse and Taylor of PSR B1913+16, the first pulsar in a binary system ever observed, its companion being another neutron star [26]. Precise measurements of the time derivative of the pulsar orbital period were achieved by observing the Doppler shifting of the pulsar signals due to orbital velocity variations. They confirmed that this system in strong gravitational interaction was behaving exactly as Einstein’s theory dictates: it showed an orbital velocity increase, due to gravitational potential energy loss, which matched the general relativistic prediction of energy loss due to gravitational wave emission (see Figure 1.1). The up to date experimental precision of the agreement between measurements and General Relativity is about 0.2%[4]. Hulse and Taylor where awarded with the Nobel Prize in 1993; their binary system, as well as systems PSR B1534+12 and J0737-3039 [27], continues to be observed and to provide an incontrovertible proof of the existence of gravitational radiation.

1.2 Gravitational Wave Emission

The only way to really predict the GW signal emitted by a body is to solve the Einstein equations with the body energy-momentum tensor as a source term and to look at what reaches infinity. In most cases, however, this cannot be done analytically because the Einstein equations are highly non-linear; even from a numerical point of view, this is not an easy task. Nevertheless, one can determine the main properties of the oscillatory solutions without having to solve the equations exactly. It is instructive to present some simple basic features of GWs and their emission by looking for differences and similarities between GR and electromagnetism. Electromagnetic waves are emitted whenever a distribution of electric charges evolves in time with a breaking of spherical symmetry. This fact is connected to the vectorial nature of the electromagnetic field (the photon is a massless spin-1 particle) and to Gauss theorem. If the spherical symmetry is preserved during the time evolution of the charge distribution, the electric field remains constant, or, in other words, there is no physical scalar part in the electromagnetic field. An equivalent statement is that time variations of the electric multipoles higher or equal to the dipole are necessary for electromagnetic wave emission to happen. In an analogous fashion, one may show that the rank 2 tensor field describing the gravitational field in Einstein’s theory does not include any physical vectorial or scalar parts. This means (1) that the graviton is a massless spin-2 particle and (2) that for GWs to be emitted a 1.2 Gravitational Wave Emission 9

Figure 1.1. Orbital period decrease of the PSR B1913+16 system measured by the cumulative shift in the time at periapsis (vertical axis) as a function of time (horizontal axis) from 1975 to 2005. The continuous curve represents the prediction of General Relativity for energy losses from gravitational radiation. (Figure taken from [4].)

mass-energy distribution must posses time evolving mass multipoles of quadrupole order at least.

Let’s push the parallelism with electromagnetism further. In Maxwell’s theory, electromagnetic wave emission is achieved not only through a non-spherical time evolution of a charge distribution (i.e. electric multipole variation), but also through time variations of magnetic multipoles. Since magnetic multipoles are equivalent to moving electric multipoles, the analogy between electromagnetism and GR intuitively shows that a mass-energy distribution keeping the same shape but having (at least) quadrupolar internal motions also 10 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

Characteristic Electromagnetic Waves Gravitational Waves Medium Space Spacetime Source Incoherent dipole motions of Coherent quadrupole motions charged particle distributions of mass-energy distributions Frequency > 107 Hz < 104 Hz Interaction with matter Absorption and scattering; negligible; could probe early information only from outer Universe, superdense NS layer of objects is carried cores, supernovae centres... Detectors Unidirectional Omnidirectional Gauge boson (*) Photon, spin 1 Graviton, spin 2 Table 1.1. Comparison of electromagnetic and gravitational waves. (*): in the absence of experimental evidence and a coherent theory of quantum gravity, it is unknown whether the graviton is a gauge boson or not.

emits GWs through current multipoles or gravitomagnetic multipoles1. A geometrical criterion for gravitational radiation emission is not enough though: if one wants a source to be relevant for detection, emissivity must be considered too. Using post- Newtonian calculations, the gravitational emissivity of a body with mass M, characteristic radius R and characteristic frequency of its internal motions ν is found to be

dE G ∼ N M 2R4ν6 (1.1) dt c5

5 −53 3 −2 −1 where GN is Newton’s constant. The factor GN/c ' 2.74 · 10 s m Kg clearly dominates for terrestrial/human size objects: no GW emitter may be built in a lab. However, 2 introducing in Eq. (1.1) the Schwarzschild radius of the source (RS = 2GNM/c ) and the typical velocity of its internal motions v ∼ νR we get

dE c5 R 2 v 6 ∼ S . (1.2) dt GN R c

Therefore the ∼ 10−53 factor is replaced by its inverse ∼ 1053, i.e. the GW emission is important, if the candidate source

• is a compact object (high M/R ∼ RS/R), as we have previously mentioned

• has relativistic internal motions2 (v ∼ c).

As a consequence, binaries containing black holes and neutron stars and isolated rotating and oscillating neutron stars are good candidate sources: this thesis addresses the modelling of such sources in Part I and Part II respectively.

1Think of a spherical ball of ﬂuid with quadrupolar internal motions). 2The internal motions must also be coherent in order to avoid destructive interferences between various contributions to the time variations of the multipoles. 1.3 Gravitational Wave Detection 11

1.3 Gravitational Wave Detection

Since Weber’s first attempts to detect GWs, two distinct experimental research areas and technologies have developed: resonant (“bar”) detectors and interferometric detectors. With the former, one monitors the vibration modes of a test-mass at very low temperature. Since a gravitational wave varies proper lengths, if we model the test-mass as an harmonic oscillator we see that a wave may put the oscillator in resonance and hence make it vibrate. With interferometric detectors, on the other hand, one monitors the relative distance between suspended mirrors that play the role of test-masses: such distance varies when gravitational radiation crosses the detector and it is therefore monitored by measuring with interferometric techniques the optical path of laser light circulating in the detector [28]. At the moment, several cylindrical bar detectors are in function: ALLEGRO (Louisiana State University), EXPLORER (CERN) and the more recent third generation resonant detectors AURIGA (INFN, Laboratori Nazionali di Legnaro, Italy) and NAUTILUS (INFN, Lab- oratori Nazionali di Frascati, Italy). A second class of resonant antennas has spherical/quasi- spherical shape and may be full or empty. Among such detectors we mention MiniGRAIL (Leiden, The Netherlands), and OMNI-1 and TIGA in design phase in Brazil and the United States respectively. As far as interferometric detectors are concerned instead, there exist four ground-based interferometric antennas, which are operative or undergoing upgrades: Virgo (Italy-France, Italian based) [9], LIGO (United States) [7], GEO600 (U.K.-Germany, German based) [6] and TAMA300 (Japan) [8]. Altogether, they form, along with the various bar detectors, a whole international network of GW detectors in order to reduce the false detection rate and improve the source sky localization. The main advantage of interferometric antennas is that they are sensitive to a large range of frequencies, whereas resonant detectors operate in a narrow frequency band. Ground-based interferometers are sensitive in the interval 10 Hz ≤ ν ≤ 104 Hz with a peak in sensitivity around 200 Hz (see AppendixA), while the narrow bands of the resonant antennas are centred around 1 kHz. By 2010, planned upgrades on the LIGO and Virgo detectors are expected to improve their amplitude sensitivities by a factor of ∼ 2. Additional upgrades, planned to be completed by the end of 2014, will give us advanced detectors about 10 times more sensitive to gravitational wave strain than current detectors, and therefore with about 1000 times the reach in terms of volume. These final upgraded detectors are named Advanced Virgo and Enhanced LIGO. Below the ranges just mentioned, the noise due to time varying Newtonian gravitational fields generated close to Earth’s surface (e.g. motions in the atmosphere and seismic waves) makes a non-terrestrial observation necessary. In this context, NASA and ESA are collaborating in order to realise the space interferometer LISA (Laser Interferometer Space Antenna) [10]. This detector, which will orbit around the Sun following Earth with a lag angle of 20◦, is designed to have a five million kilometers optical path running among its three orbiting stations disposed on the vertexes of an equilateral triangle. The frequency range that LISA will address is 10−4 Hz −10−1 Hz, with a maximum sensitivity around 10−2 Hz. Finally, other two interferometric antennas have been proposed to cover a third frequency interval (10−2 Hz −10 Hz with a sensitivity peak around 10−1 Hz): these are Japan’s DECIGO (DECIhertz Gravitational wave Observer) [29] and NASA’s BBO (Big Bang Observer) [30]. At the end of 2009, no direct detection of GWs has been achieved yet. This is clearly one of the main goals of interferometric GW antennas, but it is worth reminding that, even 12 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

Detector Location Date arm length [m] Virgo Cascina, Italy Operational 3000 L1 Livingston, LA, USA Operational 4000 LIGO H1 Hanford, WA, USA Operational 4000 H2 Hanford, WA, USA Operational 2000 GEO600 Hannover, Germany Operational 600 TAMA300 Mitaka, Japan Operational 300 LISA Space-based ∼ 2018 5 × 109

Table 1.2. Main existing or planned interferometric gravitational wave detectors.

if the ﬁrst direct detection of a GW signal will probably come from one or several of the mentioned experiments, they have been designed as gravitational telescopes (working as whole) and not as detectors to prove the existence of GWs.

1.4 Compact Objects

In our comments about Eq. (1.2), we stated that good candidate sources involve compact objects. Formation, structure and evolution of compact objects are complex subjects [31]. Compact objects are astrophysical objects essentially static over the Universe lifetime3 which are “born” when normal stars “die”, that is, when most of their nuclear ﬂuid has been consumed. Generally speaking, there are three classes of compact objects: white dwarfs (WDs), neutron stars (NSs) and black holes (BHs). In the rest of the thesis, however, we will follow the common jargon of ground-based GW detector science and use the terms “compact binary” to indicate a binary containing BHs and NSs and not WDs. Compact objects differ from normal stars in two main aspects:

1. they cannot contrast gravitational collapse with thermal pressure because they have no nuclear fuel to burn; WDs are supported by electron degeneracy pressure; NSs are supported by neutron degeneracy pressure; BHs, instead, are completely collapsed stars that had no way of supporting self-gravity and avoiding to become a spacetime singularity;

2. they are extremely small; the radius of a compact object of mass M is much smaller than the radius of a normal star having the same mass, and therefore its surface gravity is much larger (see Table 1.3).

In this section, particular attention is dedicated to neutron stars and compact binaries, the two types of GW sources addressed by this thesis. Moreover, a short account on gamma-ray bursts is given since these seem to be linked to compact object progenitors.

3With the exception of mini-black holes. 1.5 Neutron Stars 13

Object Mass Radius Mean density Surface Potential 3 2 [g/cm ] GN M/(Rc ) −6 Sun M R 1 10 −2 7 −4 WD . M ∼ 10 R 10 10 −5 15 −1 NS . (1 − 2)M ∼ 10 R 10 10 2 3 BH Any 2GN M/(Rc ) ∼ M/R 1 Table 1.3. Main features of the Sun compared to those of the three classes of compact objects.

1.5 Neutron Stars

In 1934, soon after Chadwick discovered the neutron [32, 33] and Landau thought of stars entirely composed of neutrons [34], Baade and Zwicky suggested that neutron stars could be formed in a supernova event due to the collapse of the massive stellar core [35]. In 1939 Oppenheimer, Volkoff and Tolman first derived the equations of stellar structure from Einstein’s field equations, assuming that neutron stars were gravitationally bound states of a neutron Fermi gas [36, 37]. The first observational evidence of neutron star existence arrived thirty-four years later when Hewish and Bell [38] discovered the first radio pulsar and confirmed what Pacini had predicted in 1967 [39]: neutron stars might be observable at long radio wavelengths if they were magnetised with misaligned magnetic and rotation axes. More than 1100 pulsars have been found since Hewish and Bell’s discovery. Two additional pulsars were discovered shortly later, the Crab and Vela pulsar; both were found within supernova remnants, the Crab and Vela nebulae, and both have very short periods. Their periods of 33 ms and 89 ms provided convincing evidence that pulsars are rotating neutron stars which form in supernova explosions. Today, neutron stars are one of the most interesting objects in our Universe as they are the most compact known objects without event horizons and therefore serve as extraordinary laboratories for dense matter physics, e.g. [40, 41, 42, 43, 44, 45]. Neutron stars have central densities of the order of 1015 g/cm3 (i.e. 1014 times denser than Earth) and most of them are known to have a magnetic field of the order of 1012 Gauss. Understanding the properties of matter in the interior of such extreme objects is one of the most complex tasks in physics and astrophysics. Their global properties, such as their mass and size, are regulated by nucleon interactions at such high densities. The neutron star radius, for example, is controlled by the properties of nuclear forces in the vicinity of the equilibrium density of nuclear matter, i.e. 14 3 −3 ρs ' 2.7 × 10 g/cm or ns ' 0.16 baryons/fm , in particular by the density dependence of the nuclear symmetry energy. This means that the radius and many other predictions about neutron stars are microphysics-dependent, that is, they depend on the nuclear equation of state (EOS) used to model the neutron star at microscopic level. However, the behaviour of matter at such high densities and, therefore, the internal composition of the cores of neutron stars is currently still poorly understood. We thus expect the comparison between theoretically determined values of these quantities and their observation to yield information on the state and composition of matter at supranuclear densities, which prevail in the cores of neutron stars and are unobtainable in laboratory. 14 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

1.5.1 Neutron Star Observed Properties Nowadays, the identification of pulsars as highly magnetized rotating neutron stars is very well established. Due to a misalignment of their rotational and magnetic axes, these stars emit magneto-dipole radiation (at the expense of their rotational energy) in the form of radio waves. An observer lying on the radiation cone that is swept out by the rotating star has the chance of observing it as a pulsed source if it is not too distant. Several hundred pulsars are known today and were discovered in large surveys conducted in the 1970s at Arecibo, Jodrell Bank and Parkes4. All known pulsars lie within our Galaxy and in the nearby Large Magellanic Cloud galaxy; indeed, most observed pulsars lie within a third of the Galaxy radius from the Sun. We expect other galaxies with populations of massive stars (M > 8 M ) to also contain pulsars, but they are too distant for us to detect the dim radio signals. The first radio binary system was found by Hulse and Taylor [26], who were able to determine many parameters, such as the masses, the orbital period, the separation, the orbital inclination and the inward spiraling of the neutron stars due to the emission of gravitational waves (Figure 1.1), which provided the first (indirect) evidence for the existence of gravitational waves. Several recent astronomical observations of neutron stars in binary systems (with a white dwarf, a main-sequence star or a neutron star companion) and celestial mechanics allow us to determine their gravitational masses. The most accurately measured masses come from timing observations of the radio binary pulsars. The mass is usually determined from measurements of the orbital parameters of the binary system, such as the orbital period P , the projection x of the binary semi-major axis on the line of sight, the eccentricity, etc. The observed parameters are related to the masses m1 and m2 of the binary constituents through a mass function fM = fM (m1, m2, P, x). Relativistic corrections to the orbital parameters are usually parametrized in terms of one or more post-Keplerian parameters describing, for instance, the advance of the periastron, the orbital decay due to gravitational wave emission, the gravitational redshift. In the context of General Relativity, the measurements of the mass function fM and of one of the post-Keplerian parameters is sufficient to uniquely determine m1 and m2. A sufficiently well-observed system can have masses determined to impressive accuracy. The textbook case is the binary pulsar PSR 1913 + 16, in which the masses are (1.3867 ± 0.0002) M and (1.4414 ± 0.0002) M , respectively [4]. Figure 1.2 shows the measured neutron star masses as of November 2006. An important comment about Figure 1.2 is that mass determinations in binaries with white dwarf companions seem to show a broader range of neutron star masses than binary neutron star pulsars. An exceptionally intriguing case is that of PSR J0751 + 1807 in which the estimated mass with 1σ error bars is (2.1 ± 0.2) M [46]; additionally one of the two pulsars Ter 5 I and J has a reported mass larger than 1.68 M to 95% confidence [47]. Perhaps when double neutron star binaries are formed, a particular combination of evolutionary circumstances occurs (leading to a restricted range of masses) which is relaxed for other neutron star binaries [48]. Significant developments in the mass determinations for neutron star-white dwarf binaries and for neutron stars in general are very much awaited for. Most masses are presently known only with large uncertainties and the measurements are consistent with a range of masses spanning (1−1.8) M ; in the meantime, the evidence of large neutron star masses should be handled with extreme caution. Continued observations are necessary to clarify this situation.

4Arecibo is a Radio Observatory located in Puerto Rico, Jodrell Bank is in England and Parkes is in Australia. 1.5 Neutron Stars 15

Figure 1.2. Measured and estimated masses of neutron stars in radio binary pulsars (gold, silver and blue regions) and in X-ray accreting binaries (green). For each region, simple averages are shown as dotted lines; weighted averages are shown as dashed lines. (Figure from [40] where bibliographical references for the data are provided.)

Measurements of the neutron star radius are far less precise than mass measurements. Their small size and the large distance from us make the direct determination of radii very difficult to achieve. There are, however, several techniques and methods used to estimate the actual radii of neutron stars. Recent observations include upper limits from rapidly rotating neutron stars, estimates from the thermal emission of cooling neutron stars, estimates from 16 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs the properties of sources with bursts or thermonuclear explosions on their surfaces, and estimates of crustal properties from (1) glitches of pulsars, (2) “star-quakes” occurring in the aftermath of giant flares from soft gamma repeaters, and (3) cooling timescales during periods of quiescence in between X-ray bursts from accreting neutron stars in low-mass X-ray binaries5 [40]. For example, from thermal observations of the neutron star surface in low-mass X-ray binaries, one may obtain the so-called radiation radius R R = , (1.3) ∞ p 2 (1 − 2GN M/Rc ) which results from a combination of flux and temperature measurements, both redshifted due to the strong gravitational field on the neutron star surface. For a specific star, a given value of R∞ implies R < R∞. Since R∞ is clearly mass-dependent, its measurement sets upper limits on both the neutron star radius and its mass, but without an independent mass estimate, only limited constraints on the radius value are possible. Precious additional information may come from recent studies that are aimed at determining the neutron star mass-radius ratio from redshift measurements. Cottam et al. [49] have reported that the Iron and Oxygen transitions observed in the spectra of 28 bursts from the X-ray binary EXO0748-676 correspond to a gravitational redshift z = 0.35, where 2GM −1/2 z = 1 − − 1 , (1.4) Rc2 yielding in turn a mass-radius ratio M/R = 0.153 M /km. Later on, 45 Hz burst oscillations in the average power spectrum of 38 thermonuclear X-ray bursts from this same source were reported and this frequency has been interpreted as the NS spin frequency [50]. The authors in [50] have shown that the widths of the lines reported in [49] are consistent with a 45 Hz spin frequency as long as the star radius is in the range (10 − 15) km. However, the identification of spectral lines in [49] is still controversial [51]. A second example of possible accurate radius estimates is offered by the the double- pulsar system PSR J0737-3039A&B [41, 42], which has also opened a new window for testing fundamental physics under extreme conditions thanks to its very short orbital period (Pb = 2.45 h) [52]. After a few years of high precision pulsar timing, this system could provide, a measurement of higher order relativistic corrections to the advance of periastron ω˙ SO, due to spin-orbit coupling and this could eventually lead to a determination of the moment of inertia IA of the star A through the relation

IA ω˙ SO ∝ 2 , (1.5) Pb M

Pb is the orbital period and M is the total gravitational mass [52]. The measurement of IA would have an enormous importance on discriminating among families of equations of state. Moreover, the radius of a neutron star could already be constrained using moment of inertia measurements having a 10% uncertainty [40]. The general consensus is that the radius of a neutron star is of about 9 − 16 kilometers. In Figure 1.3 we show the dependence of the neutron star mass upon its radius for the models of EOS employed in this thesis. APR2 and APR2 (II) only differ in the description of the crust: as mentioned in Section 1.5.2, this causes slight changes in the stellar structure.

5These are accreting binary systems in which the orbital companion of a neutron star is a low-mass main- sequence star or a white dwarf. The donor component usually ﬁlls its Roche lobe and therefore transfers mass to the compact star. 1.5 Neutron Stars 17

2.2 2 1.8 1.6 1.4 1.2 1 0.8 APR2 M [Solar Masses] BGN1H1 0.6 BPAL12 GNH3 0.4 APR2 (II) 0.2 9 10 11 12 13 14 15 R [km]

Figure 1.3. Neutron star mass versus radius curves for the ﬁve equations of state adopted in this thesis and described in Section 5.4 (also see Section 1.5.2 for nomenclature). APR2 indicates the crust-core combination BPS+HP94+Sly4+APR2, BGN1H1 the BPS+HP94+Sly4+BGN1H1 combination, BPAL12 the BPS+HP94+Sly4+BPAL12 combination, GNH3 the BPS+HP94+Sly4+GNH3 combination and APR2(II) the BPS+PRL+APR2 combination. 18 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

1.5.2 Neutron Star Anatomy As one moves inward from the surface of a neutron star to its centre, the density increases 15 3 enormously (up to ∼ 10 g/cm ) and very rapidly (RNS ∼ 10 km). This creates a very rich and complex inner structure which is so far only partially understood. A neutron star contains a non-uniform crust (generally subdivided into outer and inner crust) above a uniform liquid mantle (or outer core) that, in turn, is located above a possibly exotic core. The whole star is enclosed in a magnetosphere. Figure 1.4 displays what is believed to be an accurate rendition of a neutron star [53].

Figure 1.4. Pictorial representation of the structure and phases of a neutron star. (Figure taken from [53].)

• The magnetosphere is a plasma, the density of which depends on the magnetic field of the star. In the case of a standard neutron star, the magnetic field may be of the order of 1012 G; when the field reaches very intense values such as 1015−16 Gauss, one talks of magnetars (magnetic neutron stars). The mangetic field may be measured by detecting the cyclotronic resonances of protons and electrons in X-ray spectrum observations. Millisecond pulsars have fainter magnetic fields of about 109 Gauss. 1.5 Neutron Stars 19

This seems to indicate that they are very old neutron stars (i.e. they are billions of years old) which have been spun up by mass accretion in binary systems: this is known as pulsar recycling.

• The outer crust of a neutron star is a ∼ 300 m thick non-uniform region whose density spans the interval (104 − 1011) g/cm3. At such densities, the electrons move freely throughout the crust; they are present in order to preserve the star charge neutrality. On the other hand, at the lower end of this density interval, the density is too small for nucleons to form stable clusters heavier than 56F e: therefore the nucleons cluster into iron nuclei which in turn arrange themselves in a crystalline face-centered-cubic lattice in order to minimize their overall Coulomb repulsion. As the density increases (and one moves deeper into the star), 56F e is no longer the most energetically favored nucleus. This happens because, as the density rises, the increase of the electronic contribution to the energy is more rapid than the corresponding nuclear contribution, which “pushes” towards a symmetry in the number of protons (Z) and neutrons (N). As a result, the most energetically advantageous process that may occur is the capture of energetic electrons by protons, the excess energy being carried away by neutrinos. The nuclear lattice, therefore, starts presenting nuclei with a neutron excess larger than that of 56F e. The neutron excess of the Coulomb lattice 11 3 progressively grows as the density continues to increase. ρd = 4 × 10 g/cm is the critical density for this process: it is the neutron drip line and at such density the nuclei are unable to hold any more neutrons. All in all, the physics of the outer crust is dominated by the competition between an electronic contribution that attempts to drive the system toward more neutron-rich nuclei (i.e. smaller proton fractions) and the nuclear symmetry energy that opposes such a change. Both the neutron- rich nuclei sequence and the neutron-drip density depend critically on the symmetry energy, which imposes a “price to pay” as the system departs from the symmetric (N = Z = A/2) limit. In order to describe the outer crust, in Section 5.4 of this thesis, we shall use the Baym-Pethick-Sutherland (BPS) equation of state [54] or alternatively a combination of the BPS equation of state and the Haensel-Pichon (HP94) equation of state [55].

• The inner crust of a neutron star is about 500 m thick; it spans the density interval which goes from the neutron-drip density (∼ 4 × 1011 g/cm3) to the density at which 14 −3 uniformity in the system is restored, i.e. 1/3ρs to 1/2ρs, where ρs = 2.7×10 g/cm is the normal nuclear matter saturation density. At these densities, the length scales of short-range nuclear attraction and long-range Coulomb repulsion become comparable and start competing dynamically: the result is a so-called Coulomb frustration6 which promotes rich and complex structures. This transition region from the highly- ordered crystal to the uniform liquid mantle (where the aforementioned length scales are separated again) is complex and not well understood. It has been speculated that the transition to the uniform phase must go through a series of changes in the dimensionality and topology of these complex structures known as nuclear pasta. The formation of pasta phases (mainly spheres, rods and slabs) is intimately related to the density dependence of the symmetry energy, a poorly known quantity: it requires models with a stiff symmetry energy [56]. Even though the dynamics and structure of

6Frustration develops from the inability of a physical system to satisfy all its elementary interactions. 20 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

the crust (especially the inner one) is intriguing, its structural impact on the neutron star is rather modest as it only makes up about 10% of the size of the neutron star and contains about 2% of its mass only. In Section 5.4 of this thesis, we shall describe the inner crust in two ways: with the Pethick-Ravenhall-Lorenz (PRL) equation of state [57] or with the Douchin-Haensel (SLy4) equation of state [58, 59].

• The mantle or outer core spans a range of densities which go from about 1/2ρs up to about 2.5ρs. It is a few kilometers thick and is made up of asymmetric nuclear matter with electrons and muons in β-equilibrium. Muons start to appear once the chemical potential of the electrons exceeds the muon rest energy (105 MeV). • The (inner) core is one to two kilometers thick and very little about it is known. As Figure 1.5 shows, there are lots of conjectures on the composition of this part of the neutron star; existing models include meson (kaon or pion) condensates [60], transitions to a quark-gluon plasma [61], the appearance of hyperons (in different kinds and amounts) [62]. The large number of models follows from the present lack of knowledge in strong interactions in superdense matter and of experimental evidence for nucleon-hyperon and hyperon-hyperon interactions [40]. Experiments are, in fact, strongly limited by the instability of hyperons at terrestrial densities. Since the lifetime (from the birth to the merger) of observed binary neutron stars is longer than 100 Myrs [63], the thermal energy per nucleon in each neutron star will be at least two orders of magnitude lower than the Fermi energy of neutrons at the

Figure 1.5. Pictorial representation of a the structure of a neutron star and of the several suggested hypothesis for its core composition. 1.5 Neutron Stars 21

onset of the merger [31]: this implies that when modelling binary neutron stars just before the merger, it is appropriate to use cold nuclear equations of state, as we will do in Part I of this thesis.

In Section 5.4 we shall adopt four different core descriptions: the Akmal-Pandharipande- Ravenhall (APR2) EOS [64], the Balberg-Gal (BGN1H1) EOS [65], the Bombaci-Prakash et al. (BPAL12) EOS [66, 67] and the Glendenning (GNH3) EOS [68]. These four barotropic equations of state were chosen because they span the set of state equations currently available in the mass-radius plane (e.g. [40]). More details about them will be given in Section 5.4.1 and 5.4.2. The neutron star mass versus radius curves they yield in combination with the aformentioned crust equations of state are displayed in Figure 1.3.

1.5.3 Neutron Stars as Continuous Sources Astrophysical sources of gravitational waves are generally grouped into four (standard) categories: compact binary coalescences, bursts, continuous sources and stochastic backgrounds. In this section we shall discuss neutron stars as continuous (i.e. almost monochromatic) gravitational wave sources, whereas a discussion on compact binary coalescences is postponed to Section 1.6.1. We shall not examine burst sources and stochastic backgrounds instead, as these two topics are not relevant for this thesis. No gravitational wave source is truly monochromatic, as the emission of gravitational waves removes energy from the source which is producing them by affecting the characteristic frequency itself. In the case of a point-mass binary system, the total energy reservoir is given by −m1m2/(2r) and thus energy loss by GW emission induces a reduction of the orbital radius (−m1m2/[2(r − ∆r)] < −m1m2/(2r)) and therefore decreases the orbital period; the energy reservoir for a rotating star is, instead, its rotational kinetic energy Mω2R2/2 which responds to an energy loss by decreasing its value as M(ω − ∆ω)2R2/2, that is, by increasing its period. Binary star systems formed by ordinary stars are a source for continuous nearly monochromatic gravitational waves; they will in fact emit gravitational waves at twice their orbital frequency which is typically less than 10−3 Hz: these sources will be available to LISA. Spinning neutron stars form a more promising class of continuous sources for current ground-based detectors and their future upgrades; they may in fact emit gravitational waves if

1. non-axisymmetric distortions form

2. free precession around the spin axis occurs

3. unstable oscillation modes build up in the stellar ﬂuid.

In all cases the spinning neutron star is non-axisymmetric and therefore possesses a spinning quadrupole and emits gravitational waves. The following are typical axial symmetry deviation scenarios.

• Bumpy neutron star: a large deformation (“mountain”) forms on the crust of an accreting neutron star.

• Strong internal magnetic ﬁelds produce deformations (“magnetic mountains”). 22 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

Assuming the star spin rate ν0 to be constant, the deformations result in a continuous gravitational wave emission at the frequency 2ν0. The gravitational wave strain is instead given by [69] −26 ν0 10 kpc ε h0 ∼ 2 · 10 −6 , (1.6) 1 kHz dobs 10 where dobs is the distance to the spinning star and I − I ε = xx yy (1.7) Izz is the equatorial ellipticity, which is deﬁned in terms of the moment of inertia tensor in the principal frame. The value of ε is very uncertain. Neutron stars showing a misalignment of a principal axis and the spin axis are known as wobbling neutron stars; the misalignment may be due, for example, to strong magnetic ﬁelds and would lead to gravitational wave emission at the frequencies ν0 + νprec and 2ν0 with a strain amplitude of 2 −27 θw 10 kpc ν0 h0 ∼ 10 , (1.8) 0.1 dobs 500 Hz where the “wobble angle” θw is the angle between the spin axis and the precessing principal axis [70]. It is believed that at birth or during accretion, rapidly rotating neutron stars may be subject to various non-axisymmetric instabilities, which would lead to gravitational wave emission. The CFS [71, 72, 73, 74] and the r-mode instability [75] have been proposed as mechanisms which turn neutron stars into sources of gravitational waves. Searches for gravitational waves from spinning neutron stars are referred to as “pulsar searches”; they have three main advantages over other gravitational wave searches:

1. the spin rate ν0 is precisely known for gravitational wave searches involving known pulsars

2. integration over entire gravitational wave detector science runs is possible, allowing much smaller gravitational wave strain amplitudes to emerge from the noise than other types of searches

3. it is much easier to produce a detector calibration at a single frequency than over the entire band of the detector. Observations of pulsars show that their spin rates decrease over time (spin-down). This decrease is thought to be due to a combination of mechanisms: magnetic dipole radiation, particle acceleration in the magnetosphere, and emission of gravitational waves [76]. An upper limit on gravitational wave emission may hence be set by measuring the spindown rate via electromagnetic observation.

1.6 Compact Binaries

It should be clear from the ﬁrst paragraph of Section 1.4 that there can be three kinds of compact binaries: 1.6 Compact Binaries 23

1. binary black holes (BBHs, or BH-BH binaries)

2. black hole-neutron star binaries (BH-NS, or mixed binaries)

3. binary neutron stars (BNSs, or NS-NS binaries).

In the final moments of their evolution, compact binaries are among the leading potential sources for detection by gravitational wave observatories: this issue will be considered in Section 1.6.1. Moreover, as we shall see in Section 1.7, black hole-neutron star and neutron star-neutron star binary mergers are now thought to be the leading candidates for explaining short gamma-ray bursts (SGRBs). Since SGRBs will be one of our main interests in Part I of this thesis, we will now discuss binaries containing at least a neutron star and leave out binary black holes, i.e. the third class of compact binaries. NS-NS binaries were first detected in 1975, when pulsar PSR1913+16 [4] was observed in close orbit around an NS companion [26]. This discovery eventually led to the 1993 Nobel Prize in physics for Hulse and Taylor. Several such systems are now known and monitored; the most famous amongst them is J0737−3039 [41], a binary consisting of two observed pulsars, which allows for the prospect of very stringent tests of GR [52]. On the other hand, there have been no confirmed observations of BH-NS binaries: in addition to a strong likelihood that they are less numerous than NS-NS systems, this also reflects the lower probability of detecting such systems given current detection limits. Such systems are an expected byproduct of binary stellar evolution, and properties of their population may be inferred from the results of population synthesis calculations tuned to the observed NS-NS sample (see, e.g., [77]). Close BNSs and BH-NS binaries — i.e., those for which the merger timescale is smaller than a Hubble time — are believed to typically form through similar evolutionary channels 7 in the field of galaxies [77]. Starting as a high-mass binary system (M1,M2 & (8−10)M in order to ensure a pair of supernovae), the more massive star, i.e. the primary, evolves over millions of years, eventually abandons the main sequence, passes through giant phases and, after undergoing a Type Ib, Ic or II supernova, leaves behind what will become the heavier compact object, i.e. the BH for a BH-NS binary or the more massive NS for a BNS (first four steps in Figure 1.6). When the secondary evolves off the main sequence, it forms a common envelope around the primary compact object when it reaches the giant phase (fifth and sixth step in Figure 1.6). Dynamical friction shrinks the orbital separation dramatically, until sufficient potential energy is converted into thermal energy to evaporate the envelope: without this step, binaries would remain too wide to merge through the emission of GWs within a Hubble time. Eventually, the core of the secondary undergoes a supernova (left leg of Figure 1.6): depending on the magnitude and orientation of the supernova kick, the system becomes either unbound or it is turned into a tight binary. During the common envelope phase, there is also the possibility for the primary to merge with the helium core of the secondary to form a still hypothetical Thorne-Zytkow object (right leg of Figure 1.6); the fate of these stars remains unclear: single (possibly massive) NSs or BHs should descend from them [79]. This evolutionary pathway has important consequences for the physical parameters of NS-NS and BH-NS binaries, leading to preferred regions in phase space. For example, since the primary accretes matter during the common envelope phase, it will be spun up to rapid

7They may also both form through many-body processes in the high-density regions of globular clusters, but the contribution of this evolutionary channel to the total population is small and poorly constrained [78]. 24 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

T~3 Myr, N~104 Two OB main-sequence stars

T~104 yr, N~30 More massive star (primary) overfills Roche lobe

5 Helium-rich WR star T~2·10 yr, N~500 with OB-companion

−2 -1 Primary explodes as ν~10 yr type Ib Supernova and becomes a neutron star or black hole

4 Secondary is close to Roche lobe. T~10 yr, N~100 Accretion of stellar wind results in powerful X-ray emission

4 Helium core of the secondary T~10 yr, N~30 with compact companion inside mass-losing common envelope

4 T~2·10 yr, N~50 Components merge. Wolf-Rayet star with compact Red (super)giant with neutron companion surrounded by star or black hole core expanding envelope (Thorne-Zytkow object)

8 Secondary explodes as type T ~10 Gyr, N~10 −4 -1 Single neutron star Ib Supernova, ν∼10 yr or black hole

5 T~10 Gyr, N~10 Supernova explosion Binary relativistic disrupts the system. star Two single neutron stars or black holes Merger of components with a burst of emission of gravitational waves, 53 -5 -1 E~10 erg, ν~10 yr

Figure 1.6. Evolutionary scenario for the formation of neutron stars or black holes in close binaries (figure taken from [79]). T indicates the duration of the evolutionary phase, N the expected number of systems in that phase in our Galaxy. Wolf-Rayet (WR) stars are evolved, very hot, massive −5 stars (> 20 M ), which lose mass rapidly (10 M /yr) by means of a very strong stellar wind (up to 2000 km/s). OB refers to the two hottest classes of the Morgan-Keenan system of stellar classification. 1.6 Compact Binaries 25 rotation. In NS-NS binaries, one expects this process to reduce the magnetic field down to levels seen in “recycled” pulsars, which is typically one or two orders of magnitude lower than for young, unrecycled pulsars. The secondary, on the other hand, never undergoes an accretion phase: it is therefore expected to spin down rapidly from its nascent value and is more likely to maintain a strong magnetic field. Several binary evolutionary parameters may be constrained in similar ways, but others (such as the maximum allowed NS mass, the nuclear EOS, the supernova kick velocity distribution and especially the mass transfer process efficiency during the common envelope phase) remain extremely uncertain or poorly understood. Assuming the current general consensus on all these parameters to be correct, BH masses in close BH-NS binaries are likely to fall primarily at values near MBH = 10 M . Previous calculations assuming large accreted masses in the common envelope phase typically favored binary mass ratios closer to unity.

1.6.1 Compact Binary Coalescences

Compact binary coalescence (CBC) events, the inspiral and merger of binary systems of compact objects, are a primary target for gravitational wave searches and one of the most promising for a ﬁrst direct gravitational wave detection. The energy of gravitational radiation emitted in CBC events is, in fact, large and the frequency of the gravitational waves emitted in the last moments of life of the binary may typically reach the interval in which current interferometric detectors are mostly sensitive (see AppendixA). Searches for gravitational waves from CBC events have been and are performed for binary neutron stars (BNSs), binary black holes (BBHs), black hole-neutron star systems (BH-NS) and primordial black holes. The gravitational wave emission from some parts of the life cycle of compact binaries is well-modelled if compared to burst sources like supernovae and the expected frequencies of some systems are near the sweet spot of ground-based interferometric gravitational wave detectors. For example members of a BNS pair can orbit at frequencies of up to ∼ 1500 Hz, thus sweeping through (most of) the detector sensitivity band [70]. Compact binary coalescence may be subdivided into three distinct stages :

• the inspiral stage

• the merger stage

• the ringdown stage.

In each stage, gravitational waves are emitted. During the inspiral stage, the members of the binary system are well separated in space and the system evolves in an orderly fashion as the binary orbit decays due to loss of energy via gravitational wave emission. Gravitational wave emission from the inspiral stage is modelled well enough for templated searches to take place. For non-spinning systems the inspiral strain waveform at the Earth may be written as

1 Mpc h(t) = A(t) cos(φ(t) − φ0) , (1.9) deff where A(t) and φ(t) depend on the masses of the binary constituents, φo is an unknown phase parameter, and deff is the effective distance, that is, the distance at which a merger event could be detected if the binary system were optimally oriented and located relative to 26 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

8 the detector . deff is given by d deff = q > d , (1.10) 2 2 2 2 2 F+ cos ι + F×(1 + cos ι) /4 where F+,× are the antenna responses of the detector to the plus and cross polarizations of incoming gravitational waves and ι is the inclination angle between the binary system and the detector. In the approximation of point mass non-spinning constituents the parameter phase space for templated merger searches is defined by the two masses. Extrinsic parameters like the source effective distance, the inclination angle and the unknown orbital phase do not increase the dimensionality of the template space. The frequency and amplitude of the gravitational wave signal given by Eq. (1.9) increase in time and therefore inspiral signals are known as “chirp” signals. The merger stage is difficult to model and is the focus of much of the ongoing research in numerical relativity9. For the final fate of compact binaries, there are the following possibilities. • BBH case: the two BHs fall into each others event horizons and merge into a single black hole, which, according to the No Hair Theorem [80], is a Kerr black hole [81]. • BH-NS case: depending on the mass ratio and on the NS compactness, the NS may be tidally disrupted or not; again, in either case, the remnant eventually settles downs to a Kerr black hole. • BNS case: depending on the mass ratio and on how compact the NSs are, one of the NSs may undergo tidal stripping; the remnant may be a hyper-massive NS which leads to a Kerr black hole or directly a Kerr black hole. During the ringdown stage, the black hole resulting from the merger is in an excited state and decays through gravitational wave emission from damped non-spherically-symmetric ringdown modes (Section 7.2). Merger rates are assumed to depend on the rate of star formation in a volume, which is measured by the blue luminosity in that volume. While several steps of the evolutionary sequence leading to a compact binary remain poorly constrained, overall population estimates may still be made to within 1 − 2 orders of magnitude. Population synthesis calculations are matched to the observed supernovae rate and BNS population [82]; BNS merger rates can therefore be estimated from four observed binary pulsar systems, while BBH and BH-NS merger rates are based on theoretical populations studies since there are no observations available yet [79]. Converting these rates into detection rates for a specific detector is complicated, as the detection range depends on the choice of the signal to noise ratio (SNR) threshold, on the detector sensitivity as a function of frequency and on the component masses which set the frequencies tracked by the inspiral waveform. To grasp the sense of merger rates for a specific detector, horizon distance is used: this is the distance at which the detector would detect an optimally oriented and located binary merger with SNR of 8. Table 1.4 gives the horizon distances for LIGO and Advanced LIGO for archetypal CBC events, while Table 1.5 reports on the detection rates of CBC events for current and advanced detectors. 8At a sky position directly at the zenith or the nadir and orbiting in a plane parallel to the detector plane. 9More about the status of the art in this field — especially about NS-NS and BH-NS coalescences since we shall address BH-NS coalescences ourselves — will be said in the opening chapter of Part I. 1.6 Compact Binaries 27

Horizon Distance (Mpc) Detector BNS BHNS BBH LIGO 20 43 100 Adv. LIGO 300 650 1600 Table 1.4. Horizon distances of LIGO and Advanced LIGO for compact binary coalescence events of archetypal systems: (1.4 + 1.4)M BNSs, (10 + 1.4)M BH-NSs and (10 + 10)M BBHs [70].

˙ −1 ˙ −1 ˙ −1 Detector Source NLow [yr ] NRe [yr ] NHigh [yr ] NS-NS 2 × 10−4 0.02 0.2 Initial BH-NS 9 × 10−5 0.006 0.2 BH-BH 2 × 10−4 0.009 0.7 NS-NS 0.4 40 400 Advanced BH-NS 0.2 10 300 BH-BH 0.5 20 1000 ˙ ˙ ˙ Table 1.5. Pessimistic (NLow), optimistic (NHigh) and intermediate (NRe) detection rates of initial and advanced ground-based GW detectors. The table is extracted from https://nrda2009.aei.mpg.de/program/Mandel_Monday.pdf.

When CBC events will ﬁnally be observed by gravitational wave detectors, we will be able to extract information about the source system parameters: the masses of the compact objects, their spins, the eccentricity of the orbits and the equation of state of superdense nuclear matter (see Section 5.4.3 and, for example, [5] and [40] for a review on these topics). 28 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs

1.7 Gamma-Ray Bursts

Gamma-ray bursts (GRBs for short) are the most electromagnetically luminous events in the universe after the Big Bang. They are brief flashes of γ-radiation detected in the energy range (0.1 − 100) MeV, with typical photon fluxes of (0.01 − 100) photons/s/cm2 and durations of (0.1 − 1000) s; they appear on average once a day at an unpredictable time and from an unpredictable direction in the sky. Ever since their accidental discovery in 1967, thousands of bursts have been detected and by now we know that their sky distribution is completely uniform (Figure 1.7), and that they do not appear to come from the Milky Way. For about thirty years after their discovery, however, studying these short-lived outbursts of the most energetic electromagnetic radiation was impossible as their positions in the sky were known with a very limited precision. The distance uncertainty, for example, was so big that astronomers did not know if GRBs occurred in our Solar System, in our Galaxy, or in the distant Universe. The situation dramatically changed in 1997, when the Italian-Dutch satellite BeppoSAX provided more precise positional information about GRBs and at a very rapid rate. On May 8, 1997 BeppoSAX detected a GRB and five days later Very Large Array (VLA) observers discovered radio emission coming from this source: this first example of radio afterglow allowed scientists to rule out some of the (many) theoretical models existing at that time. The VLA detected several other GRB radio afterglows subsequently and also revealed the expansion speed and the size of the GRB fireball. The May 8, 1997 GRB, for example, was a tenth of a light year across at its detection time and its expansion velocity was very close

Figure 1.7. Sky distribution of GRBs as observed by the Burst And Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (CGRO) (http://www.batse.msfc.nasa.gov/batse/grb/skymap/). 1.7 Gamma-Ray Bursts 29 to the speed of light [83]. Spectroscopy observations, on the other hand, have shown that GRBs occur at large distances from us, i.e. beyond the Milky Way Galaxy. Astronomers, therefore, now know that the most violent events in the current Universe occur in galaxies far from Earth. The big open question is presently “what is at the origin of GRBs”? The isotropic GRB sky-distribution (see Figure 1.7) may in principle be explained as

1. a cosmological distribution similar to that of distant galaxies and clusters, i.e. hundreds to thousands of Mpcs away

2. a distribution in an “extended halo” of our galaxy which is so large that the small dipole moment associated with our off-center locations is not noticeable, i.e. & 200 kpc away

3. a “galactic disk” distribution in which objects are sufﬁciently faint to be detectable only out to distances smaller than the width of the disk, i.e. a few kpcs away.

Interpretation number 3 clearly has difficulty in explaining the high GRB event rate (few/day), whereas interpretation 2 additionally has the difficulty of explaining the physical origin of GRBs in this “extended halo”. This leaves us with the cosmological interpretation which is also validated by the observations and analyses of afterglows and counterparts. We are thus led to two possible sources: binary neutron star or mixed binaries (i.e. BH-NS) mergers driven by gravitational wave loss and collapsars events (aka failed supernovae), in which a star undergoes core collapse to a BH with an incomplete explosion [84]. Both types of GRB progenitors should occur at a rate of 10−5/MWEG/yr and produce (1050 − 1051) ergs, so that the typical frequency and fluence are easily explained. Another very important piece of observational evidence supports the cosmological interpretation and its two-progenitor prediction. After the 1993 influential work of Kouveliotou et al. [85], based on 222 GRBs detected by the Burst And Transient Source Experiment (BATSE1) on board the Compton Gamma-Ray Observatory (CGRO), a hint that had already emerged 20 years ago became the common view: the duration distribution of GRBs is bimodal suggesting that the GRB population might not be monolithic but is most likely composed out of two major distinctive sub-populations. Kouveliotou et al.’s analysis also found that the minimum of the GRB duration distribution is around 2 s. They further showed that bursts with durations shorter than 2 s are on average composed of higher energy (harder) photons than longer bursts. It took a dozen additional years before recent observations confirmed that the two sub-populations, defined in duration-hardness space, indeed represent two distinctive physical phenomena10 [86]. Nowadays one therefore talks of:

• long-soft GRBs, whose duration is greater than 2 s and which may be associated with a collapsar progenitor

• short-hard GRBs (SGRBs), whose duration is shorter than 2 s and which may be associated with a compact binary merger progenitor.

The nature of the stellar progenitor of long GRBs has been a matter of debate for many years, until 2003, when a detection of a long GRB at a relatively low redshift (z = 0.1685) by High Energy Transient Explorer (HETE-2) led to the identiﬁcation of a type Ic supernova

10The afterglow provides a wealth of information about the physics of the burst. 30 1. An Overview on Gravitational Waves, Neutron Stars, Compact Binaries and GRBs spectrum superposed on the afterglow of this burst [87, 88]. The consensus today, which is supported by several additional evidence, is that most, and probably all, long GRBs are produced by the collapse of very massive stars that release a vast amount of energy 2 (& 0.001 M c ) in a compact region (< 100 km) on time scales of seconds to minutes. This energy source, the “central engine”, accelerates an ultra-relativistic outflow to a Lorentz factor & 100 and this outflow generates the observed γ-ray emission, and later the afterglow, at large distances from the source. SGRBs, being shorter and harder, eluded accurate localization and no SGRB afterglow was detected despite the effort until the breakthrough during the spring-summer of 2005 when Swift and HETE-2 succeeded in localizing several SGRBs, leading to afterglow detections and to the determination of their redshifts. The first conclusions were that SGRBs are cosmological, but unlike long GRBs, their progenitors are not massive stars, thereby confirming that these two observationally defined classes are distinct phenomena; on the other hand, the comparable luminosities and roughly similar afterglows of long and short GRBs, suggest that similar physical processes are involved. The leading progenitor candidate, as mentioned, is the coalescence of compact binaries with at least one NS. This idea appeared in the mid-80’s [89, 90, 91, 92] and was first explored in detail by Eichler et al. in [93]. It has successfully survived 20 years of observations and does get some support from the properties of SGRBs afterglows and of the identified host galaxies which appear to be elliptical and therefore old. Let us quickly discuss the CBC progenitor mechanism, as this scenario is strictly connected to the work reported in Part I of this thesis. The remnants of both BH-NS and NS-NS mergers may result in a black hole with negligible baryon contamination along its polar symmetry axis and surrounded by a hot massive accretion disk: before the disk gas is accreted to the black hole, intense neutrino fluxes are emitted which, through energy transfer, trigger a high-entropy gas outflow off the surface of the accretion disk (“neutrino wind”); at the same time, energy deposition by νν¯ annihilation in the baryon-free funnel around the rotation axis, powers relativistically expanding e±γ jets which can give rise to gamma-ray bursts [94, 95]. Other progenitor models are still viable (see [86, 96, 97]). An intriguing possibility, for example, is that a portion of short bursts could be due to soft gamma repeater (or SGR) giant flares; the spectral characteristics and energetics of some observed short GRBs and their afterglows seem to contradict this hypothesis in most cases, however statistical analyses indicate that at most 15% of known short GRBs can be accounted for as soft gamma repeaters [98]. The soft gamma repeater hypothesis is partially supported by the LIGO gravitational wave search coincident with GRB 070201 which excluded, at > 99% confidence, the possibility that it was a CBC event occurring in the Andromeda Galaxy [99]. We have emphasized the word “partially” because the uncertainty on whether or not the GRB event occurred in the Andromeda Galaxy is big: therefore, excluding through GW data analysis a CBC progenitor in the Andromeda Galaxy does not mean excluding CBCs as progenitors in general, nor for this specific case. However, this analysis is noteworthy as it is one of the first cases of GW astronomy. An interesting consequence of the cosmological origin of GRBs is that, if they may be exploited as distance indicators11, they may be used in cosmography in combination with type Ia supernovae. Supernovae Ia, in fact, do not allow us to go beyond a certain redshift value (z = 1.7, the redshift of the most distant supernova yet seen). The present redshift

11None of the existing models for GRB formation and emission is capable of connecting all the GRB observable quantities [96] and therefore GRBs cannot be considered as standard candles. 1.7 Gamma-Ray Bursts 31 record for GRBs is instead z = 8.2 (GRB 090423, [100]), so that there are several efforts to frame them into the standard of the cosmological distance ladder (see [101, 102] for recent proposals on GRB cosmography). A second consequence of the cosmological origin of GRBs is, of course, that they would also emit gravitational waves in bursts with an energy of about a solar mass. The detection of such GWs would also help in conﬁrming the nature of the progenitors. From this point of view GRBs are the electromagnetic counterparts of strong gravitational wave sources.

Part I

Analytic Models of Mixed Binaries

Chapter 2

BH-NS and NS-NS Binaries

The original work in this part of the thesis (Chapters4,5 and6) is devoted to BH-NS binaries. Since BH-NS and NS-NS binaries are both possible SGRB progenitors (see page 30) and since the progress in calculations on mixed binaries is strictly connected to the skills acquired over the years with BNSs, we shall now review the recent work on BH-NS coalescence as well as NS-NS coalescence. The recent breakthrough in BBH simulations [81, 103, 104] has also had important benefits for BH-NS and NS-NS binary simulations; however, we will not discuss BBHs: the reader may refer to [105] and [106] for a review of the status of simulations of BBHs and a quick introduction to their modelling. The coalescence of BH-NS and NS-NS may be divided into three phases, as mentioned on page 25, i.e. inspiral, merger and ringdown: each phase presents its distinct challenges for modelling and for detection. During their inspiral, binary systems may be accurately described by quasi-equilibrium formalisms until the gravitational radiation timescale and the dynamical timescale become comparable. When this occurs, the merger phase begins and fully general relativistic simulations are required in order to understand the complicated (magneto-)hydrodynamical processes that take place. Eventually the system settles into its final configuration, i.e. a rotating black hole, but before this happens, intermediate steps are possible: BNSs may go through a hyper-massive neutron star phase, and both mixed binaries and BNSs may evolve into a rapidly rotating black hole surrounded by a massive disk. Long-term GW emission is well understood for BHs in vacuum through semi-analytic ringdown formalisms, whereas non-vacuum configurations are more poorly constrained given the wide variations in magneto-hydrodynamic processes that may occur in different regimes. Our best understanding of the final fate of various systems currently comes from the same (magneto-)hydrodynamical calculations used to study the merger itself.

2.1 Quasi-Equilibrium and Pre-Merger Simulations

BH-NS and NS-NS binaries are well described by quasi-equilibrium conﬁgurations until the last moments of their existence, when two instabilities may drive them to the merger. If the total binary energy reaches a minimum at some separation, which occurs at the innermost stable circular orbit (ISCO) for point masses orbiting black holes, the binary becomes dynamically unstable and plunges towards the merger. Alternately, the NS in mixed binaries, or the lower-density secondary NS in BNSs, ﬁlls its Roche Lobe, mass transfers onto the primary and the system merges as the NS is tidally disrupted. The line of separation

35 36 2. BH-NS and NS-NS Binaries between the two scenarios in the BH-NS case will be investigated in Chapter5. Generally speaking, the second case is favoured by comparable masses in BH-NS systems [107, 21] and by soft equations of state in NS-NS binaries [108, 109]. A third possibility would be a stable mass transfer, in which mass loss from a less massive object onto a heavier one causes the binary orbit to widen, thereby regulating the mass transfer rate. This is a process which is observed to occur in binaries containing white dwarfs; even though preliminary models and calculations indicated that such processes might occur in BH-NS and NS-NS binaries (e.g. [110]), all dynamical relativistic calculations performed to date have showed that mass transfer is always unstable (e.g. [111]). In GR, the most widely used technique to construct BH-NS and NS-NS binary quasi- equilibrium sequences is the conformal thin-sandwich method: one assumes that the spacetime metric is conformally flat and chooses maximal slicing, thus obtaining five coupled non-linear elliptic equations for the various components of the metric and an elliptic equation for the matter enthalpy whose form is determined by the velocity profile one assumes for the matter1. Since the orbital timescales of stellar-mass binaries are very short during the final moments of their lives, the NS fluid may be treated as essentially as irrotational and viscous effects are not expected to tidally lock the NS(s) to the orbital motion [113, 114]. BNS quasi-equilibrium sequences have been mainly constructed by the Meudon group by developing the Lorene code2 which uses multi-domain pseudospectral methods to solve the elliptic equations [115, 116, 109, 117]. The Lorene libraries have been adapted by the Meudon [118] and the Illinois [119, 120, 121, 107] groups in order to handle BH-NS binaries; this was done by using an excised region with suitable boundary conditions to represent the BH horizon. The conformal thin-sandwich approach has three main drawbacks: since it is explicitly time-symmetric, (1) it is difficult to include an inspiral velocity component and (2) NS tidal lags, and since it ignores the spacetime radiation content (3) it leads to a great deal of “junk radiation” in the initial solution [11]. Thus, several alternative techniques were developed to construct quasi-equilibrium data, such as waveless formulations [122], helically symmetric models [123] and modified Kerr-Schild metric forms replacing the conformally flat metric description. This last technique was used for BH-NS binary sequences containing a spinning BH in [124], where the BH boundary conditions where modified in order to reduce the initial binary eccentricity. Several other methods were introduced over the last years to study mixed binaries and binary neutron stars by re-adapting the experience gained in the field of binary black holes and especially in the use of punctures3 [125, 126, 127].

2.2 Inspiral and Tidal Disruption

Quasi-equilibrium sequences and semi-analytic post-Newtonian expressions (e.g. [128]) may be used to identify the key frequencies at which instabilities play a notable role — possibly leading to the SGRB progenitor scenario described on page 30 — which in turn may be used to constrain the NS EOS ([129] and Chapter5), and can tell us a great deal about where dynamical runs should be started: this is what the original work in this part

1We refer the reader to Section 8.3 of [112] for further details. 2http://www.lorene.obspm.fr/. 3Within puncture methods, a special pole-like structure of the singularity (the “puncture”) inside the black 3 hole is assumed and one takes it into account with a speciﬁc ansatz for the initial data. R is therefore the relevant space for the constraint equations. 2.3 Dynamical Calculations of Mergers 37 of the thesis is dedicated to. Fully dynamical simulations, in fact, are required to probe the merger of BH-NS and NS-NS binaries from the onset of instability until the production of a quasi-stable remnant product. NS-NS and BH-NS mergers present a characteristic difference as far as GW observations are concerned: the former show the effects of tidal and dynamical instabilities at frequencies & 1 kHz, whereas the latter show them at frequencies typically falling in the interval [100, 500] Hz, depending on the mass ratio. This means that BH-NS instabilities/tides are within the “LIGO-Virgo band”, i.e. the frequency region where GW interferometers are mostly sensitive (AppendixA), while for NS-NS it may be necessary to operate GW detectors using narrow-band modes4 to observe signs of the instabilities. The dimensionality of the potential parameter space of all possible NS EOS models is unknown; however, it takes as few as four parameters to roughly approximate all known EOS models by breaking up the EOS into piecewise polytropic segments and to place constraints on the nature of neutron-star matter by astrophysical observations [130, 131]. BNS and BH-NS inspirals may yield other information about NS structure which goes beyond the imprint of tidal disruption and dynamical instability on the GW. NS have a wide variety of oscillation modes (see Section 7.3) which may be triggered by resonances with the orbital frequency as the system sweeps upward in frequency [132, 133]: the excitation of a particular oscillation mode would serve as an energy sink for the system which would dramatically change the phase evolution of the binary.

2.3 Dynamical Calculations of Mergers

BH-NS and NS-NS binaries are both fundamentally general relativistic systems. The current generation of simulations now generally evolves both the hydrodynamic matter and the GR metric fields self consistently, with some groups also incorporating a magneto- hydrodynamic evolution scheme that evolves the magnetic field of the fluid under the assumption of infinite conductivity. All the various codes make use of the insight gained in binary black hole evolutions since its breakthrough [81, 103, 104], but their spatial meshes, evolution schemes and numerical techniques differ. The two schemes used in all recent fully relativistic binary merger calculations are the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [134, 135] and the generalized harmonic formalism [136]. The former was used by the Tokyo groups, e.g. [137, 138, 125, 126, 139, 140, 141], the AEI group, e.g. [142] and the Illinois group, e.g. [143, 144, 145], whereas the latter was used by the Cornell/Caltech group, e.g. [146] and by the BYU/LSU/LIU collaboration [147] as well as by Pretorious in the aforementioned first binary black hole calculations [81]. We now shortly turn our attention to three important issues in numerical evolution of compact binaries containing neutron stars.

• With fluids, as opposed to metric fields, there is the problem of using simple finite differencing operators across discontinuities induced by shocks; non-linear codes must therefore include a shock-capturing technique in order to take care of these jumps. Conservative schemes are a popular solution: the “primitive” fluid variable set P~ — i.e. fluid density, pressure, velocity and, possibly, magnetic field — is transformed

4The sensitivity is optimized in a narrow frequencies range. 38 2. BH-NS and NS-NS Binaries

into a new set U~ whose evolution equations take the form

∂tU~ = ∇ · F~ + S,~ (2.1)

where the ﬂux functions F~ (P~ ) and source terms S~(P~ ) may be expressed in terms of the primitive variables only, and not their derivatives. Magnetic ﬁelds, moreover, must be evolved so that they remain divergence free. We refer the reader to [148] for a review on the attempts and techniques used in general relativistic (magneto- )hydrodynamics.

• Another computational issue is the one of grid schemes: one must resolve small-scale features in the central regions of a merger but also extend grids out to the point where the GW signal has taken on its roughly asymptotic form. Almost every code, therefore, incorporates some means of focusing resolution on the high-density regions; however, the approaches differ widely. We mention the Carpet toolkit5 which is based on the idea of adaptive mesh reﬁnement (AMR), that is, the level of reﬁnement on the grid is allowed to evolve dynamically to react to the binary evolution. It was successfully implemented by the AEI group to perform NS-NS merger calculations [142], in the Tokyo code SACRA for BNS and BH-NS mergers [140], in the BYU/LSU/LIU code [147] and in the Illinois code [145].

• A final issue which is worth mentioning is the treatment of boundary conditions for grid-based simulations; these must preserve the GR constraints, lead to well-posed initial boundary value problems and minimize spurious reflection of GWs off the outer boundaries. The first requirement is especially delicate as no constraint-preserving formulation has yet been proposed for the BSSN system [11].

2.3.1 NS-NS Mergers BNS merger simulations have established a certain number of accepted results which we shall now briefly go through; some of their quantitative details, however, are still under investigation and continue to be explored as technical and technological improvements become available. NS-NS merger studies indicate that some systems promptly collapse to a black hole surrounded by a disk, while others produce hyper-massive neutron star remnants in which the differential rotation is sufficient to stabilise the remnant against immediate gravitational collapse. The dividing line between initial configurations that lead to the formation of a hyper-massive neutron star and those leading to a prompt collapse to a black hole depends primarily on the ratio of the total mass of the system and the maximum mass of a single stationary neutron star; the value of this ratio depends on the EOS used and decreases as the EOS softens [137, 149, 150]. Another important result is that the hyper-massive neutron star is heated up and may become a neutrino source, whereas neutrino emission from the disk seems hard to obtain in the case of a prompt collapse to a black hole. The GW signal produced by these mergers is a standard “chirp” signal followed by a short transition phase during the merger, and then a signal whose form depends on the nature of the merger remnant. Hyper-massive neutron stars with a stiff EOS produce a very strong remnant oscillation signal, at a magnitude big enough for the radiation damping to be the

5http://www.carpetcode.org/ 2.3 Dynamical Calculations of Mergers 39 primary mechanism for damping rotation and leading to the final collapse to the BH, after about 30 ms to 100 ms. The frequency spectrum of hyper-massive neutron star remnants shows a clear mass dependence. Unfortunately, the peaks in the spectrum tend to be at frequency ranges which are too high to be seen by current and planned interferometric GW detectors [150]. Significant differences in the GW signal yielded by BNS merger simulations occur if one includes shock heating, as this “absorbs” kinetic energy and delays the collapse to a black hole [142]. Another reason to study BH-NS and NS-NS mergers is their possible connection to SGRBs (Section 1.7). NSs with stiffer equations of state are found to produce smaller disks around the central remnant with respect to softer ones; mass ratios of 0.7 − 0.8 seem to be ideal for producing massive disks around the remnant, thus leading to the best SGRB progenitor candidates [151, 140]. Simulations of mergers of magnetized NSs have also been performed and compared to the unmagnetised case results, e.g. [147, 144, 152]. As the two stars come into contact, their magnetic fields begin to interact and slow down the merger by repulsion, leading to a much longer period of gravitational wave emission; in the hyper-massive neutron star formation case, the amplitude and phase of the signal seem to be particularly affected. Moreover, the slowing down of the merger prevents a stable bar-mode from forming and the remnant is thus more circular. We mention that the formation of a Kelvin-Helmholtz unstable vortex sheet at the surface of contact between the two neutron stars has drawn very much attention, e.g. [153, 152], as these vortices may greatly enhance the local magnetic field strength.

2.3.2 BH-NS Mergers

The history of BH-NS merger simulations is significantly shorter than that of NS-NS simulations. This is due to the unique challenge of evolving the NS while handling the BH singularity at the same time. The first calculations by Lee and Kluzniak [154, 155], who used a quasi-relativistic potential designed to reproduce the proper relativistic ISCO and Newtonian dynamical terms to describe the binary evolution, delineated the following situation: models with soft NS equations of state gave rise to unstable mass transfer, with the star being disrupted and accreted to the BH, whereas a stiff NS and low-mass BH would produce stable mass transfer, thus leading to a widening of the binary orbit with a smaller NS still in orbit around the BH. This scenario was rejected by subsequent BH-NS merger calculations. Already in [156], by using a smoothed-particle hydrodynamics code with a conformally flat approximation to GR and a non-spinning black hole fixed in space, it was found that prompt disruption was inevitable and the GW signals were not long lived. The follow-up paper [111] showed that disks hot and massive enough to lead to SGRBs could be produced, but the mass ratio range for this to happen was limited (1 : 10 being already too much). The first Full-GR head-on collision calculations reported in [157] supported this result, as the relativistic potential well around the BH proved to be extremely effective in swallowing the NS. With the aid of the moving puncture formalism, the first Full-GR BH-NS merger simulations in which the (non-rotating) BH could move across the computational grid were presented [125, 126, 139]. All the simulations performed showed that the GW signal damps away rapidly after the merger, producing a ringdown in line with theoretical predictions. Moreover the signal could be attributed almost completely to the BH, with the disk producing no long-term quasi-periodic signal at detectable levels. A very important outcome was that 40 2. BH-NS and NS-NS Binaries determining the disk mass is a very difficult task, as it does not only sensitively depend on the binary physical parameters but also on the computational grid resolution. More indications about the disk masses for BH-NS mergers with spinless black holes were given in [143] and such masses turned out to be extremely small: 1 : 3, 1 : 2 and even 1 : 1 mass ratios could not produce disk masses larger than 0.04 M , with results essentially consistent with no disk mass at all when extrapolated with respect to numerical resolution and large initial separations. Waveforms computed in [143] and in [140] were consistent with respect to the aforementioned earlier results; signals from mergers with mass ratios away from unity were seen to produce deviations from point-mass inspirals and BBH signals which should be measurable by Advanced LIGO up to 100 Mpc. An important confirmation of the results discussed so far came with [158], in which the code used has substantial differences with respect to the ones used to produce the results previously discussed6. The general consensus nowadays is, therefore, that mergers of neutron stars with non-spinning black holes are incapable of producing a massive disk — the mass of the disk decreasing with the increase of the BH mass — and are thus extremely unlikely to be SGRB progenitors. The post-merger GW signal, moreover, is rapidly damped. First predictions about large disk masses as possible outcomes of neutron star mergers with Kerr black holes with prograde spin come from the approximate relativistic calculations of [159], where the neutron stars were seen to be disrupted outside the ISCO in the presence of prograde black hole companions. As we shall see in Chapter5, our calculations confirm this prediction (see also [21]). Very recent new Full-GR results [145] for a rapidly spinning prograde BH show that disk masses up to 0.2 M are possible: this is sufficient to power a SGRB through neutrino-antineutrino pair annihilation.

6Fields are evolved using a spectral system, while ﬂuid quantities are calculated on a ﬁnite-differencing grid; interpolations and/or spectral expansions are used to transfer variables back and forth. Chapter 3

The Afﬁne Model

A reasonable (approximate) approach to the dynamics of BH-NS binaries is the following:

1. assume the neutron star to be a classical self-gravitating ﬂuid

2. let m be the mass of the neutron star and M be the one of the black hole, with M m; the centre of mass of the star will thus orbit the black hole as a test-particle would and it will not perturb the black hole;

3. the equations of motion for a stellar ﬂuid element to be solved take the form

∂P ∂ΦN ρX¨i = − − ρ − ρfi (3.1) ∂Xi ∂Xi

where Xi indicates the coordinates of the fluid element in a coordinate system set up inside the star, ρ is the fluid density, P the fluid pressure, ΦN the Newtonian self-gravity potential and fi represents all other forces possibly acting on a fluid element.

This is an infinite set of PDEs in which the fi’s still have to be specified with further assumptions (and calculations). In this chapter we shall outline the affine model which served as a starting point for this part of the thesis. Its main building blocks are:

• the assumption that the NS radius is much smaller than the background curvature of the BH spacetime

• the afﬁne hypothesis, that is, the assumption that the star maintains the shape of an ellipsoid.

The first ingredient implies that the external forces fi’s are solely of tidal origin: these may be rigorously treated in the formalism developed by Fishbone [160], Mashhoon [161] and Marck [162] which is based on the tidal tensor (Section 3.1). The second ingredient, on the other hand, reduces the PDEs and their infinite degrees of freedom to a set of five ODEs in five dynamical variables (Section 3.2). The problem of studying the behaviour of an extended star orbiting a point mass or a rigid sphere is known as the Roche problem. More specifically we will be considering the

41 42 3. The Affine Model compressible Roche-Riemann problem, in which the extended body is a compressible ellipsoid with uniform vorticity parallel its rotation axis. For a thorough analysis of ellipsoidal figures of equilibrium, the reader should refer to Chandrasekhar’s book [163]; for the affine model, its use in the Roche-Riemann problem and in the Darwin-Riemann problem (both binary constituents are extended spinning ellipsoids), see instead the papers by Carter, Lu- minet and Marck [12, 13, 14, 15, 16], Lai, Rasio and Shapiro [164, 165, 166, 167, 168, 169], Shibata [170] and Wiggins and Lai [171].

3.1 Kerr Geometry: Parallel Transport and Tidal Tensor

We will open this section with a quick overview of the tidal tensor in Newtonian gravity; then we shall derive the tidal tensor in Kerr spacetime following [162]. In order to do so, we will have to introduce two tetrad ﬁelds: the well known Carter symmetric tetrad [172] and the tetrad which is parallel transported along timelike Kerr geodesics.

3.1.1 Newtonian Tidal Tensor

Let m be the mass of a particle which orbits another mass M m; let Xi, with i = {1, 2, 3}, be the Cartesian coordinates of m with respect to an inertial observer OM centered in M. Moreover, let xi be the Cartesian coordinates of the displacement vector that identiﬁes the position of a second (massive) particle relative to m, i.e. with respect to OM the position of this second particle is given by

ri = Xi + xi. (3.2)

The Newtonian gravitational ﬁeld at ~r is therefore

∂Φ 1 ∂2Φ ~ N N ΦN(~r) = ΦN(~r = X) + xi + xixj + ... (3.3) ∂ri ~r=X~ 2 ∂ri∂rj ~r=X~ where the expression for the Newtonian potential (in geometric units) is M Φ (~r) = − (3.4) N r

This means that with respect to Om, the second particle undergoes the acceleration

∂Φ (~r) ∂Φ (X~ ) ¨ N N N xï =r ï − Xi = − + = −Cij xj (3.5) ∂xi ~r=X~ ∂Xi where ∂2Φ N N Cij = (3.6) ∂ri∂rj X~ is the Newtonian tidal tensor field. If we consider equatorial orbits on the Z = 0 plane, the tidal tensor components are

 X2 XY  1 − 3 R2 −3 R2 0 M 2 CN =  −3 XY 1 − 3 Y 0 . (3.7) ij R3  R2 R2  0 0 1 3.1 Kerr Geometry: Parallel Transport and Tidal Tensor 43

Notice that out of equations (3.3), (3.5) and (3.6) one may build the (Newtonian) tidal potential 1 ΦT = C x x . (3.8) N 2 ij i j If we consider a massless rod, whose ends are labelled 1 and 2, and stick two equal masses at its ends, with respect to the centre of the rod the two masses will have displacement vectors ~x|1 = ~η and ~x|2 = −~η. We now let the rod undergo free fall in the gravitational ﬁeld of the mass M; we see that

N x¨i = −Cij ηj = −x¨i (3.9) 1 2 and therefore a torque of tidal origin acts on the rod [173].

3.1.2 Relativistic Tidal Tensor If we consider General Relativity, in presence of a mass M the spacetime is governed by a metric g. A massive test-particle, of course, follows the timelike geodesics of the metric g and we call u its 4-velocity. We now consider a second test-particle whose position with respect to the test particle with 4-velocity u is identiﬁed by the displacement 4-vector ξ. The relative acceleration between the two particles is governed by the geodesic deviation equation; if, once again, the observer OM centered in M sets up a coordinate system, this equation may be written as [174]

β γ α α β γ δ u u ∂β∂γξ + R βγδu ξ u = 0, (3.10)

α where R βγδ is the Riemann tensor of the metric gµν, that is,

α α α α µ α µ R βγδ = Γ βδ,γ − Γ βγ,δ + Γ γµΓ βδ − Γ δµΓ βγ , (3.11)

α Γ βγ being the Christoffel symbols. In 1956, Pirani showed that the geodesic deviation equation (3.10) may be cast in the form d2ξ(i) + C(i) ξ(j) = 0 (i, j = 1,..., 3) (3.12) dτ 2 (j) if one sets up an orthonormal tetrad field e(µ) (µ = 0,..., 3) associated with the reference frame Ou centered in the test-particle of 4-velocity u, parallel transported along its geodesics (i) and such that e(0) ≡ u [24]. The physical meaning of the tensor field C (j) is straightforward if one compares (3.10) and (3.5): it is the relativistic tidal tensor. In his work Pirani defines such tensor as follows:

def α β γ δ C(i)(j) = Rαβγδe(0)e(i)e(0)e(j) . (3.13) The tidal tensor is therefore a projection of the Riemann tensor on the orthonormal tetrad field. Notice that this setup is equivalent to considering Fermi coordinates in the case of vanishing (or negligible) Fermi velocity (see for example [175]). Since we will be dealing with BH-NS binaries, in the following sections we will focus on the Kerr metric and its tidal field. When modelling mixed binaries, in fact, we will 44 3. The Affine Model assume that the NS orbits its BH companion without perturbing its spacetime. Under this assumption, the black hole therefore plays the same role as the mass M does in the general discussion we have just seen. We will therefore have to build the orthonormal tetrad e(µ) and the tidal tensor C(i)(j) for the Kerr spacetime. A final remark is useful for the latter scope: in vacuum, as is the case of a black hole solution, one may substitute the Weyl tensor Cαβγδ to the Riemann tensor in definition (3.13), thus obtaining the equivalent expression

def α β γ δ C(i)(j) = Cαβγδe(0)e(i)e(0)e(j) . (3.14)

This is possible because the equation1 relating the Weyl tensor to the Riemann tensor, to γ µν the Ricci tensor Rµν = R αγβ and to the Ricci scalar R = g Rµν [176], i.e. 1 C = R − g R + g R + g g R, (3.15) αβγδ αβγδ α[γ δ]β β[γ δ]α 3 α[γ δ]β reduces to

Cαβγδ = Rαβγδ (3.16) since the Einstein equations in vacuum tell us that Rµν and R both vanish.

3.1.3 The Kerr Metric and the Symmetric Tetrad The Kerr metric in the Boyer-Lindquist coordinate frame is given by 2Mr 4Mr Σ A ds2 = − 1 − dt2 − a sin2 θdtdφ + dr2 + Σdθ2 + sin2 θdφ2, (3.17) Σ Σ ∆ Σ where

Σ = r2 + a2 cos2 θ (3.18) ∆ = r2 + a2 − 2Mr (3.19) A = (r2 + a2) − ∆a2 sin 2θ. (3.20)

In performing calculations within the Kerr spacetime, it is convenient to use the canonical symmetric orthonormal tetrad introduced by Carter [172]: s ¯ ∆ ω0 = (dt − a sin2 θdφ) (3.21) Σ s ¯ ∆ ω1 = dr (3.22) Σ ¯ √ ω2 = Σdθ (3.23) ¯ sin θ ω3 = √ [adt − (r2 + a2)dφ] (3.24) Σ

1Square brackets denote index antisymmetrization, i.e. 1 X A ≡ δ A [a1···al] l! π aπ(1)···aπ(l) π where we sum over all possible permutations π of the l indices and where δπ is +1(−1) for an even(odd) number of index permutations. 3.1 Kerr Geometry: Parallel Transport and Tidal Tensor 45 where we denote indices relative to this tetrad with a bar in order to avoid confusion with the tetradic indices relative to e(µ), which we will build in the next section in terms of Carter’s tetrad. The convenience of the symmetric tetrad relies upon the form the metric takes, i.e.

2 µ¯ ν¯ ds = ηµ¯ν¯ω ω , (3.25) where ηµν = diag(−1, 1, 1, 1) is the metric tensor of Minkowksi spacetime. In the tetrad ﬁeld we have just set up, the curvature 2-form has components:

1¯ 1¯ 2¯ 0¯ 3¯ Ω 2¯ = −I1ω ∧ ω + I2ω ∧ ω (3.26) 0¯ 0¯ 3¯ 1¯ 2¯ Ω 3¯ = −I1ω ∧ ω − I2ω ∧ ω (3.27) 0¯ 0¯ 1¯ 2¯ 3¯ Ω 1¯ = 2I1ω ∧ ω + 2I2ω ∧ ω (3.28) 3¯ 2¯ 3¯ 0¯ 1¯ Ω 2¯ = −2I1ω ∧ ω + 2I2ω ∧ ω (3.29) 0¯ 0¯ 2¯ 1¯ 3¯ Ω 2¯ = −I1ω ∧ ω + I2ω ∧ ω (3.30) 3¯ 1¯ 3¯ 0¯ 2¯ Ω 1¯ = I1ω ∧ ω + I2ω ∧ ω (3.31) where

def M I = (r2 − 3a2 cos2 θ) (3.32) 1 Σ3 def Ma cos θ I = (3r2 − a2 cos2 θ) (3.33) 2 Σ3 and where the wedge (∧) denotes the antisymmetric tensor product. This 2-form is related to the Weyl tensor by 1 Ωα¯ = Cα¯ ωµ¯ ∧ ων¯ (3.34) β¯ 2 β¯µ¯ν¯ and it thus allows us to easily read off the components of the Weyl tensor, which we need in order to determine the tidal tensor through Eq. (3.14). Exploiting the symmetry properties of the Weyl tensor and explicitly indicating through the use of our index notation that we 2 shall express the parallel transported tetrad e(µ) in terms of the symmetric tetrad , we obtain

n 0¯ 1¯ 1¯ 0¯ 1¯ 0¯ 0¯ 1¯ C(i)(j) = 3I1 e(0)e(i) − e(0)e(i) e(0)e(j) − e(0)e(j) 2¯ 3¯ 3¯ 2¯ 2¯ 3¯ 3¯ 2¯ o + e(0)e(i) − e(0)e(i) e(0)e(j) − e(0)e(j) n 0¯ 1¯ 1¯ 0¯ 2¯ 3¯ 3¯ 2¯ − 3I2 e(0)e(j) − e(0)e(j) e(0)e(i) − e(0)e(i) 0¯ 1¯ 1¯ 0¯ 2¯ 3¯ 3¯ 2¯ o + e(0)e(i) − e(0)e(i) e(0)e(j) − e(0)e(j) (3.35) when (i) 6= (j) and

2 2 0¯ 1¯ 1¯ 0¯ 2¯ 3¯ 3¯ 2¯ C(i)(i) = I1 1 − 3 e(0)e(i) − e(0)e(i) − e(0)e(i) − e(0)e(i)

0¯ 1¯ 1¯ 0¯ 2¯ 3¯ 3¯ 2¯ − 6I2 e(0)e(i) − e(0)e(i) e(0)e(i) − e(0)e(i) (3.36) when (i) = (j).

2Expressions (3.35) and (3.36) for the components of the relativistic tidal tensor actually hold for any orthonormal tetrad but, given Pirani’s equation (3.12), we are only interested in considering the tetrad ﬁeld parallel transported along an arbitrary timelike geodesic of Kerr spacetime. 46 3. The Afﬁne Model

3.1.4 The Parallel Propagated Tetrad

The ingredient we are missing in order to determine the tidal tensor of a rotating Kerr black hole, Eqs. (3.35)-(3.36), is the tetrad e(µ) expressed in terms of the symmetric tetrad. The parallel propagated orthonormal tetrad is built as follows [162]. One considers the worldline C of a massive test-particle that moves along a timelike Kerr geodesic and ﬁxes, say at proper time τ = 0,

e(0) = u . (3.37)

This unit vector is obviously parallel transported and its components are:

2 2 ¯ E(r + a ) − aL e 0 = √ z (3.38) (0) ∆ Σ s ∆ e 1¯ = r˙ (0) Σ √ 2¯ ˙ e(0) = Σθ aE sin θ − L sin−1 θ e 3¯ = √ z (0) Σ

where dots indicate derivatives with respect to the proper time τ and where E and Lz are respectively the energy and the angular momentum about the axis of symmetry per unit mass of the orbiting test-particle. Moreover, the Kerr metric possess a Killing-Yano tensor of order two fµ¯ν¯ out of which one may build the unit vector orthonormal to e(0)

µ¯ ν¯ f ν¯e(0) e µ¯ = √ , (3.39) (3) K

where K is “Carter’s fourth constant”. The parallel transport of e(3) is therefore automatically guaranteed; its components are

s ¯ Σ e 0 = a cos θr˙ (3.40) (3) K∆ a cos θ[E(r2 + a2) − aL ] e 1¯ = √ z (3) KΣ∆ r(aE sin θ − L sin−1 θ] e 2¯ = − √ z (3) KΣ s Σ e 3¯ = r ˙cosθ . (3) K 3.1 Kerr Geometry: Parallel Transport and Tidal Tensor 47

The orthonormal basis is completed by the two vectors

s ¯ Σ e˜ 0 = α rr˙ (3.41) (1) K∆ αr[E(r2 + a2) − aL ] e˜ 1¯ = √ z (1) KΣ∆ βa cos θ(aE sin θ − L sin−1 θ] e˜ 2¯ = √ z (1) KΣ s Σ e˜ 3¯ = β a cos θθ˙ (1) K and

2 2 ¯ αr[E(r + a ) − aL ] e˜ 0 = √ z (3.42) (2) Σ∆ s Σ e˜ 1¯ = α r˙ (2) ∆ √ 2¯ ˙ e˜(2) = β Σθ aE sin θ − L sin−1 θ e˜ 3¯ = β √ z (2) Σ where

s K − a2 cos2 θ α = (3.43) r2 + K s r2 + K β = . (3.44) K − a2 cos2 θ

However, e˜(1) and e˜(2) are not parallel propagated. In order to achieve this feature one has to rotate them as follows:

e(1) = cos Ψe˜(1) − sin Ψe˜(2) (3.45)

e(2) = sin Ψe˜(1) + sin Ψe˜(2) , (3.46) the angle Ψ being governed by

√ " # K E(r2 + a2) − aL L − aE sin2 θ Ψ˙ = z + a z . (3.47) Σ r2 + K K − a2 cos2 θ

3.1.5 The Tidal Tensor Field for Equatorial Geodesics

In the next chapters we shall consider equatorial orbits, so we may set θ = π/2 and θ˙ = 0 in Eqs. (3.38), (3.40)-(3.46) before turning to the expressions (3.35) and (3.36) for the 48 3. The Afﬁne Model tidal tensor components:

2 2 ¯ E(r + a ) − aL e 0 = √ z (3.48) (0) r ∆ rr˙ e 1¯ = √ (0) ∆ 2¯ e(0) = 0 aE − L e 3¯ = z (0) r √ 2 2 2 −1 ¯ cos Ψr r˙ − sin Ψ K[E(r + a ) − aL ]r e 0 = z (3.49) (1) p∆(r2 + K) √ cos Ψ[E(r2 + a2) − aL ] − sin Ψ Krr˙ e 1¯ = z (1) p∆(r2 + K) 2¯ e(1) = 0 √ sin Ψ r2 + K(aE − L ) e 3¯ = − √ z (1) r K √ 2 2 2 −1 ¯ sin Ψr r˙ + cos Ψ K[E(r + a ) − aL ]r e 0 = z (3.50) (2) p∆(r2 + K) √ sin Ψ[E(r2 + a2) − aL ] + cos Ψ Krr˙ e 1¯ = z (2) p∆(r2 + K) 2¯ e(2) = 0 √ cos Ψ r2 + K(aE − L ) e 3¯ = √ z (2) r K

0¯ e(3) = 0 (3.51) 1¯ e(3) = 0 aE − L e 2¯ = √ z (3) K 3¯ e(3) = 0 with

2 K = (aE − Lz) (3.52) and

√ E K + a Ψ˙ = . (3.53) r2 + K 3.2 The Afﬁne Model Equations 49

The substitution of Eqs. (3.48)-(3.51) into Eq. (3.35) and Eq. (3.36) finally yields ! M r2 + K C = 1 − 3 cos2 Ψ (3.54) (1)(1) r3 r2 ! M r2 + K C = 1 − 3 sin2 Ψ (3.55) (2)(2) r3 r2 M K C = 1 + 3 (3.56) (3)(3) r3 r2 3M(r2 + K) C = − cos Ψ sin Ψ , (3.57) (1)(2) r5 which is the result we were looking for. Notice that if one considers a non-rotating black hole (a = 0) and its tidal field — Eqs. (3.54)-(3.57) — in the Newtonian limit, which is obtainable by taking r MBH , r l and Ψ = ϑ where ϑ is the anomaly of the Newtonian orbit, one finds ! M M X2 C ≈ 1 − 3 cos2 ϑ = 1 − 3 (3.58) (1)(1) r3 r3 r2 ! M M Y 2 C ≈ 1 − 3 sin2 ϑ = 1 − 3 (3.59) (2)(2) r3 r3 r2 M C ≈ (3.60) (3)(3) r3 M M C ≈ −3 sin ϑ cos ϑ = −3 XY (3.61) (1)(2) r3 r5 and the Newtonian tidal tensor, Eq. (3.7), is thus recovered.

3.2 The Afﬁne Model Equations

3.2.1 The Afﬁne Constraint So far we have worked on the consequences of the following assumptions: • the mass of the neutron star m is much smaller than the black hole mass M, so that the centre of mass of the star follows a timelike geodesic of the black hole spacetime

• the neutron star does not perturb the black hole

• the neutron star radius is much smaller than the background curvature of the black hole spacetime. This led us to studying tides in Kerr spacetime which, as we have seen, are governed by the equation3

d2ξi + Ci ξj = 0 (3.62) dτ 2 j 3From now on we shall suppress all parentheses in the index notation for sake of simplicity; we will no longer be switching to Carter’s symmetric tetrad or to to Boyer-Lindquist coordinates, so this will not generate any confusion. 50 3. The Afﬁne Model the tidal tensor C being given by Eqs. (3.54)-(3.57). We are now ready to introduce the other two assumptions anticipated at the beginning of the present chapter. The ﬁrst one is that

• the neutron star is modelled as a classical self-gravitating ﬂuid, so its internal motions obey the hydrodynamics equations4 ∂P ∂Φ ρX¨ i = − − ρ N − ρCiXj. (3.63) ∂Xi ∂Xi j The second assumption is the cornerstone of the afﬁne model [12, 13, 14, 15, 16]:

• the position vector of an infinitesimal star fluid element relative to the centre of mass in an orthonormal parallel propagating frame is specified by the linear relation

i i ˆj ξ = q jξ , (3.64)

where ξˆ is the corresponding position vector in an initial spherical equilibrium conﬁg- uration5, and q is the 3 × 3 deformation matrix which is uniform, i.e. its elements only depend on the proper time τ and not on the ξˆi’s.

Since the idealised constraint we have postulated is non-working and does not contribute directly to the energy content of the model, and since we moreover postulate that no non- conservative force is acting in the stellar interior, the star dynamics is represented by the i evolution of the nine degrees of freedom q j through a set of Lagrangian equations of the form

d ∂LI ∂LI i = i . (3.65) dτ ∂q˙ j ∂q j

In order to build the Lagrangian LI for the star interior (“I”), we will have to bare in mind that it must reproduce the left hand side and the first two terms on the right hand side of Eq. (3.63). The last term in Eq. (3.63) will then be included by adding to LI a suitably constructed Lagrangian term LT for the tides (“T”). Now that the dynamics of the original infinite degrees of freedom of the fluid has been reduced to that of the nine components of i q j by means of the affine constraint, we may moreover assume that: • while interacting with the BH, the NS may rotate only around its axis that is perpendicular to the orbital (or BH equatorial) plane.

This means that the deformation matrix takes the form

 1 1  q 1 q 2 0  2 2  q 1 q 2 0  , (3.66) 3 0 0 q 3 where — following Eqs. (3.49)-(3.51) — we have dubbed with a “3” the direction orthogonal to the orbital plane, i.e. the θ direction, and with a “1” and a “2” the directions belonging to the orbital plane; this form of the matrix follows from the fact that the tetrad vector e(3)

4See Eq. (3.1). 5Throughout this part of the thesis, a hat indicates quantities calculated for the spherical star at equilibrium. 3.2 The Afﬁne Model Equations 51

(Eqs. (3.51)), has a non-vanishing component only in the θ-direction, i.e. perpendicularly to the orbital plane, whereas e(0), e(1) and e(2) (Eqs. (3.48)-(3.50)) have a vanishing component only along the θ-direction. The NS internal degrees of freedom are hence reduced to five. We are thus dealing with ellipsoids that admit rotation of the ellipsoid figure (spin) and internal rotation of the fluid (vorticity) only around one axis. We shall say more about this point in the next section. Before writing out LI, LT and the evolution equations for the star interior, let us clarify the physical meaning of the constraint (3.64). If we name RNS the NS radius in the initial spherical configuration and define the configuration matrix

def S = qqT , (3.67) under the afﬁne constraint given in Eq. (3.64) we obtain

−1 i j 2 Sij ξ ξ = RNS . (3.68) In other words the three eigenvalues of the configuration matrix are interpretable as the squared values of the three principal axes of an ellipsoid: the affine constraint therefore implies that the (initially spherical) NS is deformed by the BH tidal field into an ellipsoid. Since the deformation is allowed to affect only the geometry of the star configuration, the affine constraint may be stated as follows in physical terms [166]: • assume that the surfaces of constant density within the star can be approximated as self-similar ellipsoids • assume that the density profile ρ(m ˜ ), where m˜ is the mass inside an isodensity surface, is identical to that of a spherical with the same volume and equation of state.

3.2.2 The Principal Frame Equation (3.68) has an important implication: with an appropriate rotation (the one which diagonalises the configuration matrix) one may switch from the orthonormal parallel- propagated frame to a frame whose axes follow the principal axes of the stellar ellipsoid. This special reference frame is called principal frame. The degrees of freedom of the affine model considered in this frame have a direct physical interpretation (as opposed to the components of the deformation matrix q). As stated in the previous section, we must have five degrees of freedom. Three of them are of course the three principal axes; we are left with only two degrees of freedom to discuss: these are linked to the star spin and the fluid vorticity (see the comment following Eq. (3.66)) and are analysed in the rest of this section. Thanks to the last assumption of the previous section, in order to switch from the parallel-propagated frame to the (rotating) principal frame — or equivalently to diagonalise the configuration matrix given by Eq. (3.67) — it is sufficient to perform a rotation around the “3”-direction by using the matrix  cos ϕ sin ϕ 0 def   T = − sin ϕ cos ϕ 0 (3.69) 0 0 1 with dϕ = Ω , (3.70) dτ 52 3. The Affine Model where Ω is the star spin, i.e. it is the ellipsoid angular velocity measured in the parallel- transported frame associated with the star centre of mass. This one parameter rotation yields our fourth degree of freedom, ϕ. The ellipsoidal shape of the fluid star, moreover, does not change under a rotation of the initial spherical configuration along the “3”-direction (which is of course common to the parallel-propagated frame and to the principal frame) [170]. This implies that the star can have another degree of freedom for the internal motion. Since we are considering uniform internal fluid motions around the “3”-axis, we will have a second rotation matrix  cos λ sin λ 0 def   S = − sin λ cos λ 0 (3.71) 0 0 1 which acts on6 Tξˆ, the angle λ being governed by dλ = Λ , (3.72) dτ where Λ is the angular frequency of the internal fluid motions in the (rotating) principal frame. If one assumes the rotating ellipsoid to be a Riemann-S ellipsoid, the ratio between spin and vorticity is constant [163] and Λ is related to the uniform vorticity ζ as measured in the principal frame by [166] a2 + a2 ζ = − 1 2 Λ . (3.73) a1a2 Summarizing, the five degrees of freedom in the principal frame are the three principal axes a1, a2, a3 and the two rotation angles ϕ and λ, whose proper time derivatives are the star spin and vorticity. Having discussed the meaning of the degrees of freedom of the model, we would like to say more about Riemann-S ellipsoids and about Eq. (3.73). In the rotating principal frame, the velocity field of a fluid element of a Riemann-S ellipsoid is a linear function of the coordinates in the principal frame (which we denote with xi, i = {1, 2, 3}) and is given by

v = vs + ve, (3.74) where the ﬁrst contribution comes from the internal rotation, while the second one takes into account expansions or contractions of the ellipsoid. They are respectively given by

a1 a2 vs = Λx2n1 − Λx1n2 (3.75) a2 a1 and

a˙ 1 a˙ 2 a˙ 3 ve = x1n1 + x2n2 + x3n3 (3.76) a1 a2 a3 where n1, n2 and n3 are the basis unit vectors along the instantaneous directions of the principal axes of the ellipsoid (with n3 along the rotation axis). In this reference frame, for 7 an ellipsoid at equilibrium (i.e. ve = 0) the vorticity is deﬁned as

def ζ = (∇ × v) · n3 (3.77)

6See Eq. (3.64) for the meaning of ξˆ. 7 ∇ = (∂x1, ∂x2, ∂x3) is intended in the principal frame. 3.2 The Afﬁne Model Equations 53 which indeed yields Eq. (3.73). Since it will soon be useful, we conclude this section by writing the velocity ﬁeld in the (inertial) parallel-propagated frame starting from v (compare with Eq. (3.74)):

u = us + ue, (3.78) where the expansion/contraction term is unaffected by the rotation relating the two frames, i.e.

a˙ 1 a˙ 2 a˙ 3 ue ≡ ve = x1n1 + x2n2 + x3n3 , (3.79) a1 a2 a3 as opposed to the rotation/spin term a1 a2 us = vs + Ω × x = Λ − Ω x2n1 + − Λ + Ω x1n2 . (3.80) a2 a1

3.2.3 The Neutron Star Internal Dynamics

We are now ready to write out the two contributions LI and LT to the Lagrangian of the NS ﬂuid within the afﬁne model, i.e.

L = LI + LT , (3.81) to be used in the Euler-Lagrange equations

d ∂(L + L ) ∂(L + L ) I T = I T , (3.82) dτ ∂Q˙ i ∂Qi where the set of Lagrangian variables is, of course, Qi = {a1, a2, a3, ϕ, λ}. The internal Lagrangian is given by

LI ≡ TI − U − V, (3.83) where TI is the total kinetic energy of the ﬂuid, U is its internal energy and V is the self-gravitational energy. These three quantities are deﬁned as

def 1 Z T = d3xρu · u (3.84) I 2 def Z Z U = d3x ≡ dM (3.85) ρ 0 def 1 ZZ ρ(x) − ρ(x ) V = − d3xd3x0 , (3.86) 2 |x − x0| where the integrals are performed over the whole stellar ellipsoid, ρ is the mass density, u is given by Eqs. (3.78)-(3.80), is the energy density and x and x0 are position vectors inside the star. The tidal interaction Lagrangian, on the other hand, has only one term due to the coupling between the BH tidal field and the NS fluid; this coupling may be expressed as (see Eq. (3.8)) Z 1 Z 1 L ≡ −V = − ρΦ d3x = − ρc x x d3x ≡ − c I , (3.87) T T T 2 ij i j 2 ij ij 54 3. The Affine Model

where the cij’s are the components of the Kerr tidal tensor — Eqs. (3.54)-(3.57) — expressed in the principal frame, i.e. rotated by an angle ϕ, " # M r2 + K c = BH 1 − 3 cos2(Ψ − ϕ) (3.88) 11 r3 r2 " # M r2 + K c = BH 1 − 3 sin2(Ψ − ϕ) (3.89) 22 r3 r2 M K c = BH 1 + 3 (3.90) 33 r3 r2 " # M 3 r2 + K c = c = BH − sin 2(Ψ − ϕ) , (3.91) 12 21 r3 2 r2 while I is the tensor of inertia, with components Z def 3 Iij = ρxixjd x , (3.92) which in the principal frame is given by 2 aj I = Mcdiag , (3.93) RNS Mc being the scalar quadrupole moment

def 1 Z Mc = ξbiξbidM . (3.94) 3 As a ﬁrst step towards writing the equations of motion (3.82), we use Eqs. (3.78)-(3.80) to express the ﬂuid kinetic energy in such a way that it will be easy to derive it with respect to the Lagrangian variables: 1 Z Z 1 T = d3xρ(u + u ) · (u + u ) = d3x ρ(u2 + u2) = T + T , (3.95) I 2 s e s e 2 s e e s where the expansion/contraction kinetic energy and the spin kinetic energy are respectively given by Z Z " 2 2 2 # def 1 3 1 3 a˙ 1 2 a˙ 2 2 a˙ 3 2 Te = d xρ~ue · ~ue = d xρ x1 + x2 + x3 = 2 2 a1 a2 a3 1 Z a˙ 2 1 a˙ 2 Z = X d3xρ i x2 = X i d3xρx2 = 2 a i 2 a i i i i i 2 1 X a˙ i = Mc (3.96) 2 R i NS Z Z " 2 2 # def 1 3 1 3 a1 2 a2 2 Ts = d xρ~us · ~us = d x Λ − Ω x2 + Ω − Λ x2 = 2 2 a2 a1 " 2 Z 2 Z # 1 a1 3 2 a2 3 2 = Λ − Ω d xρx2 + Ω − Λ d xρx1 2 a2 a1 " 2 2 2 2# 1 a1 a2 a2 a1 = Mc Λ − Ω + Ω − Λ 2 a2 RNS a1 RNS

1 2 2 Mc = I(Λ + Ω ) − 2 ΛΩa1a2 , (3.97) 2 RNS 3.2 The Afﬁne Model Equations 55 where

Mc 2 2 I = I11 + I22 = (a1 + a2) (3.98) RNS is the moment of inertia of the stellar ellipsoid which is allowed to rotate around its axis “3”. We also need to be able to derive of U with respect to the ai’s. The afﬁne constraint formalised in Eq. (3.64) implies that

3 a1a2a3 3 d x = 3 d xb (3.99) RNS and we thus also have that dρ d(a a a ) = − 1 2 3 . (3.100) ρ a1a2a3 A second useful equation is the ﬁrst law of thermodynamics in differential form for barotropic equations of state, i.e. P dρ d = , (3.101) ρ ρ ρ where P is the ﬂuid pressure. If we derive the internal energy U by ai, we thus obtain

dU Z d(/ρ) Z P dρ Z P 1 d(a1a2a3) = dM = dM 2 = − dM = dai dai ρ dai ρ a1a2a3 dai 1 Z Π = − d3xP = − , (3.102) ai ai where

def Z Π = P d3x . (3.103)

Finally we may turn our attention to the Newtonian self-gravitational energy which we will need to derive with respect to ai. A useful relation for this was found by Chandrasekhar and reads [163] 1 Z ∞ dσ V = VRb NS p , (3.104) 2 0 ||S + σ1|| where Vb is the value of V for the spherical equilibrium configuration, S is the configuration matrix defined in Eqs. (3.67)-(3.68), 1 is the 3 × 3 identity matrix, the double vertical bars indicate that the determinant must be taken and the variable σ, which is integrated out, has the dimensions of the inverse of an area. Using such formula one finds

dV Vb Z ∞ dσ = − RNSai 2 , (3.105) dai 2 0 (ai + σ)∆(σ) where we have deﬁned q def 2 2 2 ∆(σ) = (a1 + σ)(a2 + σ)(a3 + σ) . (3.106) 56 3. The Afﬁne Model

With Eqs. (3.87)-(3.91), (3.93), (3.96)-(3.98), (3.102) and (3.105), we have all the elements to write out explicitly the equations of motion (3.82). These take the form

2 2 2 1 Vb 3 ˜ RNS Π a¨1 = a1(Λ + Ω ) − 2a2ΛΩ + RNSa1A1 + − c11a1 (3.107) 2 Mc Mc a1 2 2 2 1 Vb 3 ˜ RNS Π a¨2 = a2(Λ + Ω ) − 2a1ΛΩ + RNSa2A2 + − c22a2 (3.108) 2 Mc Mc a2 2 1 Vb 3 ˜ RNS Π a¨3 = RNSa3A3 + − c33a3 (3.109) 2 Mc Mc a3 ˙ Mc 2 2 Js = 2 c12(a2 − a1) (3.110) RNS C˙ = 0 (3.111) where

Mc 2 2 Js ≡ Pϕ = 2 [(a1 + a2)Ω − 2a1a2Λ] (3.112) RNS

Mc 2 2 C ≡ Pλ = 2 [(a1 + a2)Λ − 2a1a2Ω] , (3.113) RNS and where we have deﬁned Z ∞ ˜ def dσ Ai = 2 . (3.114) 0 ∆(σ)(ai + σ)

Js and C are, respectively, the conjugate momenta of ϕ and λ; physically they are the spin angular momentum of the star and a quantity proportional to its circulation in the locally non-rotating inertial frame8. Notice that C is a constant of motion: this is due to the fact that the model does not include viscosity. When deﬁning Riemann-S ellipsoids (page 52), we stated that they have a constant ratio between the vorticity ζ and the spin Ω. There are two special cases one may consider: 1. uniform synchronized rotation — i.e. the ﬂuid appears to be still in the principal frame — which means taking Λ = 0 and, because of Eq. (3.73), ζ = 0 and therefore ζ/Ω = constant = 0; this is the case of Jacobi ellipsoids;

2 2. irrotational fluid, which means choosing C = 0 which in turn yields Ω = (a1 + 2 a2)/(2a1a2)Λ and therefore ζ/Ω = constant = −2. The second choice is more indicated for the final phases of compact binary inspirals [113, 114], whereas the first one seems to be more indicated for black hole-white dwarf binaries (see discussion and references in [171]). In this thesis we will consider irrotational models.

3.2.4 Polytropic EOS and Newtonian Self-Gravity In order to determine the self-gravitational energy of the spherical NS and the EOS- dependent quantities which appear in the equations of motion (3.107)-(3.111), assumptions

8Henceforth, we shall use the shorthand name “circulation” for C even though the actual circulation for a 2 Riemann-S ellipsoid is C = πa1a2(2Ω + ζ) = −πRNS C/Mc. 3.3 Orbit Descriptions within the Affine Model 57 regarding local gravity and the equation of state are necessary. The mentioned quantities are Vb , RNS, Mc and Π. In all papers adopting the affine model, stars (and thus NSs too) were always modelled with Newtonian self-gravity and by specifying a polytropic equation of state [12, 13, 14, 15, 16, 164, 165, 166, 167, 168, 169, 170, 171, 177, 178, 179]. As we shall see in the next chapter, it is possible to drop both choices and to improve the model [21]. In this section we will complete the discussion on the affine model by providing expressions for Vb , Mc and Π in the case of Newtonian polytropic stars; RNS may instead be determined numerically by solving the Newtonian stellar structure equations for a chosen polytrope and central density. For the scalar quadrupole moment we simply quote the result 1 Mc = κ MR2 , (3.115) 5 n NS where κn is a constant which depends on the polytropic index n = 1/(Γ − 1) (see equation (7.4.9) of [31]). In the case of polytropes, by using Eq. (3.99), one obtains for Π

Z !1−Γ 3 a1a2a3 Π = P d x = 3 Πb (3.116) RNS where Πb is simply Π calculated for the spherical conﬁguration and is given by 1 M 2 Πb = . (3.117) 5 − n RNS

It is related to the self-gravity potential Vb of a spherical star by the following consequence of the virial theorem (see page 64, where we dedicate special attention to this theorem):

Vb = −3Πb . (3.118)

The virial theorem therefore yields 3 M 2 Vb = . (3.119) n − 5 RNS

3.3 Orbit Descriptions within the Afﬁne Model

So far we have thoroughly discussed the affine model internal dynamics of a star orbiting a black hole and tidally interacting with the black hole itself. We have given both the Newtonian tidal tensor components, Eq. (3.7), and the Kerr tidal tensor components, Eqs. (3.54)-(3.57). In particular, in working out the latter, we assumed that the centre of mass of the star follows a timelike geodesic and we had to deal with parallel transport. We shall now shortly discuss what has been done over the years with the orbital motion when the affine model and the tidal tensor formalism were adopted to study binary systems. When the Newtonian tidal tensor was used, the authors (Carter and Luminet) were interested in investigating the tidal disruption of a main sequence star after the deep penetration into the gravitational field of a massive black hole, i.e. they truly considered physical cases in which MBH Mstar [12, 13, 14, 15]. Therefore, in their work, the black hole did not move whereas the star followed unbound Newtonian equatorial orbits around it. 58 3. The Affine Model

Luminet and Marck tackled the same problem, but instead of considering a Newtonian black hole9 they worked with a massive Schwarzschild black hole [16]. In their case, the unbound equatorial orbits the star followed were determined by dt E = (3.120) dτ 2MBH 1 − r ! dr 2M L2 = E2 − 1 − BH 1 + z (3.121) dτ r r2 dθ π = 0 θ = (3.122) dτ 2 dφ L = z (3.123) dτ r2 so that the hypothesis made when constructing the tidal tensor were respected. In Lai, Rasio and Shapiro’s papers [166, 167, 168, 169], coalescing mixed compact binaries and binary neutron stars were considered. The latter case is actually a Darwin- Riemann problem and not a Roche-Riemann one (see page 42). The main difference is that each star responds to the tidal field of the other star: there is a mutual tidal interaction instead of a one way one as in the Roche-Riemann problem. For both Roche-Riemann and Darwin-Riemann problems, the tidal fields were modelled by means of Newtonian tidal tensors10, while the orbital motion was included by using the Newtonian two-body problem Lagrangian with leading order gravitational wave orbit decay terms (i.e. 2.5PN dissipative terms). Coalescing compact systems of this kind, in fact, have constituents for which the hypothesis M1 M2 no longer holds. With this modelling of the orbit and internal dynamics, the authors allowed a very interesting physical aspect to emerge: the influence of tides on the orbit. However, the Newtonian tidal tensor oversimplifies the physics and moreover it does not distinguish between spinning and non-spinning tidal field sources. A following paper by Ogawaguchi and Kojima [178] considered the Darwin-Riemann problem of coalescing neutron star binaries again. The main novelty introduced, with respect to Lai, Rasio and Shapiro’s work, was a more accurate description of the orbit: a 2.5PN expanded Hamiltonian was adopted for the orbit dynamics. Once again, though, the use of the Newtonian tidal tensor oversimplifies the physics of the tidal dynamics. Finally, in Shibata’s paper [170] and in Wiggins and Lai’s paper [171], black hole- neutron star binaries were studied. The black hole tidal field was described by means of the Kerr tidal tensor to which the neutron star was coupled as it moved around the black hole on equatorial circular geodesics. For such orbits one has √ r2 − 2M r + a M r E = BH√ BH (3.124) r P √ 2 √ 2 MBH r(r − 2a MBH r + a ) Lz = √ , (3.125) r P where 2 p P = r − 3MBH r + 2a MBH r (3.126) 9There is no such thing as a black hole in Newtonian gravity; what we mean is that Newtonian orbits and tidal tensor components are used. 10The tidal tensor was not actually named, but the gravitational potential was expanded to second order as in N Eq. (3.3) and coupled to the tensor of inertia Iij which is equivalent to using LT = −Cij Iij /2. 3.3 Orbit Descriptions within the Affine Model 59

and the constant K is related to E and Lz by

2 K = (aE − Lz) . (3.127)

Moreover, the equation of motion (3.47) for the angle Ψ reduces to s dΨ M = BH . (3.128) dτ r3

The use of these orbits is certainly coherent with the apparatus set up to build the Kerr tidal tensor, but for coalescing mixed compact binaries with stellar mass constituents, the hypothesis MBH MNS is not a solid one. In our opinion, this kind of approach is therefore useful if one is interested in tidal aspects of the problem (for example in calculating the orbital separation at which the neutron star is disrupted), since the Kerr tidal tensor is adopted; on the other hand, if one is aiming at an overall description of the dynamics — within the precision that one may hope to achieve with analytic models, of course — the orbit is sacriﬁced for two reasons:

1. MBH and MNS are comparable so that actually the black hole and the neutron star will orbit around the binary centre of mass

2. tidal effects on the orbit are neglected.

Summarizing, Kerr tidal ﬁelds are the best apparatus we have, given their relativistic nature and the fact that they take into account the spin of the source; on the other hand, because of the way they are built, they turn down any orbit description that goes beyond timelike Kerr geodesics. We anticipate that

• when interested only in tidal disruption, we will sacriﬁce a good description of the orbits in favour of the best possible modelling of tides (Chapter5), whereas

• when considering the overall dynamics, we shall somehow sacriﬁce total rigour by dropping the MBH MNS hypothesis when it comes to the orbit dynamics (Chapter 6).

Chapter 4

Two Important Improvements of the Afﬁne Model

The most systematic use of the afﬁne model and of the Kerr tidal tensor in studying mixed compact binaries is due to Wiggins and Lai [171]. As discussed at the end of the previous chapter, they considered the neutron star to be on an equatorial circular orbit around its black hole companion. Moreover, they demanded quasi-equilibrium, that is, they set all time derivatives to zero in the ODEs of the model (3.107)-(3.111): this reduces the system to a set of coupled algebraic equations, so that for every orbital radius r a value for the neutron star principal axes {ai} is determined. This way, one may construct equilibrium sequences and determine the critical orbital radius rtide at which the neutron star is disrupted by tidal forces, i.e. equilibrium is no longer possible for r < rtide. The importance of rtide resides in the fact that black hole-neutron star mergers, as well as neutron star-neutron star mergers, have been invoked as possible engines of short gamma-ray bursts (see page 30 for a short explanation and [97], for example, for a thorough discussion). The fate of BH-NS binaries, in particular, depends on the relative values of rISCO, the radius of the innermost stable circular orbit, and rtide:

• if rISCO < rtide the star is disrupted and then swallowed by the BH,

• if rISCO > rtide it is swallowed without disruption. This is therefore a crucial issue for the SGRB mechanism we are considering: only if rISCO < rtide the merger may result in the black hole + hot massive accretion disk + baryon- free funnel SGRB scenario. Notice that the condition rISCO < rtide is necessary but not sufficient: one has to follow the neutron star fluid after the disruption in order to find out whether or not the accretion disk forms. Compact binaries coalescences are certainly the realm of relativistic non-linear numerical integrations, but, as reported in Chapter2, the history of BH-NS merger simulations is significantly shorter than that of binary neutron star and binary black hole simulations since one must come to grips with difficulties due to the NS and to the BH at the same time. Right now, much of the vast parameter space of mixed binaries (mass ratio, neutron star spin, black hole spin, neutron star microphysics, etc.) is still inaccessible to numerical dynamical simulations and the length and the number of runs performed are shorter than their binary neutron star and binary black hole counterparts. The words of Faber picture the situation clearly: “[...] as we continue to observe SGRBs, it is up to simulations to finally determine

61 62 4. Two Important Improvements of the Affine Model which systems are realistic progenitors, and to quantify how various features of the observed emission reflect the underlying physical parameters of the system” [11]. In the meantime, however, given the fact that numerical relativity is still moving its first steps with mixed binary mergers, analytic models play an important role since they may provide the desired information at least in a semi-quantitative way. This is exactly what Wiggins and Lai did: they faced the problem of determining which BH-NS could be SGRB progenitors with the affine model. With all this in mind and with the goal of studying SGRB progenitors, we would like to improve Wiggins and Lai’s work under two aspects which we believe to be weak points of the model:

1. the self-gravity potential of the neutron star is Newtonian, whereas the star is indeed a relativistic object, and

2. the only EOS that may be and has been used is polytropic.

This chapter is therefore devoted to our intention of

1. including relativistic effects in the neutron star self-gravity and

2. making the use of any barotropic EOS possible.

The second feature, moreover, is something that is still unexplored in numerical relativity simulations and would thus provide us with a first tool capable of starting to span equations of state in the field of mixed compact binary coalescence. Obviously, both points arise from the desire of building an analytic model which has as many “realistic” features as possible; however there is more. The NS disruption, which may lead to the SGRB mechanism described in Section 1.7, occurs when the BH tidal force starts prevailing on the NS self-gravity; this condition may be approximately 2 3 expressed as αMNS/RNS ' MBH RNS/rtide, where α is a dimensionless coefficient, MNS and RNS are the NS mass and radius, and MBH is the BH mass. rtide hence depends essentially on two parameters: the mass ratio1 and the NS compactness at equilibrium C = MNS/RNS. If Newtonian self-gravity is adopted one of the two key parameters is thus treated inappropriately.

4.1 Pseudo-Relativistic Self-Gravity

Let us return to the self-gravitational energy of an ellipsoid expressed in the principal frame, which is given by Eq. (3.104):

1 Z ∞ dσ V = VRb NS q . (4.1) 2 0 2 2 3 (a1 + σ)(a2 + σ)(a3 + σ)

It is evident that in order to include general relativistic effects in V , we need to work on the way it is deﬁned and calculated for spherical equilibrium conﬁgurations of the star, as

1 The mass ratio is generally deﬁned as q = MNS /MBH so that it runs from 0 to 1; in Section 5.3, however, we will adopt the deﬁnition used in [107], i.e. q = MBH /MNS , since we will be testing our results against the results of that paper. 4.1 Pseudo-Relativistic Self-Gravity 63 the Newtonian character of V resides in Vb . This of course involves modifying the stellar structure equations and therefore RNS will be affected too. In General Relativity, the equations governing a spherical star at equilibrium are the Tolman-Oppenheimer-Volkoff (TOV) equations (see Appendix B.1 and Section 1.5). To include relativistic effects in the star self-gravity, our starting point is an effective scalar gravitational potential, which we call ΦTOV, that stems from the TOV equations. This pseudo-relativistic potential2 was adopted and tested in the context of stellar core collapse and post-bounce evolution simulations:

• in [180], Rampp and Janka present the VERTEX code for supernova simulations; in order to approximate relativistic gravity, this code makes use of the potential ΦTOV in place of the usual Newtonian potential ΦN in all Newtonian hydrodynamics equations;

• in [181], Liebendörfer et al. perform a comparison between results of the VERTEX and AGILE-BOLTZTRAN codes, the latter being a fully relativistic (1D) hydrodynamics code; it is shown that both codes produce qualitatively very similar results except for some small (but growing) quantitative differences occurring in the late post-bounce evolution;

• in order to achieve a better agreement than the one reported in [181], different improvements of the aforementioned effective relativistic potential are explored and tested by Marek and collaborators in [182]; the Newtonian equations of hydrodynamics remain untouched.

The way we apply the prescription ΦN −→ ΦTOV of [180]-[182] in the present, completely different, context is the following: the Eulerian hydrodynamics equations — which govern the star in the afﬁne model and are Newtonian — are left formally unchanged, but the gravitational potential ΦTOV is used to calculate the gravitational self-energy, and the pressure and density proﬁles of the star at equilibrium (on which Mc and Π implicitly depend) are built by numerically integrating the TOV equations. The pseudo-relativistic potential ΦTOV is determined by the following equations:

dP ( + P )(m + 4πr3P ) = − (4.2) dr r(r − 2m) dm = 4πr2 (4.3) dr dΦ 1 dP TOV = − . (4.4) dr ρ dr

The ﬁrst two are the TOV stellar structure equations; the last one is the Newtonian hydrostatic equilibrium equation or, equivalently, it is the weak ﬁeld limit of the third TOV equation 3

dν 1 dP 1 dP = − −→ − . (4.5) dr + P dr ρ dr

2 The gravitational potential is a Newtonian concept, but ΦTOV “mimics” general relativistic effects in Newtonian dynamic equations. 3 2ν g00 = −e ' −1 − 2ν ≡ −1 − 2Φ. 64 4. Two Important Improvements of the Afﬁne Model

ΦTOV is therefore given by Z r 03 0 ( + P )(mTOV + 4πr P ) ΦTOV(r) = dr 0 0 (4.6) ∞ ρr (r − 2mTOV) Z r 0 02 mTOV(r) = dr 4πr . (4.7) 0

In virtue of equations (4.2)-(4.4), ΦTOV approximates relativistic equilibrium and the name “pseudo-relativistic gravitational potential” is thus justified. The integral definitions of Mc and Π (but not their values!) are unaffected by the fact we use a TOV neutron star. Vb requires a little bit of care instead. In order to ensure equilibrium, in fact, we must perform the substitution ΦN −→ ΦTOV in a definition of the self-gravitational energy Vb which is general enough to satisfy the virial theorem whatever the scalar gravitational potential is. The virial theorem provides a general equation relating the average total kinetic energy of a stable system, bound by conservative forces, with its average total potential energy:

N X 2hT i = − hFk · rki. (4.8) k=1

Notice that the statement is independent of the potential generating the forces Fk and therefore must still hold if we replace ΦN with ΦTOV. At spherical equilibrium, the velocities are null, therefore the kinetic energy T vanishes and only the gravitational and pressure forces are acting; this yields

(∇P ) · r = −ρ(∇Φ) · r (4.9) and therefore

∇P = −ρ∇Φ , (4.10) which in spherical symmetry reduces to the Newtonian hydrostatic equilibrium condition (see Eq. (4.4)). So far, in these local equations, we have not speciﬁed the form of the gravitational potential Φ. If we now integrate both sides of Eq. (4.9) over the volume of a spherically symmetric star at equilibrium, we obtain 4

Z Z dP Z RNS dP (∇P ) · rd3x = rd3x = 4πr3dr dr 0 dr RNS Z RNS Z 3 2 3 = 4πr P − 3 P 4πr dr = −3 P d x 0 0 = −3Πˆ , (4.11) where we have used spherical symmetry and the fact that the pressure vanishes on the star surface, and Z Z Z 3 3 − ρ(∇Φ) · rd x = − (∂rΦ)rρd x ≡ − dMTr[∂i(Φ)rj] . (4.12)

The last equivalence holds because the trace is an invariant when changing from a coordinate system to another: in this case we are switching from spherical coordinates, in which

4For sake of simplicity we drop the hat (ˆ) everywhere except from the last step. 4.1 Pseudo-Relativistic Self-Gravity 65

Tr[∇iΦrj] ≡ ∂rΦr because of spherical symmetry, to Cartesian coordinates, in which ∇iΦrj = ∂iΦrj. The last expression is related to Vb . A truly general (i.e. “Φ-independent”) way to deﬁne Vb is in fact [163]

def Vb = TrVbij , (4.13) where

def Z Vbij = − dM∂i(Φ)rj (4.14) is the self-gravitational energy tensor written in Cartesian coordinates and Φ is the scalar gravitational potential which is not necessarily Newtonian. The deﬁnition (valid for spherical equilibrium)

Z RNS def dΦN 3 VbN = −4π rb drb (4.15) 0 drb is thus general enough to operate the substitution ΦN −→ ΦTOV in order to obtain Vb without violating the virial theorem. Therefore the crucial relation for spherical equilibrium

Vb = −3Πb (4.16) keeps holding no matter what Φ we choose. If we were to replace the gravitational potential in a “naive” deﬁnition of Vb , such as

1 Z Vb = ρΦ d3x , (4.17) 2 N the virial theorem would not be preserved: the reason is that this deﬁnition relies on the form of the (Newtonian) gravitational potential, while the virial theorem does not. Summarising, the afﬁne model equations (3.107)-(3.111) are left formally unchanged but we have improved the model by incorporating relativistic effects in the neutron star self-gravity as follows:

• the equilibrium conﬁguration of the star is determined with the TOV stellar structure equations;

• the pseudo-relativistic scalar potential ΦTOV given in Eq. (4.6) is used in Eqs. (4.13)- (4.14) which deﬁned the self-gravitational energy at equilibrium Vb ;

• through Eq. (4.1) the relativistic effects are automatically included in the self-gravitational energy of the star when it is deformed by the black hole tidal ﬁeld;

• the values of the three integral quantities Vb , Mc and Π, which appear in the affine model equations, are calculated numerically once the density, energy density and pressure profiles of the star are determined (more about this will be said in the next section). 66 4. Two Important Improvements of the Affine Model

4.2 Using any Barotropic EOS

We now turn our attention to the EOS-dependent integral quantities appearing in the afﬁne model equations (3.107)-(3.111); these are Vb , Mc and Π. Since they were determined analytically for polytropic equations of state in all previous papers employing the afﬁne model, what we want is to be able to calculate them for a generic barotropic EOS, i.e. for any (analytic or tabulated) functional dependence P = P (ρ) between the local pressure and mass density. In order to do so, we make the following hypothesis5:

• the local EOS does not change with the deformations, i.e. the functional dependence of P on ρ is left unchanged.

i i j 3 We recall that the affine model prescription r = q jrb of Eq. (3.64) holds, and thus d r = 3 3 3 a1a2a3/RNSd rb and ρ = RNS/(a1a2a3)ρb. We therefore have ! Z Z RNS a1a2a3 3 a1a2a3 ρb 2 Π = 3 P (ρ)d rb = 4π 3 P 3 rb drb (4.18) RNS RNS 0 a1a2a3/RNS Z Z RNS 1 i 3 4π 4 Mc = rb rbiρdb rb = rb ρdb rb (4.19) 3 3 0 Z RNS 3 dΦ Vb = −4π rb drb , (4.20) 0 drb where Φ = ΦN if one wishes to work with Newtonian self-gravity, or Φ = ΦTOV if one wants to include relativistic effects in the stellar structure as described in the previous section. Once the neutron star spherical equilibrium configuration is determined, one has the tabulated pressure, mass density and energy density profiles inside the star and is therefore able to calculate Π, Mc and Vb . For example, to determine Π numerically one has to:

1. use rb as the (independent and sampled) integration variable

2. read ρb(rb) for every rb from the mass density proﬁle table 3 3. calculate ρ = ρ/b (a1a2a3/RNS) 4. read P (ρ) from the pressure proﬁle table

5. add up all the contributions to the integral.

To test the numerical routines, the analytic results reported in Eqs. (3.115), (3.117) and (3.119) for Mc, Πb and VbN in the case of a polytropic EOS may be used.

4.3 A Note About Solving the EOS-Dependent Integrals Numer- ically

Instead of using r as the independent integration variable to determine the neutron star spherical equilibrium conﬁguration, we used ln P . This improves the integration of the TOV stellar structure equations since the condition to end the integration and to determine the

5This hypothesis has actually been implicitly used in the previous chapter, but it is worth stressing it now. 4.3 A Note About Solving the EOS-Dependent Integrals Numerically 67

neutron star radius is P = 0 (see Appendix B.1). Moreover, this strategy has the advantage “hiding” the Φ-dependence of Vb in the pressure proﬁle, since

dVb = −4πr3P. (4.21) d ln P

To determine ΦTOV and VbTOV with r as an integration variable, see Appendix B.2.

Chapter 5

BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

5.1 Formulation

In this chapter we consider coalescing black hole-neutron star binaries. We solve the afﬁne model equations in the quasi-equilibrium approximation, that is, we take vanishing time derivatives in the ODEs (3.107)-(3.110) which therefore reduce to

2 2 2 1 Vb 3 ˜ RNS Π 0 = a1(Λ + Ω ) − 2a2ΛΩ + RNSa1A1 + − c11a1 (5.1) 2 Mc Mc a1 2 2 2 1 Vb 3 ˜ RNS Π 0 = a2(Λ + Ω ) − 2a1ΛΩ + RNSa2A2 + − c22a2 (5.2) 2 Mc Mc a2 2 1 Vb 3 ˜ RNS Π 0 = RNSa3A3 + − c33a3 (5.3) 2 Mc Mc a3

Mc 2 2 0 = 2 c12(a2 − a1) . (5.4) RNS

Given expression (3.91) for c12, the last equation implies the requirement

ϕ = Ψ (5.5) and thus

Ω ≡ ϕ˙ = Ψ˙ , (5.6) where ϕ is the angle that brings the parallel-propagated frame in the principal frame by a rotation (see Eq. (3.69)) and Ψ is the rotation angle that guarantees that the the former frame is indeed parallel-propagated as the star orbits around the black hole. Ψ˙ is then speciﬁed by assuming that the neutron star centre of mass follows circular orbits, i.e. s M Ψ˙ = BH . (5.7) r3 The system is now a set of three coupled algebraical equations which we solve with respect to the three principal axes of the stellar ellipsoid a1, a2 and a3 by using a 3D Newton-Raphson

69 70 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach scheme [183] to obtain BH-NS quasi-equilibrium sequences. Summarising, within our approach, one has to first build a neutron star at spherical equilibrium (by selecting an EOS and integrating the Newtonian or TOV stellar structure equations), then choose a mass ratio q and a black hole spin parameter a and solve the affine model quasi-equilibrium equations (5.1)-(5.3) with respect to a1, a2 and a3 starting from a (sufficiently) large orbital separation and gradually reducing the separation r. The quasi-equilibrium sequence ends when the neutron star surface crosses the black hole event horizon or when

∂(a /a )−1 2 1 = 0 . (5.8) ∂r

This equation denotes that the distortion of the ellipsoid (a2/a1) becomes infinite if the orbital separation (r) is reduced further. The orbital separation at which this condition is verified thus identifies the tidal disruption separation and we indicate it with rtide: beyond this distance, quasi-equilibrium configurations are no longer possible. Once again, we stress the fact that, as explained at the beginning of Chapter4, if rtide > rISCO the binary is a possible SGRB progenitor. In a first moment we implemented this condition by monitoring the left hand side of Eq. (5.8) and ending the calculation when it reached a small, user specified, value; we then observed that the Newton-Raphson scheme routine itself would fail in solving (5.1)-(5.3) at the same rtide yielded by the implementation of Eq. (5.8). By failure we mean that the NS axis a1 would suddenly shrink, whereas a2 and a3 would expand: this is clearly an unphysical behaviour because, for example, the value of a1 should grow as r decreases. Therefore, instead of implementing Eq. (5.8) in order to determine rtide, it is sufficient to monitor the axes and to interrupt the sequence when a contraction of a1 is encountered along with an expansion of a2 and a3 since, as explained, such a behaviour is unphysical. In Section 5.3 we will show that this method does yield reliable results. If we use polytropes and the Newtonian gravitational potential ΦN, this setup is like the one of Wiggins and Lai [171] (see also Section 3.3). However we have the advantage — as explained in the previous chapter — of being able to use any barotropic EOS and to take into account relativistic effects in the neutron star self-gravity by replacing ΦN with ΦTOV. A code was written from scratch in order to obtain BH-NS quasi-equilibrium sequences by implementing the strategy just discussed. As explained in the next section, with the newly written code our plan was to proceed as follows:

1. we ﬁrst used Newtonian self-gravity and polytropes in order to test our code with the aid of Wiggins and Lai’s work;

2. we then switched to pseudo-relativistic self-gravity for the neutron star and used non-rotating black holes to compare the results yielded by our model to the relativistic quasi-equilibrium results of Taniguchi, Baumgarte, Faber and Shapiro [107];

3. once our model was reinforced by the previous comparison, we considered black hole rotation and different barotropic nuclear equations of state for the neutron star ﬂuid; this allowed us to systematically explore for the ﬁrst time the

q × C × a × EOS (5.9)

parameter space — where C is the neutron star compactness — and to determine which binary systems are possible SGRB progenitors. 5.2 Code Calibration and Tests 71

5.2 Code Calibration and Tests

Once the code for the quasi-equilibrium sequences is written and is working, we must make sure it works correctly. The very first thing to do is to calibrate its two free parameters: the initial orbital separation rstart and the step ∆r by which the orbital separation r is reduced every time the Newton-Raphson routine is called. If rstart is not large enough, placing the spherical neutron star in the black hole tidal field too close to the black hole itself — or, equivalently, suddenly turning on a tidal field which is too strong — may lead to unphysical behaviour: the star may be disrupted right away or shrink, for example. In other words, the initial spherical neutron star must be placed in a “safe” region where its ellipsoidal equilibrium configurations does not depart much from sphericity. rstart must, therefore, be large enough, i.e. the equilibrium sequence yielded by the code must be stable when increasing the value of rstart. ∆r must instead be small enough: if it is not, the Newton-Raphson scheme fails in finding physically acceptable solutions of Eqs. (5.1)-(5.3) and the quasi-equilibrium sequence changes when one decreases ∆r, which is obviously an undesired behaviour. As an example, in Figure 5.1 we show the quasi-equilibrium sequences obtained for a neutron star — with a mass of 1.4 M , an n = 1 polytropic EOS and a 15.7 km radius — orbiting a Schwarzschild black hole of mass ∼ 3.66 M for various choices of ∆r (normalised to the choice of the first sequence, indicated with ∆r0). Notice that the curves of the quasi-equilibrium sequences converge as ∆r decreases and that they are almost coincident for ∆r = ∆r0/6

0.9

0.8 1 / a 2

a ∆ r0/8 0.7 ∆ r0/6 ∆ r0/4 ∆ 0.6 r0/2 ∆ r0 0.5 4 5 6 7 8 9 10 r/MBH

Figure 5.1. Calibration of ∆r. As a function of the orbital separation in units of black hole masses r/MBH , we plot the ratio a2/a1 of the axes of a neutron star of mass MNS = 1.4 M , radius RNS = 15.7 km and n = 1 polytropic equation of state orbiting a Schwarzschild black hole of mass MBH ' 3.66 M . Each quasi-equilibrium sequence corresponds to a different choice of the step ∆r; the various choices are indicated in terms of the largest step choice ∆r0, which belongs to the red sequence. 72 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

and ∆r = ∆r0/8. The ﬁnal result of our calibration analysis is that

rstart & 10(MBH + MNS) (5.10) −3 ∆r . 10 MBH (5.11) correspond to safe values for the two free parameters. This conclusion holds for the kind of binaries we are interested in, of course: if one wanted to use the afﬁne model for systems containing a white dwarf, for example, or a massive black hole, the two parameters would have to be re-tuned.

The calibrated code may now be tested; in order to do so, we use polytropes for the NS fluid and Newtonian self-gravity and compare our results to the ones obtained by Wiggins and Lai [171]. Since the NS fluid in BH-NS inspirals is better described as an irrotational fluid (see page 56), we focus on their results for C = 0. This means looking at Table 1 and Figure 2 of [171]. In the former, the authors provide with four significant figures the value of the tidal disruption radius in the normalised form

rtide r¯tide = , (5.12) R¯NS where the normalization factor is 1/3 MNS R¯NS = RNS , (5.13) MBH for several values of R¯NS, for black hole spin a = {−1, 0, 1} MBH and polytropic index 3 n = {0, 0.5, 1, 1.5}, whereas in the latter they plot the axis ratio a2/a1 and RNS/(a1a2a3) against the dimensionless orbital separation r¯ = r/R¯NS for an n = 1 polytropic NS, for the dimensionless radius-mass combination R¯NS = 4 and for the three values of the black hole spin parameter a = {−1, 0, 1} MBH . With our code we are capable of reproducing all the results just described. This ensures us that we have built the code correctly and we are ﬁnally ready to see whether the replacement ΦN → ΦTOV discussed in the previous chapter yields encouraging and solid results or not.

5.3 Comparison with Recent Relativistic Results

The next step is to see if the ﬁrst of the afﬁne model improvements discussed in the previous chapter — pseudo-relativistic self-gravity — works well. Our primary intention is, in fact, to provide a quick and reliable tool which allows us (1) to determine whether a mixed binary is a possible short gamma-ray burst progenitor or not and (2) to access parameter space regions which are currently inaccessible with other methods. The results reported in this section are published in [21]. The paper by Taniguchi, Baumgarte, Faber and Shapiro [107] provides us with the perfect testbed, that is, relativistic models of black hole-neutron star binaries in quasi-equilibrium circular orbits. Their Figure 11, which we reproduce in Figure 5.2, is particularly useful: the binary mass ratio is plotted against the orbital angular frequency at tidal disruption for several general relativistic irrotational models of Γ = 2 polytropic neutron stars with 5.3 Comparison with Recent Relativistic Results 73

Schwarzschild black hole companions. A comparison with their results is ideal since it allows us to exclusively focus on the effects of the replacement ΦN → ΦTOV. A few words about [107] are necessary in order to explain the authors’ approach1. The Einstein constraint equations are solved in the conformal thin-sandwich decomposition along with the relativistic equations of hydrostationary equilibrium adopting maximal slicing and assuming spatial conformal flatness. The black hole is modelled by using excision and equilibrium boundary conditions are imposed on the excision surface (i.e. the apparent horizon). As already mentioned, the neutron star is assumed to be irrotational and its fluid is provided with a Γ = 2 polytropic equation of state. The onset of tidal disruption is identified by the formation of cusps on the neutron star surface which are monitored as follows. A mass-shedding indicator χ is defined as the ratio between the radial derivatives of the specific enthalpy h (see Section 8.2) on the neutron star surface in the direction of the companion (“eq”) and in the polar direction (“pole”), i.e.

def [∂(ln h)/∂r] χ = eq . (5.14) [∂(ln h)/∂r]pole This quantity works as an indicator since it is equal to unity for a spherical star and vanishes when a cusp is formed at the neutron star equator, i.e. where the tidal forces are strongest. χ is tabulated as a function of the orbital angular velocity and, in order to find the onset of tidal disruption, it is extrapolated to χ = 0 by using fitting polynomial functions. A few details about the physical quantities used by Taniguchi et al. in Figure 11 of [107] are required since we stuck to them. The mass ratio they adopt is irr MBH q = ADM,0 , (5.15) MNS irr ADM,0 where MBH is the irreducible mass of the black hole and MNS is the ADM mass of the spherical, isolated neutron star: these coincide, respectively, with the Schwarzschild black hole gravitational mass and with the gravitational mass yielded by the TOV stellar structure equations (Appendix B.1). We shall therefore use M q = BH . (5.16) MNS The orbital angular velocity at tidal disruption is defined as usual for circular orbits, i.e. s def MBH + MNS Ωtide = 3 , (5.17) rtide but the quantity actually used in the plot by the authors is dimensionless and is given by the product of Ωtide with the the polytropic length scale

def 1/(2Γ−2) Rpoly = κ . (5.18) Finally, a neutron star model is identiﬁed by its dimensionless baryonic mass, which in the “TOV language” is given by

Z RNS 2 ¯ NS NS 4π r ρ(r) MB = MB /Rpoly = p dr , (5.19) Rpoly 0 1 − 2mTOV (r)/r

1Their numerical code is based on the LORENE spectral-methods library routines developed by the Meudon relativity group (http://www.lorene.obspm.fr). 74 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

¯ NS ¯ MB MNS C 0.12 0.1136 0.1088 0.13 0.1223 0.1201 0.14 0.1310 0.1321 0.15 0.1395 0.1452 0.16 0.1478 0.1600 0.17 0.1560 0.1780 ¯ NS ¯ Table 5.1. Dimensionless baryonic mass (MB ), dimensionless gravitational mass (MNS) and stellar compactness (C) considered in [107] and used here to test our model (see Figure 5.2 and Table 5.2). When Newtonian self-gravity is used M¯ NS is identiﬁed with the dimensionless Newtonian ¯ N mass MNS. where the integral is obviously intended for the isolated star at equilibrium. For each value ¯ NS ¯ of MB considered, one has a value of the (dimensionless) gravitational mass MNS = MNS/Rpoly and of the compactness

def M C = NS . (5.20) RNS All values considered in [107] and here are gathered in Table 5.1. In Figure 5.2, which was published in [21], we show the results yielded by our pseudo- relativistic afﬁne approach (“Pseudo TOV”), the results of [107] (“Full GR”) and the results obtained by using Newtonian self-gravity (“Newtonian”), i.e. â la Wiggins and Lai: in this case, since using Eq. (5.19) does not make sense, the neutron star model is ﬁxed by equating N the Newtonian star mass MNS to the value of the gravitational mass MNS prescribed by the comparison one is performing. The data displayed in the six panels of Figure 5.2 show that:

• our pseudo-relativistic ellipsoidal model agrees with relativistic data much better than the Newtonian ellipsoidal model;

• for a given compactness, the agreement between our data and Full GR data improves as the mass ratio increases. This is due to the fact that in our model we assume that the centre of mass of the star follows the geodesics of the black hole spacetime and we build a parallel propagated orthonormal reference frame on this hypothesis (Section 3.1.4): such condition is better satisﬁed for larger mass ratios. For instance, NS for q = 9 and MB = 0.12Rpoly (C = 0.1088), our result and the result of Full GR calculations practically coincide;

• for lower values of the mass ratio q, the agreement between our data and the Full GR data increases as the stellar compactness increases.

For a more detailed and quantitative comparison, in Table 5.2 we provide the values of ΩtideRpoly plotted in Figure 5.2. Each data set in the table corresponds to one of the panels in Figure 5.2, which is identified by the row indicating the relativistic NS compactness C and its gravitational/Newtonian mass normalised with respect to the polytropic length ¯ scale, i.e. MNS = MNS/Rpoly. The first column gives the mass ratio q = MBH /MNS; the remaining three columns are the orbital angular frequencies at tidal disruption, normalised with respect to Rpoly/10, resulting from calculations performed with the three binary models 5.3 Comparison with Recent Relativistic Results 75 considered: once again, “Full GR” indicates data extracted from Figure 11 of [107], “Pseudo TOV” data obtained with our model and “Newtonian” data obtained with the affine model using Newtonian self-gravity. We remind the reader that the BH is always non rotating (the BHs in [107] actually have a small residual angular momentum, see the reference for further details).

10 10 NS NS 9 MB = 0.12Rpoly 9 MB = 0.13Rpoly 8 8 7 7 NS 6 6 /M

BH 5 5

M 4 4 3 Full GR 3 Newtonian Pseudo TOV Full GR 2 Newtonian 2 Pseudo TOV 1 1 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.06 0.08 0.1 0.12 0.14 0.16 0.18

10 10 NS NS 9 M = 0.14R 9 Newtonian M = 0.15R B poly Full GR B poly 8 8 Pseudo TOV 7 7 NS 6 6 /M

BH 5 5

M 4 4 3 Newtonian 3 Full GR 2 Pseudo TOV 2 1 1 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.06 0.08 0.1 0.12 0.14 0.16 0.18

10 10 NS NS 9 Newtonian MB = 0.16Rpoly 9 Newtonian M = 0.17R Full GR Full GR B poly 8 Pseudo TOV 8 Pseudo TOV 7 7 NS 6 6 /M

BH 5 5

M 4 4 3 3 2 2 1 1 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Ω Ω tideRpoly tideRpoly

Figure 5.2. Comparison between relativistic results from [107] (Full GR), results obtained with the ellipsoidal model using Newtonian self-gravity (Newtonian) and results from our improved ellipsoidal model (Pseudo TOV). Each graph shows the mass ratio (Eq. (5.16)) versus the dimensionless orbital angular frequency at tidal disruption (Eq. (5.17)). In all cases the neutron star EOS is an n = 1, κ = 1 polytrope. The baryonic mass in units of Rpoly (Eq. (5.19)) is indicated in each panel; see Table 5.1 for the corresponding dimensionless gravitational mass and compactness values.

The comparison we performed with Full GR results validates our approach of using the pseudo-relativistic scalar gravitational potential ΦTOV in the afﬁne model equations in order 76 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

q Full GR Pseudo TOV Newtonian C = 0.1088 MNS/Rpoly = 0.1136 9 0.88 0.88 0.70 8 0.89 0.91 0.71 7.5 0.90 0.92 0.72 7 0.91 0.94 0.74 6 0.93 0.96 0.75

C = 0.1201 MNS/Rpoly = 0.1223 7 0.97 0.99 0.75 6.5 0.99 1.00 0.76 6 1.00 1.02 0.77 5 1.04 1.05 0.80 3 1.08 1.16 0.87

C = 0.1321 MNS/Rpoly = 0.1310 6 1.06 1.07 0.79 5 1.08 1.11 0.82 4 1.11 1.17 0.85 3 1.16 1.24 0.89

C = 0.1452 MNS/Rpoly = 0.1395 5 1.21 1.18 0.84 4 1.22 1.24 0.87 3 1.28 1.33 0.92 2 1.31 1.45 0.99 1 1.37 1.72 1.17

C = 0.1600 MNS/Rpoly = 0.1478 4.5 1.28 1.29 0.87 4 1.31 1.33 0.89 3 1.35 1.43 0.94

C = 0.1780 MNS/Rpoly = 0.1560 3.5 1.49 1.50 0.93 3 1.53 1.56 0.96 2 1.61 1.73 1.04 Table 5.2. Data plotted in Figure 5.2: each sub-table corresponds to a panel of the figure, which is identified by the stellar compactness C and the dimensionless gravitational mass MNS/Rpoly (see Table 5.1. The mass ratio q (Eq. 5.16) is given in the first column, while the orbital angular frequency at tidal disruption resulting from each model considered here — Full GR, Pseudo TOV, Newtonian — is displayed in the form ΩtideRpoly/10 in the remaining three columns.

to calculate the tidal disruption limit rtide. We may thus use it to study BH-NS binaries as possible SGRB progenitors taking into account black hole rotation and barotropic equations of state which differ from Γ = 2 polytropes. 5.4 New Applications of the Quasi-Equilibrium Approach 77

5.4 New Applications of the Quasi-Equilibrium Approach

Having tested the validity of our approach, we employ it to study different possible BH-NS binary conﬁgurations, determining the quasi-equilibrium sequences and the tidal disruption radius rtide for equatorial circular orbits. Section 5.4.1 is based on the results published in [21].

5.4.1 Effects of Varying the NS EOS and the BH Spin In order to study the effects of the nuclear equation of state, we begin by considering NSs with MNS = 1.4 M and describe the ﬂuid forming the core of the NS with two different equations of state proposed in recent years by the nuclear physics community, which we call APR2 and BGN1H1. These equations of state were chosen because they yield quite different radii for a 1.4 M neutron star. • The Akmal-Pandharipande-Ravenhall (APR2) hadronic EOS [64] describes matter consisting of neutrons, protons, electrons and muons in weak equilibrium; it is obtained within nuclear many-body theory using a variational approach to the Schrödinger equation; its microscopic input is based on the Argonne v18 potential for nucleon-nucleon interactions [184] — which is calibrated to deuteron properties and vacuum nucleon-nucleon phase shifts for laboratory energies Elab up to 350 MeV — and on the Urbana IX (UIX) three-body potential [185]; relativistic corrections to ∗ both potentials are included [186] (which yields v18 + δv + UIX ). • The Balberg-Gal (BGN1H1) EOS [65] describes matter consisting of neutrons, protons, electrons, muons and hyperons (Σ, Λ and Ξ) in equilibrium. Assuming the mean ﬁeld approximation, an effective potential is employed; its parameters are tuned in order to reproduce the properties of nuclei and hypernuclei according to high energy experiments. This EOS is a generalization of the Lattimer-Swesty EOS [187], which does not include hyperons. Both equations of state are completed by an EOS for the crust. In the case of the table with the APR2 core, which was kindly provided by Dr. Omar Benhar, the crust was subdivided into two regions according to the density2: 1. (107 − 4 · 1011) g/cm3 where the Baym-Pethick-Sutherland (BPS) EOS [54] is used

2. (4 · 1011 − 2 · 1014) g/cm3 where the Pethick-Ravenhall-Lorenz (PRL) EOS [57] is used. For densities lower than 107 g/cm3 the BPS EOS is extrapolated. In the case of the the table with the BGN1H1 core, which is part of the Lorene ﬁles3, the crust is subdivided into three density intervals: 1. ≤ 108 g/cm3, where the BPS EOS is used

2. (108 − 1011) g/cm3, where the Haensel-Pichon (HP94) EOS [55] is used

3. (1011 − 1.6 · 1014) g/cm3, where the Douchin-Haensel (SLy4) crust [58, 59] is used.

2Notice that 4 · 1011 g/cm3 is the neutron drip density (Section 1.5.2) 3http://www.lorene.obspm.fr/ 78 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

The density of the crust-core interface, i.e. 2 · 1014 g/cm3 and 1.6 · 1014 g/cm3 for the two cases considered here, is usually indicated with ρcc. Figure 1.3 shows the mass vs radius curves obtained with this APR2 table (indicated as “APR2 (II)” to distinguish it from the APR2 EOS table used in Section 5.4.2 which has a different outer crust description) and the Lorene BGN1H1 table. With the neutron mass choice of 1.4 M , the radius that APR2 yields is of 11.53 km, whereas BGN1H1 gives RNS = 12.84 km: a BGN1H1 1.4 M neutron star is therefore less compact than an APR2 one. In this section and in the rest of the thesis, we abandon the convention inherited during the comparison with [107] and redeﬁne the mass ratio as

def M q = NS . (5.21) MBH We consider several values of the mass ratio ranging from 0.02 to 0.2. For the black hole spin parameter we shall use the three values a = {0, 0.5, 0.99}MBH . To assess whether a BH-NS binary is a possible SGRB progenitor candidate, we compare the tidal disruption orbital separations rtide with rISCO, i.e. the radius of the innermost stable circular orbit. In the Schwarzschild case, we determine the location of the ISCO through the analytic fit given in [107], where the effects of the finite mass ratio (Eq. (5.21)) and of the stellar compactness (Eq. (5.20)) are taken into account: 6(1 + q) rISCO = MBH . (5.22) 1 − 0.444q1/4(1 − 3.54C1/3)2/3 The ISCO locations from which this fit was inferred were obtained by looking for turning points along the binding energy curve (or equivalently the total angular momentum curve) [107]. Note that according to this fit the values of rISCO/MBH we give for a = 0 in Figure 5.3 depend on the mass ratio q. Since no such fit exists for the Kerr BH-NS binaries, when a 6= 0 we estimate the ISCO by using the formulae derived in [188] for a point mass in the gravitational field of a Kerr BH:

1/2 rISCO = MBH {3 + Z2 ∓ [(3 − Z1)(3 + Z1 + 2Z2)] } 2 2 1/3 1/3 1/3 Z1 = 1 + (1 − a /MBH ) [(1 + a/MBH ) + (1 − a/MBH ) ] (5.23) 2 2 2 1/2 Z2 = (3a /MBH + Z1 ) , where the upper (lower) sign holds for corotating (counterotating) orbits. Note that according to these equations the values that rISCO/MBH assumes for a 6= 0 do not depend on q. The results of our numerical integrations, which are published in [21], are displayed in Figure 5.3, where the ratios a2/a1 (continuous lines) and a3/a1 (dotted lines) among the NS axes are shown as functions of r/MBH ; the quasi-equilibrium sequences end when the tidal disruption of the NS is reached (see Section 5.1), apart from the q = 1/50 cases in the two lower panels which end when the NS surface touches the BH outer horizon + located at r = RBH . The left panels refer to the EOS APR2, the right ones to BGN1H1; the upper panels refer to the Schwarzschild case, the middle panels to a = 0.5 MBH , the lower panels to a = 0.99 MBH . The dashed, vertical lines mark the locations of the ISCO. In the a = 0.99 MBH case we also indicate the location of the black hole outer horizon with a dot-dashed vertical line. From the graphs in Figure 5.3 we extract the following information. 5.4 New Applications of the Quasi-Equilibrium Approach 79

APR2 BALBN1H1 1 1

0.9 1/10 0.9 1/10 1/5

1 0.8 1 0.8

/ a 1/6 / a

2,3 a/M = 0 2,3 1/6 a/MBH = 0 a 0.7 BH a 0.7 1/5

0.6 0.6

0.5 0.5 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 r/MBH r/MBH 1 1

0.9 1/10 0.9 1/10

1 0.8 1/5 1 0.8 1/6

/ a / a 1/5

2,3 a/M = 0.5 2,3 a/M = 0.5 a 0.7 BH a 0.7 BH

0.6 1/6 0.6

0.5 0.5 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 r/MBH r/MBH 1 1

1/50 1/50 0.9 0.9 1/20 1/6 1/20 1/6 1/10 1/5 1/10 1/5 1 0.8 1 0.8 / a / a

2,3 a/M = 0.99 2,3 a/M = 0.99 a 0.7 BH a 0.7 BH

0.6 0.6 ISCO ISCO Horizon Horizon 0.5 0.5 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 r/MBH r/MBH

Figure 5.3. BH-NS quasi-equilibrium sequences. The NS axes ratios a2/a1 (continuous lines) and a3/a1 (dotted lines) are plotted against r/MBH for a neutron star orbiting a black hole. The black hole spin parameter takes the values a = {0, 0.5, 0.99}MBH as indicated in each panel. The mass ratio q = MNS/MBH is indicated by the fractions next to each curve; for practical uses, the numerical values of such fractions are 1/50 = 0.02, 1/20 = 0.05, 1/10 = 0.1, 1/6 = 0.16¯, 1/5 = 0.2. In all graphs the neutron star mass is MNS = 1.4 M . In the left column the star is modelled using the APR2 EOS (RNS = 11.53 km), while in the right column the BGN1H1 EOS is adopted (RNS = 12.84 km). The ISCO locations are indicated by the dashed vertical lines, while the dot- dashed lines in the lower graphs mark the location of the black hole outer horizon at the equatorial plane. In the two q = 1/50 cases, the sequence does not terminate with the NS tidal disruption but + + with a plunge: it ends when r = a1 + RBH , where RBH is the size of the black hole horizon at the equatorial plane.

• In each panel, the smaller the mass ratio q is, the closer the NS can get to the BH before 80 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

being disrupted, i.e. rtide/MBH decreases as q decreases. Therefore the conditions for the BH-NS → SGRB mechanism to take place are more likely to be satisﬁed for high values of q.

• If a = 0 (first row) the star enters the ISCO before being disrupted for all the considered values of q, and for both the APR2 and the BGN1H1 EOS: therefore, our model and our calculations confirm that SGRBs cannot be ignited by mixed binaries containing non-spinning black holes, as we discussed in Section 2.3.2. On the other hand, if a = 0.5MBH (middle row) the star is disrupted at r > rISCO for q & 0.175 (APR2 EOS) and q & 0.152 (BGN1H1 EOS) and the SGRB may take place. If, finally, a = 0.99MBH (third row), the star is disrupted at r > rISCO for q & 0.036 (APR2 EOS) and q & 0.030 (BGN1H1 EOS) and, again, the SGRB may take place.

We note that as the the black hole spin increases, both rtide/MBH and rISCO/MBH decrease; however, rISCO/MBH decreases more rapidly than rtide/MBH , and consequently for a given mass ratio q, chances to develop an SGRB are higher if the black holes rotates faster. Thus, in general, the conditions for the SGRB mechanism proposed in [94] — and described on page 30 — to take place are favoured for high values of q and high values of a.

• Comparing the two graphs in each row, we see that for a more compact NS rtide/MBH is smaller. This is what one expects, since a more compact star is less deformable: hence a stronger tidal ﬁeld is needed to disrupt the star.

• As the value of the BH spin increases, the values of the axes a2 (continuous lines) and a3 (dotted lines) tend to coincide.

We also note that, if a = 0.99MBH , for q ' 0.05 the star enters the ergosphere before disruption.

5.4.2 Varying the NS Mass and Using Other Equations of State To describe the fluid forming the core of the NS, this time we consider four different archetypal barotropic equations of state developed by the nuclear physics community; we denote them APR2, BGN1H1, BPAL12 and GNH3. Since the first two have already been described in the previous section, we shall briefly review only the remaining two.

• The Bombaci-Prakash et al. (BPAL12) EOS [66, 67] can be considered as a soft extreme of the nuclear equations of state of NS matter. It describes nucleons, electrons and muons in weak equilibrium by means of a phenomenological schematic potential model. Such ﬁnite temperature potential is designed to reproduce the results of detailed microscopic calculations of both nuclear and neutron-rich matter at zero temperature. The free parameters in the potential are constrained by empirical knowledge. In order to suitably simulate ﬁnite range effects, the phenomenological approach used requires the choice of a function in the potential: the other standard choice for this function leads to Skyrme-like effective interactions.

• The Glendenning (GNH3) EOS [68] considers nucleon, electron and muon matter up to a certain density ρH ' 2ρs; beyond this point, additional baryon states (such as the ∆ and the hyperons Λ, Σ, Ξ, π) and the mesons σ, ρ, ω, K, K∗ are introduced. In the 5.4 New Applications of the Quasi-Equilibrium Approach 81

2.2 2 1.8 APR2 GNH3 1.6 1.4 1.2 BPAL12 BGN1H1 1 0.8 M [Solar Masses] 0.6 0.4 0.2 9 10 11 12 13 14 15 R [km]

Figure 5.4. Neutron star mass versus radius curves for the four equations of state used in this section of the thesis.

case of this EOS a relativistic mean ﬁeld model of baryonic matter is used. Below the hadronization density ρH the EOS is very stiff but causal; the appearance of hyperons strongly softens the EOS because they are more massive than nucleons and when they start to ﬁll their Fermi sea they are slow and replace the highest energy nucleons.

The APR2, BGN1H1, BPAL12 and GNH3 we used are available in the Lorene ﬁles4; the crust description in all four cases is the one of the BGN1H1 table described on page 78. In Figure 5.4 we show the mass versus radius curves for these four equations of state yields. The four equations of state clearly have very different behaviours and were actually chosen because they span, in the mass-radius plane, the current possible choices for a barotropic equation of state (see e.g. [40]). BPAL12 and GNH3 can be considered as soft and hard extremes respectively. APR2 has a wide mass interval in which its radius is almost insensitive to mass variations; BGN1H1 also shows this feature, albeit for a more restricted mass interval, whereas BPAL12 and GNH3 do not. BGN1H1, on the other hand, has a sudden softening which does not characterise the other three equations of state; this softening is due to the formation of hyperons in the core of the neutron star above a critical central density (see the beginning of Section 5.4.1). Above 1.2 M , i.e. in the range relevant

4http://www.lorene.obspm.fr/. 82 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach for neutron stars, the relative position of the BGN1H1 and the APR2 curves changes (around 1.6 M ) due to the combination of the two features just discussed. We consider three different values of the black hole spin, i.e. a = {0.50, 0.75, 0.99} MBH , and ﬁve values of the mass ratio, i.e. q = MNS/MBH = {1/10, 1/7, 1/5, 1/3, 1/2}. This mass ratio range is relevant for the detection of GWs emitted by mixed binaries [189]. We also vary the neutron star mass, which can be equal to 1.2, 1.4, 1.6, 1.8 and 2.0 solar masses depending on the maximum mass that the equation of state allows, so that values up to 2.0 M are used for APR2, 1.6 M for BGN1H1, 1.4 M for BPAL12 and 1.8 M for GNH3. For each combination of the parameters, rtide and rISCO are determined as discussed in Eq. (5.8) and Eq. (5.23), respectively. The results of our calculations are reported in Tables 5.3 to 5.9, where rtide and rISCO are given for each possible choice of the NS model, the binary mass ratio q and the black hole spin parameter a.

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.50 1.2 11.32 9.07 4.23 1.4 11.29 7.95 1.6 11.23 7.06 1.8 11.10 6.33 2.0 10.85 5.69 1/3 0.50 1.2 11.32 7.01 4.23 1.4 11.29 6.16 1.6 11.23 5.49 1.8 11.10 4.94 2.0 10.85 4.46 1/5 0.50 1.2 11.32 5.11 4.23 1.4 11.29 4.52 1.6 11.23 4.06 1.8 11.10 3.70 2.0 10.85 3.38 1/7 0.50 1.2 11.32 4.20 4.23 1.4 11.29 4.74 1.6 11.23 3.40 1.8 11.10 3.14 2.0 10.85 2.92 1/10 0.50 1.2 11.32 3.46 4.23 1.4 11.29 3.14 1.6 11.23 2.91 1.8 11.10 2.75 2.0 10.85 2.63

Table 5.3. Orbital separation at tidal disruption (rtide, ﬁfth column) and ISCO radius (rISCO, sixth column) obtained with an APR2 neutron star and a black hole with a = 0.50 MBH spin parameter (second column). Several values of the binary mass ratio q and the neutron star mass (and radius) are considered as indicated by the ﬁrst, third and fourth column respectively. 5.4 New Applications of the Quasi-Equilibrium Approach 83

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.75 1.2 11.32 9.01 3.16 1.4 11.29 7.88 1.6 11.23 6.99 1.8 11.10 6.25 2.0 10.85 5.59 1/3 0.75 1.2 11.32 6.93 3.16 1.4 11.29 6.07 1.6 11.23 5.39 1.8 11.10 4.83 2.0 10.85 4.33 1/5 0.75 1.2 11.32 5.01 3.16 1.4 11.29 4.40 1.6 11.23 3.92 1.8 11.10 3.53 2.0 10.85 3.19 1/7 0.75 1.2 11.32 4.06 3.16 1.4 11.29 3.58 1.6 11.23 3.21 1.8 11.10 2.91 2.0 10.85 2.66 1/10 0.75 1.2 11.32 3.28 3.16 1.4 11.29 2.92 1.6 11.23 2.65 1.8 11.10 2.44 2.0 10.85 2.28

Table 5.4. Same as Table 5.3 except for the black hole spin parameter a which is equal to 0.75 MBH . 84 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.99 1.2 11.32 8.95 1.45 1.4 11.29 7.82 1.6 11.23 6.92 1.8 11.10 6.18 2.0 10.85 5.52 1/3 0.99 1.2 11.32 6.87 1.45 1.4 11.29 6.00 1.6 11.23 5.32 1.8 11.10 4.75 2.0 10.85 4.24 1/5 0.99 1.2 11.32 4.92 1.45 1.4 11.29 4.30 1.6 11.23 3.82 1.8 11.10 3.41 2.0 10.85 3.05 1/7 0.99 1.2 11.32 3.96 1.45 1.4 11.29 4.46 1.6 11.23 3.07 1.8 11.10 2.75 2.0 10.85 2.46 1/10 0.99 1.2 11.32 3.14 1.45 1.4 11.29 2.75 1.6 11.23 2.44 1.8 11.10 2.19 2.0 10.85 1.96

Table 5.5. Same as Table 5.3 except for the black hole spin parameter a which is equal to 0.99 MBH . 5.4 New Applications of the Quasi-Equilibrium Approach 85

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.50 1.2 12.97 10.26 4.23 1.4 12.84 8.91 1.6 10.32 6.24 1/3 0.50 1.2 12.97 7.91 4.23 1.4 12.84 6.89 1.6 10.32 4.88 1/5 0.50 1.2 12.97 5.75 4.23 1.4 12.84 5.03 1.6 10.32 3.65 1/7 0.50 1.2 12.97 4.69 4.23 1.4 12.84 4.13 1.6 10.32 3.11 1/10 0.50 1.2 12.97 3.83 4.23 1.4 12.84 3.42 1.6 10.32 2.74 1/2 0.75 1.2 12.97 10.21 3.16 1.4 12.84 8.85 1.6 10.32 6.16 1/3 0.75 1.2 12.97 7.84 3.16 1.4 12.84 6.81 1.6 10.32 4.76 1/5 0.75 1.2 12.97 5.65 3.16 1.4 12.84 4.92 1.6 10.32 3.49 1/7 0.75 1.2 12.97 4.57 3.16 1.4 12.84 3.99 1.6 10.32 2.88 1/10 0.75 1.2 12.97 3.68 3.16 1.4 12.84 3.23 1.6 10.32 2.42 1/2 0.99 1.2 12.97 10.15 1.45 1.4 12.84 8.79 1.6 10.32 6.09 1/3 0.99 1.2 12.97 7.79 1.45 1.4 12.84 6.74 1.6 10.32 4.68 1/5 0.99 1.2 12.97 5.58 1.45 1.4 12.84 4.84 1.6 10.32 3.40 1/7 0.99 1.2 12.97 4.48 1.45 1.4 12.84 3.89 1.6 10.32 2.71 1/10 0.99 1.2 12.97 3.56 1.45 1.4 12.84 3.09 1.6 10.32 2.16 Table 5.6. Same as Table 5.3; the black hole spin parameter a (second column) takes the values 0.50 MBH , 0.75 MBH and 0.99 MBH . Calculations are performed with the BGN1H1 equation of state. 86 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.50 1.2 11.14 8.75 4.23 1.4 10.00 6.91 1/3 0.50 1.2 11.14 6.76 4.23 1.4 10.00 5.37 1/5 0.50 1.2 11.14 4.94 4.23 1.4 10.00 3.99 1/7 0.50 1.2 11.14 4.07 4.23 1.4 10.00 3.35 1/10 0.50 1.2 11.14 3.37 4.23 1.4 10.00 2.88 1/2 0.75 1.2 11.14 8.68 3.16 1.4 10.00 6.83 1/3 0.75 1.2 11.14 6.68 3.16 1.4 10.00 5.27 1/5 0.75 1.2 11.14 4.83 3.16 1.4 10.00 3.84 1/7 0.75 1.2 11.14 3.92 3.16 1.4 10.00 3.15 1/10 0.75 1.2 11.14 3.18 3.16 1.4 10.00 2.61 1/2 0.99 1.2 11.14 8.62 1.45 1.4 10.00 6.76 1/3 0.99 1.2 11.14 6.62 1.45 1.4 10.00 5.19 1/5 0.99 1.2 11.14 4.74 1.45 1.4 10.00 3.73 1/7 0.99 1.2 11.14 3.82 1.45 1.4 10.00 3.00 1/10 0.99 1.2 11.14 3.03 1.45 1.4 10.00 2.39 Table 5.7. Same as Table 5.3; the black hole spin parameter a (second column) takes the values 0.50 MBH , 0.75 MBH and 0.99 MBH . Calculations are performed with the BPAL12 equation of state. 5.4 New Applications of the Quasi-Equilibrium Approach 87

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.50 1.2 14.34 11.09 4.23 1.4 14.13 9.59 1.6 13.73 8.33 1.8 13.03 7.14 1/3 0.50 1.2 14.34 8.55 4.23 1.4 14.13 7.40 1.6 13.73 6.45 1.8 13.03 5.55 1/5 0.50 1.2 14.34 6.19 4.23 1.4 14.13 5.39 1.6 13.73 4.72 1.8 13.03 4.10 1/7 0.50 1.2 14.34 5.04 4.23 1.4 14.13 4.41 1.6 13.73 3.90 1.8 13.03 3.43 1/10 0.50 1.2 14.34 4.09 4.23 1.4 14.13 3.62 1.6 13.73 3.25 1.8 13.03 2.94 1/2 0.75 1.2 14.34 11.04 3.16 1.4 14.13 9.53 1.6 13.73 8.26 1.8 13.03 7.06 1/3 0.75 1.2 14.34 8.48 3.16 1.4 14.13 7.33 1.6 13.73 6.37 1.8 13.03 5.45 1/5 0.75 1.2 14.34 6.11 3.16 1.4 14.13 5.29 1.6 13.73 4.61 1.8 13.03 3.97 1/7 0.75 1.2 14.34 4.93 3.16 1.4 14.13 4.29 1.6 13.73 3.75 1.8 13.03 3.25 1/10 0.75 1.2 14.34 3.96 3.16 1.4 14.13 3.45 1.6 13.73 3.04 1.8 13.03 2.67 Table 5.8. Same as Table 5.3; the black hole spin parameter a (second column) takes the values 0.50 MBH and 0.75 MBH . Calculations are performed with the GNH3 equation of state. 88 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

q aBH [MBH ] MNS [M ] RNS [km] rtide [MBH ] rISCO [MBH ] 1/2 0.99 1.2 14.34 10.99 1.45 1.4 14.13 9.48 1.6 13.73 8.21 1.8 13.03 7.00 1/3 0.99 1.2 14.34 8.43 1.45 1.4 14.13 7.27 1.6 13.73 6.30 1.8 13.03 5.37 1/5 0.99 1.2 14.34 6.03 1.45 1.4 14.13 5.21 1.6 13.73 4.52 1.8 13.03 3.86 1/7 0.99 1.2 14.34 4.85 1.45 1.4 14.13 4.19 1.6 13.73 3.63 1.8 13.03 3.10 1/10 0.99 1.2 14.34 3.85 1.45 1.4 14.13 3.32 1.6 13.73 2.89 1.8 13.03 2.47 Table 5.9. Same as Table 5.3; the black hole spin parameter a (second column) takes the value 0.99 MBH . Calculations are performed with the GNH3 equation of state.

The data given in Tables 5.3 to 5.9 is displayed in the four 3D graphs shown in Figure 5.5. A different EOS is considered in each graph, and the ratio rtide/rISCO is plotted as a function of the mass ratio q and the dimensionless black hole spin parameter a/MBH , while a colour scale is used to indicate the neutron star mass. In all graphs a green plane is shown at rtide/rISCO = 1, so that possible SGRB progenitors are locate above this critical plane, since rtide > rISCO must hold for them. By comparing the four graphs we see that

• the lower the NS mass is — for a ﬁxed mass ratio — the greater the ratio rtide/rISCO is; this means that neutron stars with smaller masses are disrupted further away from the ISCO, so that one expects a greater fraction of their mass to be involved in the formation of the disk surrounding the BH remnant which then gives rise to the SGRB.

Moreover the data and the graphs conﬁrm that binaries with

• a higher mass ratio q

• a higher black hole spin a and

• a less compact NS model are more likely to be SGRB progenitors since they have higher values of the ratio rtide/rISCO. We then determine the intersections of the rtide/rISCO = 1 planes of Figure 5.5 with the surfaces built upon our quasi-equilibrium calculations: for a given NS model and black hole crit spin a, this yields a critical black hole mass (MBH ) below which the BH-NS binary may be 5.4 New Applications of the Quasi-Equilibrium Approach 89

Figure 5.5. Ratio of the orbital separation at tidal disruption (rtide) and the radius of the innermost circular orbit (rISCO) as a function of the black hole spin parameter a (in units of black hole masses) and the binary mass ratio q = MNS/MBH . The colour of the surfaces is determined by the neutron star mass, with red (blue) denoting the heavier (lighter) end of the colour scale. The EOS used in each graph is also indicated. The green horizontal planes are located at rtide/rISCO = 1, so that points belonging to the data-generated surfaces and lying above these planes are possible SGRB progenitors since rtide > rISCO holds for them. a SGRB progenitor. The values of these critical BH masses are calculated by performing a numerical ﬁt of our rtide/rISCO data — for a given NS model and black hole spin — as a function of MBH . In Table 5.10 we provide the critical BH masses obtained this way that fall within the black hole mass intervals considered in our simulations. 90 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

crit EOS MNS [M ] RNS [km] C aBH [MBH ] MBH [M ] APR2 1.2 11.32 0.1566 0.50 8.39 0.75 − 0.99 − 1.4 11.29 0.1831 0.50 8.08 0.75 12.13 0.99 − 1.6 11.23 0.2104 0.50 7.77 0.75 11.69 0.99 − 1.8 11.10 0.2394 0.50 7.39 0.75 11.23 0.99 − 2.0 10.85 0.2723 0.50 6.86 0.75 10.66 0.99 − BGN1H1 1.2 12.97 0.1366 0.50 10.02 0.75 − 0.99 − 1.4 12.84 0.1610 0.50 9.54 0.75 − 0.99 − 1.6 10.32 0.2289 0.50 6.43 0.75 9.78 0.99 − BPAL12 1.2 11.14 0.1591 0.50 7.97 0.75 11.97 0.99 − 1.4 10.00 0.2067 0.50 6.58 0.75 9.91 0.99 − GNH3 1.2 14.34 0.1236 0.50 11.21 0.75 − 0.99 − 1.4 14.13 0.1464 0.50 10.61 0.75 − 0.99 − 1.6 13.73 0.1720 0.50 9.90 0.75 14.86 0.99 − 1.8 13.03 0.2040 0.50 8.89 0.75 13.37 0.99 − Table 5.10. Black hole critical masses for a given NS model (columns one to four) and black hole spin (ﬁfth column); binaries with BH masses below this critical value are possible SGRB progenitors. Dashes indicate that the critical mass exceeds the maximum black hole mass considered in our simulations for that particular NS model and BH spin parameter. 5.4 New Applications of the Quasi-Equilibrium Approach 91

By ﬁxing the black hole spin parameter a, the critical black hole masses we have calculated and reported in Table 5.10 may be used to distinguish possible SGRB progenitors in the MNS-MBH plane. This is done in the three graphs shown in Figure 5.6. The triangles crit connected by solid lines represent the critical black hole masses (MBH ) of Table 5.10 and the shaded areas below them therefore denote the fact that rtide > rISCO for those values of the binary constituent masses: BH-NS binaries which fall in the shaded areas are thus possible SGRB progenitors according to the mechanism described on page 30. Four different colours are used for the shaded regions in order to distinguish the EOS used to model the neutron star. It is evident that

crit • the softer the equation of state is, the lower MBH is. This is due to the fact that softer equations of state yield NSs which are more compact and can thus reach shorter distances from the BH before being disrupted (i.e. rtide decreases, as discussed at the end of Section 5.4.1), whereas the ISCO radius grows with the BH mass: given a NS mass, therefore, the combination of these two effects reduces the BH mass range available for the conditions rtide > rISCO to hold. Figure 5.6 shows that one could start discriminating between the different nuclear equations of state if we were able to observe a BH-NS coalescence leading to a GRB and to measure its physical parameters. Suppose, in fact, that one could determine precisely the two masses and the spins5 of a BH-NS coalescence by measuring the emitted gravitational radiation; moreover, suppose that we were to observe a GRB from such binary: if, for instance, the observations would gather in the green regions of the graphs, one could start pinning down GNH3 as the EOS of matter in neutron stars belonging to BH-NS binaries. If, instead, they were to lie in the purple regions, for example, we could start excluding GNH3 and probably BGN1H1 as equations of state. An accurate analysis would of course have to take into account all existing equation of state models. Clearly, the current technology does not allow us to perform the kind of observations just discussed; however, our results point in this direction and astronomical observations of this kind could become feasible in the future, so that

1. this would potentially allow us to infer the equation of state of neutron stars in mixed binaries and

2. this exciting prospect adds a further important reason to the need of accurately exploring the parameter space of BH-NS binaries as SGRB progenitors.

5We are setting the NS spin to zero in our model; by introducing the NS spin we would have an additional dimension in the parameter space, but our argument would still hold. 92 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

a=0.5 MBH 12 GNH3 11 BGN1H1 APR2 BPAL12 10

[Solar Masses] 8 BH

M 7

6 1.2 1.4 1.6 1.8 2 MNS [Solar Masses]

a=0.75 MBH 18 GNH3 BGN1H1 16 APR2 BPAL12

[Solar Masses] 12 BH M 10

1.2 1.4 1.6 1.8 2 MNS [Solar Masses]

a=0.99 MBH

GNH3 50 BGN1H1 APR2 BPAL12

40 [Solar Masses] BH M

30 1.2 1.4 1.6 1.8 2 MNS [Solar Masses]

Figure 5.6. The data reported in Table 5.10 is organized into three graphs according to the black hole spin parameter a. In each graph the shaded regions indicate neutron star mass (horizontal axis) and black hole mass (vertical axis) values for which a BH-NS binary is a possible short gamma-ray burst progenitor, i.e. rtide > rISCO holds in these regions. The colours of the shaded regions are related to the EOS used to model the neutron star, as explained by the legends. 5.4 New Applications of the Quasi-Equilibrium Approach 93

5.4.3 Measuring the NS Radius with GWs

As we have discussed, depending on whether rtide is greater than rISCO or not, the evolution of a mixed binary may end with the tidal disruption of the neutron star or with a plunge. We have extensively explored the connections of short gamma-ray bursts to the ﬁrst case; there is also a second intriguing consequence linked to the tidal disruption scenario: the disruption of a neutron star by a black hole in a mixed binary is, in fact, expected to have a repercussion on the emitted gravitational waveform. The amplitude of gravitational waves quickly decreases after the onset of the tidal disruption and the spectrum amplitude consequently sharply falls above a cut-off frequency which is determined by the tidal disruption process [139]. As we have seen, the onset of the tidal disruption depends on the neutron star compactness and hence on the equation of state of the neutron star ﬂuid: therefore the cut-off frequency in the emitted gravitational radiation will carry information about the equation of state. If one measures the neutron star mass by analysing the the gravitational waves emitted during the inspiral phase and is able to measure the cut-off frequency yielded by the tidal disruption of the star, one is thus also able to infer the neutron star radius and to constraint the equation of state. In the following we will assume that the cut-off frequency coincides with the gravitational wave frequency at the moment of tidal disruption, i.e. when the orbital separation is equal to rtide. If we were to drop the quasi-equilibrium assumption, the picture would partially change since the two gravitational wave frequencies, νtide and νcut-off would not exactly coincide; however, the conclusions of the analysis we perform on the measurability of the neutron star radius would be left unchanged since νcut-off(& νtide) would still be equation of state dependent [139]. We begin by illustrating the dependence of rtide on the neutron star equation of state and radius. As an example, in Figure 5.7 we plot — from Tables 5.3 to 5.9 — the neutron star radius versus rtide/MBH in the a = 0.99 MBH case for all four equations of state considered so far. It is evident that rtide depends on the NS equation of state and radius. 94 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

q=1/2 a=0.99 MBH q=1/3 a=0.99 MBH

14 14 [km] [km]

NS 12 NS 12 R R APR2 APR2 BGN1H1 BGN1H1 BPAL12 BPAL12 GNH3 GNH3 10 10 5 6 7 8 9 10 11 12 4 5 6 7 8 9 rtide/MBH rtide/MBH

q=1/5 a=0.99 MBH

14 [km]

NS 12 R APR2 BGN1H1 BPAL12 GNH3 10 3 4 5 6 7 rtide/MBH

q=1/7 a=0.99 MBH q=1/10 a=0.99 MBH

14 14 [km] [km]

NS 12 NS 12 R R APR2 APR2 BGN1H1 BGN1H1 BPAL12 BPAL12 GNH3 GNH3 10 10 2 3 4 5 2 3 4 rtide/MBH rtide/MBH

Figure 5.7. The neutron star radius as a function of the binary orbital separation at the tidal disruption (rtide) in the a = 0.99 MBH case. Different points on the graph denote different neutron star masses, whereas the colours indicate the equations of state used. 5.4 New Applications of the Quasi-Equilibrium Approach 95

We now carry out a simple study which shows which binaries would allow the determination of the neutron star radius by measuring the cut-off frequency in the emitted gravitational waveform. Once again, in order for this to happen the condition rtide > rISCO must hold. In Figures 5.8 and 5.9 we plot the data reported in Tables 5.3 to 5.9 by ﬁxing the neutron star mass in each graph and showing the frequency of the gravitational wave at the tidal disruption (νtide) as a function of the black hole mass. The curves are displayed as crit continuous lines if rtide > rISCO, i.e. up until the critical black hole masses MBH given in 6 Table 5.10, and as dashed lines otherwise. νtide is calculated according to the formula q 3 −1 νtide = [π(a + rtide/MBH )] . (5.24)

The graphs therefore show the regions of the parameter space in which the gravitational wave signal is expected to be characterised by a cut-off frequency; we thus see that the measurement of neutron star radii is favoured by

• low NS masses

• low BH masses and

• high black hole spins.

Moreover, the frequency of the gravitational wave emitted when the NS tidal disruption takes place

• spans a wider interval if the NS radius is smaller, i.e. if the star is more compact, and

• is more sensitive to variations of the black hole spin if the NS mass is larger.

It is now evident that there are BH-NS binaries that may yield precious information on the NS equation of state and radius by means of measurements of the gravitational wave cut-off frequency, which — through rtide of course — is an equation of state dependent quantity.

6 We use this formula here for illustrative purposes; we will soon determine νtide in a more accurate way. 96 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

BGN1H1 M = 1.2 M APR2 MNS=1.2 M NS 1000 850

950 800 D D 900 Hz Hz @ @ 750 tide tide 850 Ν Ν a = 0.99 M 700 a = 0.99 M 800 a = 0.75 M a = 0.75 M a = 0.50 M a = 0.50 M 750 650 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16

MBH Solar Masses MBH Solar Masses GNH3 MNS=1.2 M BPAL12 MNS=1.2 M 1050 740 @ D @ D 1000 720

D 700 D 950 Hz Hz

@ 680 @ 900 tide

tide 660 Ν

Ν a = 0.99 M 850 a = 0.75 M 640 a = 0.99 M a = 0.50 M a = 0.75 M 800 620 a = 0.50 M 750 600 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16

MBH Solar Masses MBH Solar Masses BGN1H1 M =1.4 M APR2 MNS=1.4 M NS 1100 900 @ D @ D 850 1000 D D 800 Hz Hz @ @ 900 tide

tide 750 Ν Ν a = 0.99 M a = 0.99 M 800 a = 0.75 M 700 a = 0.75 M a = 0.50 M a = 0.50 M 700 650 5 10 15 5 10 15

MBH Solar Masses MBH Solar Masses GNH3 M =1.4 M BPAL12 MNS=1.4 M NS 1300 800 @ D @ D 1200 750 D D 1100 Hz Hz @ @ 700 tide tide 1000

a = 0.99 M Ν Ν a = 0.99 M a = 0.75 M 650 a = 0.75 M 900 a = 0.50 M a = 0.50 M 800 600 5 10 15 5 10 15

MBH Solar Masses MBH Solar Masses

Figure 5.8. The gravitational@ wave frequencyD at the onset of tidal disruption@ — calculatedD according to Eq. (5.24) — as a function of the black hole mass, for 1.2 M (four upper panels) and 1.4 M (four lower panels) neutron stars. The continuous curves denote that rtide > rISCO, i.e. that a short gamma-ray burst is possible, whereas dashes are used when this condition is not veriﬁed. The neutron star equation of state is indicated for each panel. 5.4 New Applications of the Quasi-Equilibrium Approach 97

APR2 MNS=1.6 M BGN1H1 MNS=1.6 M 1100 1300 1200 1000

D D 1100 Hz Hz @ 900 @ 1000 a = 0.99 M tide tide

Ν a = 0.99 M Ν 900 a = 0.75 M a = 0.75 M 800 800 a = 0.50 M a = 0.50 M 700 700 5 10 15 20 5 10 15 20

MBH Solar Masses MBH Solar Masses GNH3 MNS=1.6 M 850 @ D @ D 800

D 750 Hz @

tide 700 Ν a = 0.99 M a = 0.75 M 650 a = 0.50 M 600 5 10 15 20

MBH Solar Masses GNH3 M =1.8 M APR2 MNS=1.8 M NS 1100 950 @ 900 D 1000 850 D D

Hz 800 Hz

900 @ @ a = 0.99 M 750 tide a = 0.99 M tide

Ν Ν 800 a = 0.75 M a = 0.75 M 700 a = 0.50 M 650 a = 0.50 M 700 600 5 10 15 20 25 5 10 15 20 25

MBH Solar Masses MBH Solar Masses APR2 MNS=2.0 M 1200 @ 1100 D @ D

D 1000 Hz @ 900 a = 0.99 M tide Ν 800 a = 0.75 M 700 a = 0.50 M 600 5 10 15 20 25

MBH Solar Masses

Figure 5.9. Same as Figure 5.8 except for the value@ of the neutronD star mass: the three upper panels refer to 1.6 M neutron stars, the two middle ones refer to 1.8 M neutron stars, while the lower one to 2.0 M neutron stars. 98 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

In the rest of this section we show how the detection of the gravitational radiation emitted by a coalescing BH-NS binary may allow one to infer the radius of the neutron star in the binary from measurements of the cut-off frequency (νtide) — which is present in the gravitational waveform if the star is tidally disrupted by the black hole. We begin by considering a result of [190] displayed in Figure 5.10: the four panels report the one-sigma fractional errors in the chirpmass and in the symmetric mass ratio (η) for non-spinning binary black hole sources. These errors reduce when the dynamical evolution of spins is included, so that they may be regarded as upper limits. Since a similar analysis has not been performed in the BH-NS case, in what follows we shall adopt these errors for mixed binaries. Notice that extending the BH-BH binary errors to BH-NS binaries may be optimistic since in the latter case the presence of matter can complicate the gravitational wave signal.

Figure 5.10. One-sigma fractional errors in the chirpmass (upper row) and symmetric mass ratio (lower row) for sources with a fixed signal to noise ratio equal to 10 (left panels) and at a fixed distance of 300 Mpc (right panels). These plots are for non-spinning black hole binaries; the errors reduce greatly when dynamical evolution of spins is included. (The Figure is extracted from [5] and is a slightly modified version of a figure from [190].)

In order to extract information about the one-sigma errors from Figure 5.10, we anticipate that in this section we shall take mixed binaries with mass ratio 0.2 and neutron star mass equal to 1.2 M or 1.4 M . We shall therefore consider total masses of roughly 10 M ; by considering the black continuous curves for Advanced LIGO, we thus ﬁnd the fractional errors for the chirp mass M and the symmetric mass ratio η to be given by

∆M 0.001 (5.25) M . ∆η 0.02 . (5.26) η .

Since the mass parameters of the binary in our setup are the neutron star mass and the 5.4 New Applications of the Quasi-Equilibrium Approach 99 mass ratio, we express M and η as q η = (5.27) (1 + q)2 M M = NS . (5.28) q2/5(1 + q)1/5

By propagating the errors according to the well known formula

∂f ∆f({xi}) = ∆xi , (5.29) ∂xi we thus have the following expressions for the one-sigma fractional errors in the neutron star mass and in the mass ratio: ∆q 1 + q ∆η = (5.30) q 1 − q η ∆M ∆M 3q + 2 ∆q NS = + . (5.31) MNS M 5(1 + q) q We now choose a typical case to examine, that is, as already anticipated, a mixed binary with a neutron star of mass 1.4 M and mass ratio q = 0.2. We use the fractional errors for the chirpmass M and the symmetric mass ratio η given in Eqs. (5.25) and (5.26) — along with the values q = 0.2 and MNS = 1.4 M that characterise the binary — in the formulas (5.30) and (5.31). This way we ﬁnd that if the gravitational radiation emitted by such a system is detected, its mass ratio and the mass of the neutron star in it will be determined with accuracies

∆q ' 0.006 (5.32) and

∆MNS ' 0.02 M (5.33) respectively. We then combine each of the four equations of state (APR2, BGN1H1, BPAL12 and GNH3), with the two values of the neutron star mass MNS = {1.38, 1.42} M , the two values of the mass ratio q = {0.194, 0.206} and the two values of the black hole spin a = {0.5, 0.99} MBH and determine rtide for each combination of these four variables. The choice a/MBH ≥ 0.5 is due to the fact that that it is more likely to achieve tidal disruption with these values of the black hole spin. Once the tidal disruption orbital radii are known, we consider the post-Newtonian inspiral of two point masses up to order 2PN with 2.5PN dissipative terms and determine the gravitational wave frequency νtide at rtide: if this cut-off frequency is measured it can give us precious information about the neutron star radius and thus about its equation of state. The Hamiltonian needed to perform this calculation is built upon Eqs. (6.7), (6.8) and (6.9) and is discussed in the next chapter, whereas the gravitational radiation dissipative terms are given in Eqs. (6.23)-(6.26); explicit expressions for the equations of motion are provided in AppendixC. Within this Hamiltonian approach, νtide is given by

Pφ νtide = 2 , (5.34) πµrtide 100 5. BH-NS Coalescing Binaries: Quasi-Equilibrium Approach

where Pφ is the conjugate momentum to the angle φ of the orbit and µ = MNSMBH /(MNS+ MBH ) is the reduced mass of the binary. In order to determine νtide we therefore evolve the equations of motion of AppendixC for a binary with point-mass constituents from large orbital separations until r = rtide. In the top panel in Figure 5.11 we display our results in a graph with the neutron star radius as a function of the gravitational wave frequency. The different coloured symbols denote the equation of state used and the two points for each equation of state are obtained by taking into account the one-sigma uncertainties on the mass ratio, Eq. (5.30), and on the neutron star mass, Eq. (5.31), that are yielded by gravitational wave measurements: given an EOS, the left-most point is obtained by setting MNS = 1.42 M , a = 0.5 MBH and q = 0.194, whereas the right-most one is obtained with the parameters MNS = 1.38 M , a = 0.99 MBH and q = 0.206. The graph tells us that • if the NS is disrupted by tidal forces and one is able to determine the cut-off frequency that the disruption produces in the emitted gravitational waveform, the radius of the NS can be measured with and accuracy of roughly one kilometre.

BH-NS binaries whose evolution ends with a violent disruption of the neutron star may therefore be very precious in order to constrain the equation of state of neutron stars through gravitational wave observations. In the bottom panel in Figure 5.11 we repeat the analysis for a neutron star mass of 1.2 M . From a comparison with the top panel, we see the data relative to APR2 moves closer to the data of BPAL12: this is due to the fact that, as shown in Figure 5.4, for MNS ' 1.2 M the two equations of state almost yield the same radius. 5.4 New Applications of the Quasi-Equilibrium Approach 101

14 APR2 BGN1H1 BPAL12 GNH3

[km] 12 NS R

10 1000 1200 1400 ν tide [Hz]

APR2 14 BGN1H1 BPAL12 GNH3

13 [km] NS R 12

1000 1200 ν tide [Hz]

Figure 5.11. The neutron star radius (RNS) as a function of the gravitational wave cut-off frequency at tidal disruption (νtide) for a BH-NS binary with mass ratio 0.2, black hole spin greater than 0.5 MBH and neutron star mass 1.4 M (in top panel) or 1.2 M (bottom panel). The symbols denote the equation of state used and the two points for each equation of state are obtained by taking into account the one-sigma uncertainties on the mass ratio, Eq. (5.30), and on the neutron star mass, Eq. (5.31), that are yielded by gravitational wave measurements (see Figure 5.10).

Chapter 6

BH-NS Coalescing Binaries: Dynamical Approach

So far we have been considering the quasi-equilibrium approximation of the afﬁne model and we have seen that

• the use we introduced of the pseudo-relativistic scalar potential ΦTOV (Section 4.2) led to a signiﬁcant improvement of the afﬁne model (Section 5.3) and

• we are able to explore for the ﬁrst time the EOS × black hole spin × mass ratio × neutron star mass parameter space determining whether or not a mixed binary satisﬁes the necessary condition rtide > rISCO (page 61) for it to lead to a short gamma-ray burst progenitor.

Given these two attractive features of the quasi-equilibrium case, it is desirable to tackle the dynamic problem, which also comprehends orbital motion. Dropping the quasi-equilibrium approximation and considering orbital dynamics too may allow us to see how the tidal disruption changes and to work on gravitational wave (GW) emission. The tidal disruption radius rtide is expected to change since the quasi-equilibrium requirement given in Eq. (5.5) no longer holds: this allows the stellar ellipsoid to rotate since the rotation angle ϕ is a true degree of freedom in the dynamical case and is not “frozen” by Eq. (5.5). As far as GWs are concerned, instead, one will be looking at finite size effects on the emission: for compact binaries, the main consequence of finite size effects — i.e. the departure from the point-particle waveform — is expected to be a phase difference which mainly accumulates during the last few orbits before merging or disruption, e.g. [131], whereas repercussions on the amplitude should be negligible. In Section 3.3 we have stated that our strategy for the dynamic problem is to try to balance a good treatment of the orbit and a good treatment of the tidal interaction. The compromise we consider consists in using a post-Newtonian expanded Hamiltonian for the orbit, as Ogawaguchi and Kojima did in [178] with the Newtonian affine model and tidal field: this means that the neutron star centre of mass will not actually be following a Kerr spacetime geodesic. Total rigour is thus sacrificed in favour of a description of the orbit which is more realistic than a Kerr geodesic and in favour of the possibility to have an interplay between the orbit itself and the ellipsoid dynamics. Moreover, since this is a first step, we shall not distinguish between coordinate time and proper time: it must be noted, though, that it is possible to drop this approximation with further developments of

103 104 6. BH-NS Coalescing Binaries: Dynamical Approach our model. The approach we will build is intended to be an analytic tool for exploring the EOS × black hole spin × mass ratio × neutron star mass parameter space in the dynamic case, while fully relativistic numerical simulations are on their way: the main goal is to be able to get hold of how moving in this parameter space reﬂects on the gravitational wave emission through ﬁnite size effects.

6.1 Formulation

We consider the Hamiltonian approach preferable to set up our dynamic model, since in this case the orbital angular momentum expression is given by the “Newtonian-looking” formula L~ = X~ × P~ at any PN order, as opposed to what happens in the Lagrangian formalism [191]. Furthermore, it is straightforward to switch from the Lagrangian formulation of the afﬁne model to a Hamiltonian one, so this will not cause any difﬁculty. The Hamiltonian takes the form

Exp H = Horb + HT + HI , (6.1) where:

Exp • Horb indicates the Hamiltonian for the conservative part of the orbit dynamics which is expanded in post-Newtonian (PN) fashion; it describes the orbital motion of a binary system in terms of a point mass µ = M1M2/(M1 + M2) orbiting a central point mass MTot = M1 + M2; the inspiral caused by gravitational wave emission is obtained by adding by hand analytic expressions of the “GW-induced external forces” to the Hamilton equations of motion;

• HT is the term coupling the stellar ellipsoid to the black hole tidal ﬁeld (which follows from the Lagrangian given in Eq. (3.87) and makes use of the pseudo-TOV potential ΦTOV) and

• HI is the Hamiltonian for the star interior (which follows from the Lagrangian given in Eq. (3.83)).

The r-dependence of the tidal tensor in the term coupling the stellar ellipsoid to the black hole tidal ﬁeld, i.e.

HT = HT(r, {ai}) , (6.2) works as an interaction term between the orbit and the ellipsoid dynamics and allows for the orbit-ellipsoid interplay to take place. Once the dynamics is modelled, by assuming that the spins are aligned with the orbital angular momentum, the following formulas may be used to calculate the two polarizations of the emitted GW-form (e.g. [178]):

G d2 h = N (1 + cos2 Θ) {µr2 cos(2φ) + (I − I ) cos[2(φ + ϕ )]} (6.3) + 2c4D dt2 11 22 l G d2 h = cos Θ {µr2 sin(2φ) + (I − I ) sin[2(φ + ϕ )]} (6.4) × c4D dt2 11 22 l 6.1 Formulation 105 where (D, Θ) are the observer’s distance and angle with respect to the normal to the orbital plane, φ is the anomaly of the orbit, Iij is the tensor of inertia of the neutron star expressed in the principal axis frame — see Eq. (3.93) — and

def ϕl = Ψ − ϕ (6.5) is the lag angle. This angle expresses the misalignment of the axis a1 of the neutron star (the elongated axis in the orbit plane) and the orbit separation direction [170]: it is present since the tidal ﬁeld spins up the stellar ellipsoid (Section 3.1.1) and was vanishing in Chapter 5 because of the quasi-equilibrium requirement of Eq. (5.5). Its deﬁnition is clear if one considers that Ψ guarantees the parallel propagation of the orthonormal tetrad tied to the neutron star centre of mass with a rotation that “compensates” the evolution of the anomaly of the orbit φ along the orbit (the two angles are identical in the Newtonian limit), while ϕ brings the tetrad in the principal axis frame (see Eq. (3.69)).

6.1.1 The Orbit Hamiltonian The conservative part of the two-body Hamiltonian is known up to order 3PN, e.g. [191]; however, we shall truncate it at 2PN order since as a ﬁrst step we will use GW dissipation terms up to 2.5PN order [192]. The motion of the centre of mass will not be considered. We choose the orbital plane as the equatorial plane of the polar coordinates (r, θ, φ). Neglecting tidal interactions for now, the conservative relative motion of two masses M1 and M2 (which in our case will be MBH and MNS) is determined by the Hamiltonian Exp Horb = HN + HPN + H2PN + HSO + HSS , (6.6) the single contributions being: ! 1 P 2 G µM H = P 2 + φ − N Tot (6.7) N 2µ r r2 r !2 ( ! ) 3ν − 1 P 2 G M P 2 G2 µM 2 H = P 2 + φ − N Tot (3 + ν) P 2 + φ + νP 2 + N Tot PN 8c2µ3 r r2 2c2µr r r2 r 2c2r2 (6.8) !3 1 − 5ν + 5ν2 P 2 H = P 2 + φ 2PN 16c4µ5 r r2

 2 !2 2 !  GNMTot 2 2 Pφ 2 2 2 Pφ 2 4 + (5 − 20ν − 3ν ) P + − 2ν P P + − 3ν P  8c4µ3r r r2 r r r2 r " ! # G2 M 2 P 2 G3 (1 + 3ν)µM 3 + N Tot (5 + 8ν) P 2 + φ + 3νP 2) − N Tot (6.9) 2c4µr2 r r2 r 4c4r3 GN 3M2 3M1 HSO = 2 3 L · 2 + J1 + 2 + J2 (6.10) c r 2M1 2M2

HSS = HS1S2 + (HS1S1 + HS2S2 ) GN GN M2 = 2 3 [3(J1 · n)(J2 · n) − (J1 · J2)] + 2 3 [3(J1 · n)(J1 · n) − (J1 · J1)] c r c r M1 GN M1 + 2 3 [3(J2 · n)(J2 · n) − (J2 · J2)] (6.11) c r M2 106 6. BH-NS Coalescing Binaries: Dynamical Approach

where ν = µ/MTot is the symmetric mass ratio, Ja is the spin angular momentum of the a-th body, L is the orbital angular momentum — which in the Hamiltonian formalism is L = r × P — and n = r/r. In our case the the (relative) momentum P lies in the orbital plane and is orthogonal to r, while the spin angular momenta are perpendicular to the orbital plane, so that the 1.5PN spin-orbit (SO) and the 2PN spin-spin (SS) interaction terms reduce to s P 2 GN 2 φ 3M2 3M1 HSO = 2 2 Pr + 2 2 + J1 + 2 + J2 (6.12) c r r 2M1 2M2 and 2GN M2 M1 HSS = − 2 3 J1J2 + J1J1 + J2J2 . (6.13) c r M1 M2 Notice that with the assumption we made about the spin and orbit angular momenta, θ and its conjugate momenta Pθ do not appear in the Hamiltonian. In order to keep treating the BH as a non-dynamic object which is not influenced or modified by the presence of the NS, we consider the black hole spin angular momentum appearing in the Hamiltonian as a fix parameter and not as a degree of freedom to evolve.

6.1.2 The Hamiltonian for the NS Fluid We now need to write the total Hamiltonian for the NS Fluid, which in the afﬁne model is given by a tidal interaction term and an internal Hamiltonian. These follow of course from the Lagrangians in Eqs. (3.83) and (3.87) and are obtainable from the Legendre transform X H = PiQ˙ i − L . (6.14) i

For the neutron star fluid, we have five dynamic variables Qi = {a1, a2, a3, λ, ϕ}; the tidal ˙ Lagrangian does not contribute to their conjugate momenta, i.e. Pi = ∂LT/∂Qi = 0, and thus 1 H = −L = V = c I , (6.15) T T T 2 ij ij where the tidal tensor components cij and the tensor of inertia components Iij in the principal ˙ frame are given by Eqs. (3.88)-(3.91) and (3.93) respectively. From Pi = ∂LI/∂Qi one finds that the internal Lagrangian contributions to the conjugate momenta are, instead,

a˙ 1 Pa1 = Mc (6.16) RNS a˙ 2 Pa2 = Mc (6.17) RNS a˙ 3 Pa3 = Mc (6.18) RNS

Mc 2 2 Pλ = 2 [(a1 + a2)Ω − 2a1a2Λ] ≡ Js (6.19) RNS

Mc 2 2 Pϕ = [(a1 + a2)Λ − 2a1a2Ω] ≡ C (6.20) RNS 6.1 Formulation 107 where Ω =ϕ ˙, Λ = λ˙ , and therefore

HI = (PλΛ + PϕΩ − Ts) + Te + U + V (6.21)

where the expansion/contraction energy Te, the spin kinetic energy Ts, the internal energy U and the self-gravitational energy V are discussed in Section 3.2.3. The neutron star ﬂuid is hence governed by the Hamiltonian

HI + HT = (PλΛ + PϕΩ − Ts) + Te + U + V + VT. (6.22)

We remind the reader that Ω is the angular velocity of the ellipsoidal ﬁgure measured in the parallel transporting frame and Λ is connected to the internal ﬂuid motion by the relation 2 2 Λ = −a1a2ζ/(a1 + a2), where ζ is the uniform vorticity along the z-axis measured in principal (i.e. rotating) frame.

6.1.3 GW Dissipation

The effects of the loss of energy by gravitational wave emission may be included in the orbit dynamics by adding the four dissipative terms of order 2.5PN [192]

! 8G2 6P 2 f = − N 2P 2 + φ (6.23) r 15c5νr2 r r2 8G2 P P f = − N r φ (6.24) φ 3c5 νr4 2 2 ! 8G P P G νM 3 F = − N r φ − N Tot (6.25) r 3c5 r4 νr 5 ! 8G2 P 2G ν2M 3 2P 2 F = − N φ N Tot + φ − P 2 (6.26) φ 5c5 νr3 r r2 r

to the Hamilton equations for the variables r, φ, Pr,Pφ. The Hamilton equations of motion for the orbit are therefore modiﬁed as follows:

dr ∂H = + fr (6.27) dt ∂Pr dφ ∂H = + fφ (6.28) dt ∂Pφ dP ∂H r = − + F (6.29) dt ∂r r dP ∂H φ = − + F . (6.30) dt ∂φ φ

We will not include higher order spin-dependent dissipative terms in the equations. 108 6. BH-NS Coalescing Binaries: Dynamical Approach

6.1.4 Equations of Motion Once we have built the Hamiltonian of our model, the equations to be solved are

dr ∂H = + fr (6.31) dt ∂Pr dφ ∂H = + fφ (6.32) dt ∂Pφ dP ∂H r = − + F (6.33) dt ∂r r dP ∂H φ = − + F (6.34) dt ∂φ φ da ∂H i = (6.35) dt ∂Pai dP ∂H ai = − (6.36) dt ∂ai s dΨ M = BH (6.37) dt r3 dP ∂H λ ≡ C˙ = − = 0 (6.38) dt ∂λ dϕ ∂H = (6.39) dt ∂Pϕ dP ∂H ϕ ≡ J˙ = − (6.40) dt s ∂ϕ where the total Hamiltonian H is the sum of the contributions given in Eqs. (6.6) and (6.22), the dissipative terms fr, fφ,Fr and Fφ are given by Eqs. (6.23)-(6.26) and the index i runs from 1 to 3. The ﬁnite size effects due to the tidal interaction therefore come from the derivative ∂HT/∂r in the equation for Pr. Explicit expressions for the equations of motion are provided in AppendixC.

6.1.5 Initial Conditions To determine the initial conditions, the first thing one needs to fix is the initial separation between the black hole and the neutron star rstart. The NS fluid initial conditions are then determined by solving the quasi-equilibrium problem at rstart. One must then provide φ(0),Pr(0) and Pφ(0), and we proceed as follows in order to avoid residual eccentricity in the orbit since we are interested in circular inspiralling orbits. φ(0) does not appear in the Hamiltonian so that it can be specified freely and we set it to zero. For Pr(0) we begin by taking

2 2 2 2 64 µ MTot 64 MBH MNS Pr(0) = − 3 ≡ − 3 , (6.41) 5 rstart 5 rstart which follows if one calculates the GW luminosity of two point masses on a circular Newtonian orbit [174], while for Pφ(0) we use the Newtonian formula p Pφ(0) = µ MTotrstart . (6.42) 6.2 Results 109

These initial conditions do not obviously yield a circular orbit with the Hamiltonian we are using; however we may use them at, say, αrstart, with α 1, and integrate the orbit equations alone until we reach r = rstart: this gives the system the possibility to radiate away in gravitational waves the eccentricity which is intrinsic in the conditions speciﬁed by Eqs. (6.41)-(6.42). The values of Pr and Pφ at rstart) then complete the set of initial conditions needed to integrate of the system in Eqs. (6.31)-(6.40). A suitable choice for the value of α is 100. Summarising, to obtain the initial conditions for the dynamic problem, one

• chooses rstart,

• solves the quasi-equilibrium problem at r = rstart in order to obtain the initial conditions for the NS ﬂuid,

• sets φ = 0 and uses Eqs. (6.41)-(6.42) to integrate the orbit from αrstart — with α 1 — to r = rstart in order to radiate away the orbit eccentricity and complete the set of initial conditions. One must of course make sure that the initial conditions obtained are stable if α is increased and that the complete dynamical evolution beyond r = rstart does not change if one takes 0 rstart > rstart to produce the initial conditions.

6.2 Results

In this section we consider binaries in which the neutron star has a mass of 1.4 M and is characterised by the BGN1H1 equation of state. We consider four different mass ratio-black hole spin combinations ({0.1, 0.99 MBH }, {0.2, 0.50 MBH }, {0.2, 0.99 MBH } and {1/3, 0.99 MBH }). All our simulations begin at rstart = 10 MTot and end at rtide with the tidal disruption of the neutron star. Each system is evolved in two ways: • with a ﬁnite-size neutron star described by the ellipsoidal model and

• in the approximation of point-mass constituents. In the first case, as already mentioned, the simulation ends when the black hole and the neutron star touch: this yields an orbital separation rend which we then use to stop the point- mass evolution when r = rend. In both cases, the gravitational wave emission is calculated according to formulas (6.3) and (6.4). By comparing the gravitational wave calculated with a finite size neutron star and a point-mass neutron star, we may evaluate the effects of the finite size of the NS on the gravitational wave. The difference in the gravitational radiation will follow from the fact that in the case of a point-mass neutron star

1. the terms proportional to I11 − I22 in Eqs. (6.3) and (6.4) do not contribute and Exp 2. the Hamiltonian given in Eq. (6.1) reduces to Horb since the terms HT and HI both vanish. The results are displayed in Figure 6.1, where the “plus” polarization of the emitted gravitational wave is plotted as a function of time. Red curves refer to the signal produced by the point-mass model, whereas the green curves are given by the ellipsoidal model. By comparing the four panels of the ﬁgure, we see that 110 6. BH-NS Coalescing Binaries: Dynamical Approach

• the phase difference between the waveforms obtained with the point-mass model and the ellipsoidal model grows as the mass ratio grows, i.e. as we move towards the equal mass case;

• also the amplitude difference grows with the mass ratio;

• the phase and amplitude differences are greater if the black hole spin is larger.

In all four cases we notice that the differences in the phase and the amplitude between the waveforms calculated with the two models develop in the last few cycles. In this region we are pushing the ellipsoidal model probably too far. Drawing quantitative conclusions is therefore risky; however, the qualitative behaviours we observed are reliable and they will deserve further investigation with this or other approaches. In particular it will be interesting to determine whether or not second or third generation interferometric detectors and the known data analysis techniques will be sensitive to these effects and the differences in the gravitational waveform. A study of this kind has already been performed for binary neutron

q=0.1 a=0.99 MBH MNS=1.4MSun q=0.2 a=0.99 MBH MNS=1.4MSun

0.5 0.5 D D

+ 0 + 0 h h

-0.5 -0.5

0.16 0.17 0.04 0.05 t [s] t [s]

q=1/3 a=0.99 MBH MNS=1.4MSun q=0.2 a=0.50 MBH MNS=1.4MSun

0.5 0.5 D D

+ 0 + 0 h h

-0.5 -0.5

0.01 0.02 0.02 0.03 t [s] t [s]

Figure 6.1. h+D as a function of time for several BH-NS binaries, whose parameters are indicated above each panel. The the “plus” polarization of the gravitational wave is calculated according to Eq. (6.3) and is multiplied by the distance D from the binary source to the observer in order to get a dimensionless universal quantity. The green curves are given by the dynamic ellipsoidal model, whereas the red curves are given by a point-mass model. In each panel the origin of the time axis refers to instant in which the orbital separation is equal to 10 MTot; we plot a time interval of 0.015 s so that the phase differences in different panels may be compared directly. 6.3 Future Developments 111 stars [131]. Cases with high black hole spin and big mass ratios look particularly interesting in this respect.

6.3 Future Developments

An application of the improved affine model is currently under investigation with the collaboration of the Numerical Relativity group at the Albert Einstein Institute. The model, in the form we have used it, allows one to determine if a massive disk may form or not: the rtide > rISCO condition is a necessary but not sufficient one. The question is whether it can be used somehow to approximately determine if the disk does form, and possibly how massive it is. The idea is to use the improved affine model until the tidal disruption of the neutron star and then to simply evolve the outer layer of the star as a set of free-falling particles in a Kerr field. Intersections among the particle trajectories occurring outside the ISCO may in fact denote the formation of a disk, while the fraction of particles left outside the ISCO when an intersection takes place should allow an estimate of the mass of the disk. A second line of investigation naturally arises from the need of including in the model an initial spin for the neutron star. At large orbital radii, the figure rotation Ω vanishes; in the affine model, therefore, when the star has an intrinsic spin (Λs) one has (see Eq. (3.113))

Mc 2 2 C = 2 [(a1 + a2)Λs] (6.43) RNS and this finite and non-vanishing circulation is used to deal with the initial spin. With the introduction of the self-gravity term VbTOV, more care may be needed and it may be necessary to correct somehow the pseudo-relativistic potential ΦTOV and, consequently, VbTOV, when a rotating star is considered. If this is the case, a possible solution is suggested by the fact that, in order to correct the multi-dimensional Newtonian gravitational potential Z ρ Φ(~r) = − dr0dθ0dφ0r0 2 sin θ0 , (6.44) |~r − ~r0| the authors of [182] consider the definition of ΦTOV (Eq. (4.6)) calculated with the hydrodynamic quantities replaced by their corresponding angularly averaged values. Assessing whether this is needed and works in the case of the affine model will require further testing with relativistic quasi-equilibrium calculations. Ultimately, the Rome group has in program to repeat the analyses of Figure 5.11 with several other equations of state and with an accurate use of the one-sigma errors for the chirp mass and the symmetric mass ratio (see Eqs. (5.25) and (5.26)), in order to actually follow the data of Figure 5.10 case by case.

Part II

Towards QNMs of Rapidly Rotating Neutron Stars

113

Chapter 7

Perturbation Theory, Quasi-Normal Modes and Instabilities

In this part of the thesis we will address the problem of studying the oscillations of a rapidly rotating neutron star within a linear perturbative approach. Oscillations of stars have been extensively studied both in the framework of Newtonian gravity and in General Relativity. Stellar oscillations for most stars, the Sun first of all, can be described with a very high degree of accuracy within Newtonian gravity. Neutron stars, however, are very compact objects and have to be studied in the framework of the General Relativity since in their case relativistic effects can not be neglected. This first introductory chapter is thus devoted to stellar perturbation theory in General Relativity, a field which was launched by Thorne and collaborators in the Sixties1 [193, 194, 195, 196, 197, 198, 199] and further developed by Chandrasekhar and Ferrari [200, 201, 202, 203]. The proper modes at which a star, or a black hole, oscillates when excited by a non radial perturbation are known as quasi-normal modes (QNMs) in General Relativity. As opposed to the normal modes of Newtonian gravity, they are damped by the emission of gravitational waves (hence the “quasi” in their name) and their corresponding eigenfrequencies are therefore no longer real, but complex. QNMs were first pointed out by Vishveshwara [204] in calculations of the scattering of gravitational waves by a Schwarzschild black hole, while the term “quasi-normal frequencies” was coined by Press in [205]. It is intuitive to understand that a star may oscillate: after all it is a sphere of fluid. On the other hand the idea that also black holes — which are vacuum solutions of the Einstein equations — possess proper modes is less straightforward and it did raise considerable surprise when it was first proposed: the physical explanation resides in the fact that spacetime is a dynamical entity in GR. In order to detect the gravitational signals emitted by NSs it is important to know their pulsation frequencies. In the case of rapidly rotating neutron stars — which, as discussed on page3, have been observed — the determination of the QNM spectrum is still an open problem. This is due to the fact that the perturbative approach, which works so well in the non-rotating case, shows a high degree of complexity when generalised to include rotation, even just slow rotation, so that simplifying assumptions are introduced (see Section 7.3.2). An alternative approach consists in solving the equations describing a rotating and oscillating

1In the concluding remarks of [193], Thorne and Campollataro described their pioneering paper as “just a modest introduction to a story which promises to be long, complicated and fascinating”.

115 116 7. Perturbation Theory, Quasi-Normal Modes and Instabilities star in full general relativity, in the time domain. However, also in this approach strong simplifying assumptions are currently made, or a restriction to a particular case is considered (again, see Section 7.3.2). An innovative perturbative approach which does not make use of any simplifying assumption was very recently introduced in [20] for slowly rotating stars: it will be reviewed in Section 7.4 as it is the starting point for the calculations we develop in Chapter8 in order to calculate the QNMs of a rapidly rotating neutron star. As far as gravitational radiation is concerned, it is important to mention that due to rotation, some modes may grow unstable through the Chandrasekhar-Friedman Schutz (CFS) instability mechanism [71, 72, 73, 74] and that in general it is likely that during its life a compact star undergoes an oscillatory phase in which it may become unstable. Instabilities may have important effects on the evolution of the star itself and, for example, they may be associated with a further emission of GWs, the amount of which would depend on when and whether the growing modes are saturated by non-linear couplings or dissipative processes, such as gravitational wave emission and viscosity. An oscillating compact object is therefore a promising gravitational wave source only if an instability appears and it overcomes the damping mechanisms [206, 207, 208].

7.1 Eulerian and Lagrangian Approaches

In GR, the starting point of a perturbative calculation is a solution of the Einstein equations

Gµν = 8πTµν . (7.1) The Bianchi identities µν G ;µ = 0 (7.2) also provide us with conservation equations for the stress-energy tensor: µν T ;µ = 0 . (7.3) One then wants to study the behaviour of a solution when it slightly departs from equilibrium: physically this problem may arise, for example, from considering a neutron star that is hit by a chunk of matter during accretion and wondering how it behaves. It will obviously depart from equilibrium and oscillate (in some complicated way): one is therefore interested in the time evolution of nearby configurations having the same baryon number and total entropy, i.e. deformations of the original equilibrium configuration. From a mathematical point of view, within a first order perturbative approach this will thus mean dealing with the tensors

Gµν + δGµν (7.4) and

Tµν + δTµν (7.5) and looking for solutions of the perturbed Einstein equations and conservation equations, i.e.

δ(Gµν − 8πTµν) = 0 (7.6) µν δ(T ;µ ) = 0 , (7.7) 7.1 Eulerian and Lagrangian Approaches 117

with suitable boundary conditions. Notice that the background solutions Gµν and Tµν have been cancelled out since they obey Eqs. (7.1) and (7.3), but the background ﬁelds will still be present in the explicit expressions for δGµν and δTµν. So far we have been imprecise in specifying the meaning of the δ’s and vague about what physical quantities are being perturbed. In discussing stellar oscillations, one introduces a family of time-dependent solutions (that enter the Einstein tensor and the stress energy tensor)

µ Q(λ) = {gµν(λ), u (λ), (λ),P (λ)}, (7.8)

µ which are the metric tensor gµν, the 4-velocity field of the star fluid u , the energy density (scalar) field and the pressure (scalar) field P . When dealing with black holes Qλ will reduce to gµν(λ). The family of solutions is indexed by the parameter λ and one compares the perturbed variables Q(λ) to their equilibrium values Q(0), to first order in λ. One supposes, moreover, that the family of solutions Q(λ) may be obtained from Q(0) by an adiabatic deformation; this means that there exists a diffeomorphism χλ that maps fluid trajectories of the equilibrium model Q(0) to the fluid trajectories of the solution Q(λ). When the oscillations of a star are small compared to its radius at equilibrium, they may be treated as perturbations and may be described in two equivalent formalisms [209].

1. In the Eulerian formalism one compares at the same point in spacetime a quantity in the unperturbed configuration to its value in the perturbed configuration; the Eulerian perturbations are hence defined as:

d δQ = Q(λ)| (7.9) dλ λ=0

2. In the Lagrangian formalism one defines a displacement vector ξµ which connects fluid elements in the unperturbed equilibrium configuration to the corresponding ones in the perturbed configuration; this vector is related to the 4-velocity perturbation δuµ by

dξµ dt dξµ dξµ δuµ = = = u0 . (7.10) dτ dτ dt dt The Lagrangian variation ∆Q is deﬁned as the change in Q with respect to a frame µ µ dragged by ξ . Thus ξ is the generator of the family of diffeomorphisms χλ and the Lagrangian perturbations may be deﬁned as

d ∆Q = [χ Q(λ)]| . (7.11) dλ −λ λ=0

The Lagrangian and Eulerian variations are therefore related by

∆Q = (δ + Lξ)Q, (7.12) where Lξ is the Lie derivative along ξ (see Appendix E.3), so that one may choose which formalism to work in and switch to the other one if needed. Even if one wants to express the ﬁnal results in terms of Eulerian variations, intermediate calculations involving the ﬂuid are often easier in terms of Lagrangian variations. 118 7. Perturbation Theory, Quasi-Normal Modes and Instabilities

7.1.1 Gauge Freedom and Gauge Transformations The infinitesimal Lagrangian displacement vector ξ is highly gauge dependent since the values of its components depend on the way one chooses to identify points in the perturbed and unperturbed spacetime manifolds. In principle one may choose ξµ = 0, i.e. choose a Lagrangian (comoving) coordinate system. It is usually more convenient to choose some geometrically fixed Eulerian system and to express coordinates with respect to it [210]; the Eulerian approach, for example, has proved to be more suitable for numerical computations of mode frequencies and eigenfunctions. The freedom to alter by arbitrary infinitesimal relative displacements ζµ the way in which the spacetime manifolds of the perturbed and unperturbed configurations are identified — i.e. the freedom to perform an infinitesimal diffeomorphism generated by the vector field ζ — gives rise to gauge transformations of the first kind, under which the Eulerian variation is transformed as

δ → δ − Lζ , (7.13) so that it is physically equivalent to formulate the perturbation problem with the set of µ Eulerian perturbation variables (δgµν ≡ hµν, δu , δ, δP ) — compare with Eq. (7.8) — or with the set

0 hµν = hµν − Lζgµν = hµν − ζµ;ν − ζν;µ (7.14) µ 0 µ µ µ ν µ ν µ (δu ) = δu − Lζu = δu + u ζ;ν − ζ u;ν (7.15) 0 µ (δ) = δ − Lζ = δ − ζ ,µ (7.16) 0 µ (δP ) = δP − LζP = δP − ζ P,µ . (7.17)

This is used to simplify the linear perturbation equations, leaving the background solution unchanged. The Lagrangian variation ∆ is instead subject to gauge transformations of the second kind [211], a much more restricted group of transformations arising from inﬁnitesimal displacements with components of the form αuµ, where α is an arbitrary inﬁnitesimal scalar: these transformations, therefore, leave world lines invariant, whereas ∆ transforms as

∆ → ∆ − Lαu. (7.18)

Note that if the ﬂow vector u is parallel to the generator of an invariance group of the unperturbed motion, the Lagrangian variation of any covariant tensor orthogonal to the ﬂow will be gauge invariant. To preserve the validity of Eq. (7.12), the material displacement vector is always affected by both kinds of gauge transformations. The combined effect induces the change

ξµ → ξµ + ζµ − αuµ . (7.19)

7.2 Black Hole Oscillations

In order to see how the perturbative approach works, let us start reviewing the main results on oscillations in General Relativity by considering the simplest possible case: non-rotating black holes. 7.2 Black Hole Oscillations 119

In their outstanding pioneering paper [212] dated 1957, T. Regge and J. A. Wheeler initiated the study of compact object perturbations in General Relativity. They showed that the equations describing the perturbations of the Schwarzschild black hole solution

2M dr2 ds2 = − 1 − dt2 + + r2(dθ2 + sin2 θdφ2) (7.20) r 2M 1 − r may be separated if the metric perturbation tensor is expanded in tensorial spherical harmonics (see Appendix E.1). They also showed that the relevant equations split into two decoupled sets according to their parity:

• even or polar which have (−)l parity

• odd or axial which have (−)l+1 parity.

This is a gauge-independent result. Finally, Regge and Wheeler proved that with a suitable gauge choice — the Regge-Wheeler gauge, see Section 8.6.1 — and by expanding the perturbed functions in tensor harmonics and Fourier-transforming them, the equations for the radial part of the axial perturbations can be reduced to the single Schrödinger-like wave equation known as Regge-Wheeler equation:

2 − d Zl 2 − − 2 + [σ − Vl (r)]Zl = 0 l ≥ 2 , (7.21) dr∗ where the potential barrier is given by

def 2M l(l + 1) 6M V − = 1 − − l ≥ 2 , (7.22) l r r2 r3 the tortoise coordinate

def r r = r + 2M ln − 1 (7.23) ∗ 2M follows from

dr 2M −1 ∗ = 1 − (7.24) dr r

− and where Zl is suitably defined from the odd metric perturbations. The multipole numbers l = {0, 1} are excluded for gravitational perturbations because they are not dynamical: l = 0 gravitational perturbations are spherically symmetric and, therefore, obey the Birkhoff theorem and correspond to infinitesimal changes of the black hole mass, whereas l = 1 gravitational perturbations yield infinitesimal shifts in the black hole position. The Regge-Wheeler equation (7.21) was followed in 1970 by the Zerilli equation for the polar perturbations [213] which remarkably has the same form, but a different potential wall

def 2(r − 2M) V + = [n2(n + 1)r3 + 3Mnr2 + 9M 2nr + 9M 3] (7.25) l r4(nr + 3M)2 def (l − 1)(l + 2) n = (7.26) 2 120 7. Perturbation Theory, Quasi-Normal Modes and Instabilities

+ and of course a different solution Zl built upon the even perturbations of the Schwarzschild metric. A perturbed non-rotating black hole is therefore governed by the two wave equations 2 ± d Zl 2 ± ± 2 + [σ − Vl (r)]Zl = 0 . (7.27) dr∗ If there is a source exciting the perturbation, it will appear on the right hand side of the equations as a forcing term obtained from the harmonic expansion of the stress-energy tensor of the exciting source. With a source term, the two equations for the axial and polar perturbations therefore describe the way a Schwarzschild black hole reacts to an external perturbation. Far away from the black hole, in the so-called “wave zone”, the perturbation of the metric is well described by a gravitational wave in the transverse traceless gauge; the two polarizations of the gravitational signal emitted by the perturbed black hole can be ± calculated in terms of the Regge-Wheeler and the Zerilli functions Zl as follows [44]: " # 1 Z eiσ(t−r∗) Z− (r, σ) Xlm(θ, φ) h+(t, r, θ, φ) = X Z+ (r, σ)W lm(θ, φ) − lm dσ 2π r lm iσ sin θ lm " # 1 Z eiσ(t−r∗) Xlm(θ, φ) Z− (r, σ) h×(t, r, θ, φ) = X Z+ (r, σ) + lm W lm(θ, φ) dσ 2π r lm sin θ iσ lm where W lm and Xlm are defined in terms of angular derivatives of the spherical harmonics ± by Eq. (E.63). Notice that we have introduced the azimuthal index m in Zlm only for formal ± reasons: for spherical backgrounds there is a degeneracy in m and thus Zlm is independent of m. Also notice that we have not made any formal distinction between polar and axial eigenfrequencies, i.e. we have used the symbol σ and not σ±: physically this is due to the fact that polar and axial eigenfrequencies coincide, a property known as isospectrality of polar and axial perturbations of Schwarzschild black holes. We are now ready to discuss the fact that the σ’s are complex quantities, or in other words that black holes possess quasi-normal modes. In a classical analysis of normal modes, one usually deals with an ordinary differential equation (or a system of ODEs) and imposes boundary conditions such that the wave-function vanishes outside a given region of space. Black hole perturbations (i.e. the wave-functions in the present case) are different because they propagate throughout space and we may not require their vanishing outside a finite region. However, we want to make sure that gravitational radiation unrelated to the initial perturbation does not disturb the system: we therefore have to impose the so-called purely outgoing wave boundary condition, at radial infinity. Moreover, since we are dealing with black holes, we also require a purely ingoing wave boundary condition at the black hole horizon (r∗ = −∞) where nothing can escape [214]. This way, equations (7.27) are provided with the boundary conditions

∓iσr∗ lim Zl ∼ e (7.28) r∗→±∞ and, in fact, allow complex frequency solutions, as pointed out by Vishveshwara in 1970 [204]. The discrete complex eigenfrequencies were actually computed five years later by Chandrasekhar and Detweiler [215]. What is happening is that when radiating gravitational waves are generated by a non- radial perturbation mode, the mode itself is dissipating energy and this is why its eigenfrequency turns out to be complex: the real part of σ corresponds to the frequency of the 7.3 Oscillations of Relativistic Stars 121 oscillation mode, the imaginary part is instead the inverse of its damping time. In order to determine the quasi-normal mode spectrum of a black hole one therefore looks for the complex set of eigenfrequencies {σn} of the Regge-Wheeler/Zerilli wave equation (7.27) that satisfy the boundary conditions given in Eq. (7.28). The computation of the Schwarzschild black hole spectrum shows that it solely depends on the mass of the black hole, which for the no-hair theorem is, in fact, the only physical parameter available in this case [80]. Therefore the mass of an oscillating non-rotating black hole is in principle inferable from gravitational wave measurements. By looking at the behaviour of perturbations in the time domain (σ → ∂/∂t), one sees that their evolution may be divided into three stages: the first stage depends on the initial conditions, it is then followed by an exponential damping of the amplitude of the perturbations and finally, at asymptotically late time (t M), by the so-called tails. Once again one observes that, after the initial outburst of the first stage, the behaviour of the amplitude of the perturbations only depends on the black hole mass (and not on the initial conditions); once more, the damping law appears to be an important characteristic of the black hole and the exponential damping of the perturbations goes under the name of quasi- normal ringing. It can be of course split into the superposition of exponentially damped oscillations and the whole complex eigenfrequency spectrum may be reconstructed2. Of course one may also study QNMs of Kerr black holes, but the calculation of the QNM frequencies — which has been performed by Detweiler [216, 217, 218, 219], and subsequently by Leaver [220], Seidel and Iyer [221], Kokkotas [222] and Onozawa [223]— is more involved than in the Schwarzschild case. It is straightforward to imagine that rotation will affect the spectrum, as the black hole is now characterised by two physical parameters (its mass and its spin angular momentum). Since our goal is to determine the perturbation equations for rapidly rotating neutron stars, we will discuss the effects of rotation directly for the QNM spectrum of neutron stars which is richer than that of a black hole due to the presence of the stellar fluid. However, we want to quickly quote two important results in order to already gain some important insight on the effects of rotation. The first one is quite intuitive: the potential barrier of a Kerr black hole is complex and it depends on m and on the frequency σ (as opposed to the real, l-dependent Schwarzschild barrier). Rotation therefore removes the degeneracy in m one has for Schwarzschild modes and there is a set of eigenmodes for any couple of harmonic indices l and m. The second one is certainly less straightforward: when a → M, a being the spin angular momentum of the black hole, the imaginary part of the mode-frequencies tends to zero. This means that there is a connection between rotation and instability; however, it has been shown that when a → M the amplitude of these possibly marginally unstable modes tends to zero [224].

7.3 Oscillations of Relativistic Stars

It is likely that every compact star undergoes, during its life, an oscillatory phase: neutron star oscillations may be excited after a core collapse, for example, during a starquake induced by the secular spin-down of a pulsar or by a large phase transition, during the ﬁnal stages of a merger involving a neutron star, as transients during a gravitational collapse [225].

2Generally the functions {eiσnt} do not form a basis in the vector space of the solutions of the Schwarzschild black hole perturbation equation (completeness issue) and therefore the signal cannot be expanded in terms of these functions at all times; however, the expansion is appropriate to describe the quasi-normal ringing epoch. 122 7. Perturbation Theory, Quasi-Normal Modes and Instabilities

In this section we report on relativistic perturbation theory for non-rotating stars and we then discuss the difﬁculties associated with adding rotation. The various families of stellar quasi-normal modes are brieﬂy reviewed in AppendixF.

7.3.1 Perturbing a Non-Rotating Star The metric of a static and spherically symmetric matter distribution may be written as follows:

2 2ν(r) 2 2λ(r) 2 2 2 2 2 ds(0) = −e dt + e dr + r (dθ + sin θdφ ). (7.29) The two potentials ν and λ may be determined by solving the Einstein equations coupled to the equations of hydrostatic equilibrium which follow from Eq. (7.3). Assuming the star is composed of an isotropic perfect ﬂuid, the energy-momentum tensor may be written as

µν µ ν µν T(0) = ( + P )u u + P g(0), (7.30) and one obtains the TOV equations (Appendix B.1). The spacetime of a perturbed non-rotating star is described by the line element

2 2 µ ν ds = ds(0) + hµνdx dx (7.31) and the motion of a generic fluid element is given by the Lagrangian displacement vector ξ. An important assumption of relativistic stellar perturbation theory is that the fluid perturbations are adiabatic, i.e. the variations of energy and pressure occur with no dissipation. One has to write down the perturbed Einstein equations and the perturbed fluid equations and solve them with the following boundary conditions: 1. all (background and perturbed) quantities must be regular at the star centre 2. all variables must be continuous at the surface of the star and the Lagrangian variation of the pressure must vanish (see Figure 7.1), i.e. ∆P (R) = 0 3. at radial infinity only purely outgoing waves are allowed. Once again, in the frequency domain, the complex nature of this last boundary condition requires the characteristic values of the frequencies to be complex and thus we will have a discrete set of complex eigenfrequency for the outgoing quasi-normal oscillation modes. Therefore also relativistic stars possess a QNM spectrum. The equations governing the adiabatic perturbations of a spherical star in general relativity have been derived within different approaches by many authors [193, 198, 226, 227, 228, 229, 200, 230]. Let’s take a look at the main steps of the calculations as performed in [227, 228]. Given the symmetry of the background, the spacetime perturbations may be conveniently expanded in multipoles (Appendix E.1). Working in the frequency domain and in the Regge-Wheeler gauge (see Section 8.6.1 for more details on this gauge), one has3:

2 2ν l iσt 2 l+1 iσt ds = −e (1 + r H0Ylme )dt − 2iσr H1Ylme dtdr + (7.32) 2λ l iσt 2 l iσt 2 2 2 2 + e (1 − r H0Ylme )dr + (1 − r KYlme )r (dθ + sin θdφ ) − 2 − 2 sin θ(h dt + h dr)eiσt∂ Y dφ + (h dt + h dr)eiσt∂ Y dθ 0 1 θ lm sin θ 0 1 φ lm

3 Generally speaking there should be H2 instead of H0 in the second line, but one of the perturbed Einstein equations in this gauge yields H2 = H0. 7.3 Oscillations of Relativistic Stars 123

Figure 7.1. Pictorial explanation of the boundary condition ∆P (R) = 0. The stellar surface is determined by the condition P (R) = 0: if we consider a ﬂuid element belonging to the stellar surface (left panel), a perturbation will vary its pressure and shift it (right panel). Its pressure therefore 0 µ 0 becomes P = P (R) + δP + ξ ∂µP ; if we now require the physical condition P = 0 so that it still µ belongs to the surface of the star, we get ∆P ≡ δP + ξ ∂µ = 0.

where we have omitted the indices l and m for the ﬁve functions of r H0,H1, K, h0, h1. Once again, the properties of the spherical harmonics explicitly split the perturbed equations into polar and axial. These respectively involve the radial functions H0,H1,K and h0, h1. The displacement vector may instead be expanded as (see Appendix E.1.2)

r −λ l−1 iσt ξ = e r WYlme θ l−2 iσt ξ = −r V ∂θYlme (7.33) φ l −2 iσt ξ = r (r sin θ) V ∂φYlme .

t µ ξ then follows from the perturbation of the normalization condition u uµ = −1. Notice that we have chosen a gauge in which the axial degree of freedom of ξµ, U, is set to zero; hence, the polar perturbed Einstein equations contain all the ﬂuid perturbations (δP and δ being scalars are polar), while the axial equations do not involve any ﬂuid perturbation.

The Polar Perturbation Equations

The ﬁve perturbation functions (H0,H1, K, W, V ) are not independent. Eliminating the variable H0 with one of the Einstein equations, the following fourth-order system of linear equations is obtained [228]:

1 2m(r) H0 = − (l + 1) + e2λ + r2(P − ) H + e2λ[H + K − 4( + P )V ] 1 r r 1 0 1 K0 = {H (n + 1)H − [(l + 1) − rν0]K − 2( + P )eλW } r 0 1 " # l + 1 e−νX d l(l + 1)V H W 0 = − W + reλ − + 0 + K (7.34) r + P dP r2 2 ( l 1 1 ν0 r2σ˜2 + (n + 1)H X0 = − X + ( + P )eν − H 1 + r 2 r 2 0 2r " # ) 1 3 l(l + 1) 1 r2 + ( rν0 − 1)K − ν0 V − eλ( + P +σ ˜) − (r−2e−λν0)0 W 2r 2 r2 r 2 124 7. Perturbation Theory, Quasi-Normal Modes and Instabilities where the primes denote derivatives with respect to r, n = (l − 1)(l + 2)/2 and σ˜ = σe−ν, and H0 and V are given by the linear combinations

2r2e−νX − [(n + 1)rν0/2 − r2σ˜2]e−2λH H = 1 + 0 3m(r)/r + n + r2P [n − σ˜2r2 − ν0(3m(r) − r + r3P )]K + 3m(r)/r + n + r2P " # 1 e−νX ν0e−λW 1 V = − − H . σ˜2 + P r 2 0

The system admits four linearly independent solutions for each given couple (l, σ); moreover it is singular in r = 0 whereas the physically relevant solutions must be ﬁnite everywhere: by imposing regularity at the centre of the star only two linearly independent solutions are left. These conditions may be imposed expanding each perturbation variable in powers of r in the neighbourhood of r = 0. At the surface of the star, instead, the Lagrangian variation of the pressure must vanish and this requirement selects only one physical solution out of the two. Outside the star, only the spacetime perturbations are present and one recovers the Zerilli + equation (7.25)-(7.27) where Z is deﬁned by a suitable combination of H1 and K.

The Axial Perturbation Equations The axial perturbation equations are simpler and they may be combined into a single − wave equation for the function Z [200, 202], which is related to h0 and h1 by

i d − h0 = − (rZ ) σ dr∗ −2ν − h1 = −e (rZ ) .

In the star interior the wave equation is the Regge-Wheeler equation (7.21) with the modiﬁed potential

2ν def e V − = [l(l + 1)r + r3( − P ) − 6m(r)] , (7.35) int r3 which outside the star of course reduces to the standard Regge-Wheeler potential (7.22). The frequencies of the different families of modes (SectionF) can be easily computed by solving the equations of stellar perturbations [193, 194, 195, 196, 197, 198, 199].

7.3.2 Effects of Rotation As we have just seen, in the spherical, i.e. non-rotating, limit, perturbations decouple into purely polar and purely axial and they are degenerate in the azimuthal index m. However, all neutron stars rotate so that the non-rotating case is an academic — albeit very instructive — one. Rotation, breaks the spherical limit and increases the complexity of the physics of the inner structure. The frequency ν(i) of a mode in the inertial frame, is related to its frequency ν(r) in the frame corotating with the star at angular velocity Ω by [231]

(i) (r) 2 ν = ν − mΩ + Clm(Ω) + O(Ω ) , (7.36) 7.3 Oscillations of Relativistic Stars 125

where the term Clm(Ω) is a function that depends on the mode eigenfunction in the non- rotating star and the O(Ω2) terms come from the leading order modiﬁcations of the stellar structure due to rotation. When discussing rotation, an important physical quantity associated with a mode frequency is the pattern speed

def ν dφ σ = − ≡ , (7.37) ∓ ±|m| dt where we have written out the sign of m explicitly. This quantity expresses the apparent displacement of the wave associated with the mode with respect to the star. For positive m, the pattern speed σ− is negative, indicating that the mode is retrograde, i.e. the wave moves in the direction of decreasing φ; for negative m, σ+ > 0, meaning that those modes are prograde. The effects of rotation which are evident from the formula appearing in Eq. (7.36) are therefore: • a splitting of the non-rotating mode l in 2l + 1 different (l, m) modes just like in the case of black holes;

• a shift of the frequencies; prograde mode frequencies increase with rotation, whereas retrograde mode frequencies decrease with rotation;

• because of the second term proportional to m in Eq. (7.36), the sign of the frequency of a retrograde mode may possibly change and therefore invert the apparent direction of propagation of the mode; this phenomenon is an indicator of the onset of the CFS instability [71, 72, 73, 74]. Moreover it can be shown that because of rotation [225]: • a “polar” l mode is a sum of purely polar terms (P ) which are higher order in l (since the equipotential surfaces are no longer spherical) and purely axial terms (A) which are (l ± 1, m) or even higher order in l (since a coupling between polar and axial terms appears in the equations); for example for l = m

∞ rot X Pl ∼ (Pl+2l0 + Al+2l0±1) , (7.38) l0=0 the coupling with l + 1 axial modes being strongly favoured; similarly a rotating “axial” mode with l = m is given by the sum

∞ rot X Al ∼ (Al+2l0 + Pl+2l0±1); (7.39) l0=0

• damping times are shifted. All in all, rotation causes shifting, coupling and mixing effects. Given these effects, the following criterion was established to label the modes of a rotating star that one determines: in a sequence of rotating stars including a non-rotating member, a QNM of index l is the mode which reduces, to the QNM of the same index l in the non-rotating limit. From this discussion, it is now clear that the effects of rotation make the problem of calculating in General Relativity the QNM spectrum of spinning stars a very difficult one to 126 7. Perturbation Theory, Quasi-Normal Modes and Instabilities solve: when the perturbative approach — which works so well in the non-rotating case — is generalised to include rotation, it shows a very high degree of complexity, even if the star is only slowly rotating [232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243]. The major difficulty encountered when using the standard multipole decomposition arises form the coupling of modes with different harmonic indices (Eqs. (7.38) and (7.39)), which yields an infinite set of dynamical coupled equations. This is why further simplifying assumptions are introduced in most studies based on this approach4: for example, in [232, 233, 234, 235] the couplings between oscillations with different values of the harmonic azimuthal index l are neglected, whereas in [236, 239, 240, 241, 242] the Cowling approximation is adopted5. An alternative approach consists in solving the equations describing a rotating and oscillating star in full General Relativity, in the time domain. However, also in this approach strong simplifying assumptions are currently made (see, for instance, [244, 245, 246, 247, 248] for uses of the Cowling approximation), or a restriction to a particular case is considered. In [249], for example, only quasi-radial modes (l = 0) were considered; in [250, 251] only the neutral mode — i.e. the zero-frequency mode in the rotating frame — was studied; in [252] only axisymmetric (m = 0) modes have been analysed, using the conformal flatness condition. In [253], the frequencies of axisymmetric modes (m = 0) with 0 ≤ l ≤ 3 have been computed for rapidly rotating relativistic polytropes with polytropic index n = 1, using the Cowling approximation; a comparison of the l = 0 results obtained in [253] with those found in [249] shows that the Cowling approximation introduces large errors in the determination of the fundamental mode frequency (see AppendixF).

7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods

In this section we illustrate the main features of the perturbative approach developed in [20] in order to find the quasi-normal modes frequencies of a slowly rotating neutron star without any simplifying assumption (a part from slow rotation, of course). This is useful since the rest of this thesis contains the first steps towards an extension of such method to the case of rapidly rotating stars. The new method of [20] to find quasi-normal modes of rotating relativistic stars is based on

• studying the oscillations in the frequency domain as perturbations of a stationary, slowly rotating, axisymmetric background;

• expanding the perturbations in circular harmonics — instead of spherical harmonics, which are generally used — so that for an assigned value of the frequency and of the azimuthal index m, the perturbed equations to solve are a 2D-system of linear, partial differential equations in the radial distance r and in the polar angle θ;

4The only work on slowly rotating star oscillations which does not make use of any of these restrictive assumptions may be found in [243]; however, it investigates r-modes only. 5 One sets δgµν = 0 and calculates the pulsations described only by the perturbations in the ﬂuid variables. This works quite well for the f, p and r-modes (see the AppendixF for the QNM classiﬁcation). Another version of the approximation consists in putting to zero only δgtr; this is more suitable for the calculation of the g-modes. For f and p modes the error due to the Cowling approximation decreases as l increases; the error on f-mode frequencies diminishes as M/R increases while for the p-modes the situation is inverted. 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods 127

• generalising and implementing the standing wave boundary condition often used in the non-rotating case [196, 201, 230] in order to deﬁne the asymptotic conditions at inﬁnity and to determine the mode frequencies;

• integrating the perturbed equations by using spectral methods.

As we shall only review the essential features of this method with the intent of clarifying the foundations of some choices made in Chapter8, we refer the reader to [ 20] for further details on the method, its applications and its results.

The starting point is the stationary axisymmetric background describing a slowly rotating neutron star. This is written in the form

2 −ν(r) 2 λ(r) 2 2 2 2 2 2 2 ds(0) = e dt + e dr + r dθ + sin θdφ − 2ω(r)r sin θdtdφ . (7.40)

The metric and ﬂuid velocity perturbations can be considered as tensor ﬁelds in this background and are expanded in circular harmonics eimφ; furthermore, since one looks for −iσt quasi-normal modes, a time dependence of the form e , with σ ∈ C, is assumed. Due to the stationarity and axisymmetry of the background spacetime, the linearised Einstein equations do not couple perturbations with different values of m and σ, so that, given a couple (m, σ), the perturbed equations to solve are a 2D-system of linear differential equations in r and θ and the perturbations are of the form6

m σ imφ −iσt hµν(t, r, θ, φ) = hµν (r, θ)e e . (7.41)

The linearised perturbation equations are derived explicitly in [234]. By ﬁxing the gauge m σ for the perturbations, the ten components hµν (r, θ) reduce to six quantities denoted by m σ {Hi (r, θ)}i=1,...,6. The Regge-Wheeler gauge [212] is generalised and adopted in [20]. The linearised Einstein equations therefore give a system of partial differential equations — m σ in the six quantities Hi and in the ﬂuid velocity perturbations — which involve derivatives with respect to r and θ. At this point, two key ingredients are introduced in [20] in order to determine the QNM spectrum of a slowly rotating neutron star:

1. a suitable generalisation of the boundary conditions at the star center and at radial inﬁnity obtained from those used to determine the QNM frequencies of non-rotating stars, and

2. spectral methods, which are used to integrate the perturbed equations.

7.4.1 The Boundary Conditions Given a value of m, in order to determine the QNM frequencies, the equations are solved m σ for different values of σ in the quantities Hi (r, θ). This is done by imposing that • all metric functions are regular near the center of the star,

• the Lagrangian perturbation of the pressure vanishes at the stellar surface,

• the solution at inﬁnity behaves as a purely outgoing wave.

6Similar expressions hold for the ﬂuid perturbations. 128 7. Perturbation Theory, Quasi-Normal Modes and Instabilities

The conditions at the center and at the surface of the star may be fulfilled for every value of σ, but the outgoing wave condition at infinity is only consistent with a discrete set of (complex) frequencies {σi} which form the the QNM spectrum. The boundary condition at radial infinity thus requires particular attention. We shall therefore discuss the generalisation of the standing wave method — commonly used to integrate the equations that describe the perturbations of a non rotating star — to the rotating case.

The Standing Wave Method for Spherical Stars

This approach to handle the boundary conditions at infinity for the perturbation of a relativistic stare was first suggested by K.S. Thorne [196], and further developed by S. Chandrasekhar and V. Ferrari [201, 230]. We must stress the fact that it works only in determining the QNM frequencies of slowly damped modes; it can thus be used to find the frequencies of fluid modes, like the f-, p-, and r-modes, but it cannot be applied to highly damped modes, like the w-modes or black hole QNMs7 As discussed in Section 7.3, outside the star the equations describing the perturbations of a non-rotating star reduce to the two radial wave equations which describe the perturbations of a Schwarzschild black hole — i.e. the Regge-Wheeler [212] and the Zerilli [213] equations — for two appropriately defined functions which we shall generically denote here with Zlm(r, σ). In the frequency domain, these equations have the general form of a wave equation, as stated in Eq. (7.27). The complex mode eigenfrequencies are those values of σ for which the solutions of such equations, found by imposing the appropriate boundary conditions at the star centre and surface recently mentioned, behave as purely outgoing waves at radial infinity. This requirement means that there is no ingoing radiation and is written as8

lm lm iσr∗ Z (r, σ) = Aout(σ)e r∗ −→ ∞ . (7.42)

The standing wave approach is used to ﬁnd these eigenfrequencies and consists in the following. Assume that Zlm(r, σ) is an analytic function of the complex variable σ = f −i/T , and let σ0 = f0 −i/T0 be the frequency of a slowly damped mode, i.e. 1/T0 f0. In general, at radial inﬁnity the solution of the wave equation (7.27) is a superposition of ingoing and outgoing waves, i.e.

lm lm −iσr∗ lm iσr∗ Z (r, σ) = Ain (σ)e + Aout(σ)e . (7.43)

lm lm If σ = σ0, by deﬁnition Ain (σ) = 0. Since Z is analytic and 1/T0 f0, we can expand lm Ain (f) near the real f0 as

lm lm i lm 0 Ain (σ0) = Ain (f0) − Ain (f0) , (7.44) T0

lm from which, by imposing Ain (σ0) = 0, we ﬁnd

lm 0 lm Ain (f0) = −iT0Ain (f0) . (7.45)

7See AppendixF for a discussion on the different family of modes. 8The tortoise coordinate is deﬁned in (7.23). 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods 129

lm Near f = f0 (with f, f0 ∈ R) the function Ain (σ) thus takes the form lm lm lm 0 lm Ain (f) = Ain (f0) + (f − f0)Ain (f0) = Ain (f0) [1 − iT0(f − f0)] lm i = −iT0Ain (f0) (f − f0) + , (7.46) T0 and its square modulus is 2 1 lm 2 2 Ain (f) = B (f − f0) + 2 , (7.47) T0 where B is a constant which does not depend on f. To determine the QNM frequencies, it is therefore sufficient to integrate the wave equation (7.27) for real values of f, and to find the values fi for which the amplitude of the standing wave, given by Eq. (7.47), has a minimum: these are the QNM frequencies. The corresponding damping times Ti can be found with a 2 lm quadratic fit of Ain (f) [200].

The Standing Wave Approach for Slowly Rotating Stars In switching to rotating stars, one of the key points in the approach of [20] is to write the perturbations in such a way that their scalar, vector and tensor nature is kept explicit all the time: this is essential in order to use spectral methods. For example, after performing a gauge choice (the “generalised Regge-Wheeler gauge” choice), the metric perturbations are written out as  ν mσ mσ im mσ mσ  e H0 H1 − sin θ h0,ax sin θ∂θh0,ax  ∗ eλHmσ − im hmσ sin θ∂ hmσ  mσ  2 sin θ 1,ax θ 1,ax  hµν =  mσ 2  ,  ∗ ∗ K r 0  ∗ ∗ ∗ Kmσr2 sin2 θ and all perturbations are expanded in terms of scalar spherical harmonics in the form mσ X lmσ lm H0 (r, θ) ≡ H0 (r)Y (θ, 0) . l≥|m| mσ mσ mσ mσ mσ mσ H0 (r, θ), H1 (r, θ), H2 (r, θ), h0,ax(r, θ), h1,ax(r, θ), and K (r, θ) are therefore scalars under rotations since they are sums of products of scalars. By proceeding in a similar fashion with the ﬂuid perturbations, one is left with the set of scalar perturbation variables of m σ the form {Hi (r, θ)}i=1,...,10 (σ ∈ C). The PDEs governing the linear perturbations can then be integrated once the values of the variables at the center of the star, i.e. on a sphere of radius r0 R (R being the stellar radius), are assigned. We shall indicate these boundary conditions with m σ m σ Hi (r0, θ) = H0i (θ) . (7.48) m σ The H0i (θ)’s are subject to the constraints arising from the assumption of regularity of the spacetime as r → 0 and from the requirement that the Lagrangian pressure perturbation vanishes on the stellar surface. These constraints reduce the number of independent quantities m ω ˆ m from the ten H0i (θ)’s to a smaller number, say N, i.e. {H0j (θ)}j=1,...,N , of σ-independent quantities, which are deﬁned on a sphere of radius r0 and may be decomposed as L ˆ m X ˆ lm lm H0j (θ) = Hj Y (θ, 0) , (7.49) l=|m| 130 7. Perturbation Theory, Quasi-Normal Modes and Instabilities where the expansion is truncated at l = L. The independent solutions of the perturbed equations therefore correspond to the following set of N(L − |m| + 1) constants n ˆ lmo Hj with j = 1,...,N and l = |m|,...,L. (7.50)

m σ Once these constants are determined, the perturbed equations for the functions Hi (r, θ) may be integrated for r ≥ r0. In the wave zone, far away from the star, the far ﬁeld limit expansion of the metric describing a rotating star shows that the metric reduces to the Schwarzschild solution ([174], Chapter 19); this occurs because terms due to rotation decrease more rapidly than the “Schwarzschild-like” components. Thus, as when dealing with Schwarzschild perturbations, in this asymptotic region one may deﬁne the gauge lm lm invariant Zerilli and Regge-Wheeler functions, ZZer(r, σ) and ZRW(r, σ), in terms of the perturbed metric tensor, expanded in tensorial spherical harmonics with l ≥ 2. This tensor is found by integrating the equations describing the perturbed spacetime outside the rotating lm lm star. The asymptotic behaviour of ZZer(r, σ) and ZRW(r, σ) is

lm lm −iσr∗ lm iσr∗ ZZer(r, σ) = AZer in(σ)e + AZer out(σ)e lm lm −iσr∗ lm iσr∗ ZRW(r, σ) = ARW in(σ)e + ARW out(σ)e . (7.51)

A (complex) frequency σ0 belongs to a quasi-normal mode if, for a given m, the following condition is satisﬁed for any l: lm lm AZer in(σ0) = ARW in(σ0) = 0 ∀l , (7.52) i.e. if the set of 2(L − |m| + 1) constants

n lm lm o AZer in(σ),ARW in(σ) with l = |m|,...,L (7.53) vanishes. Notice that this is a big difference with respect to the non rotating case, where each mode belongs to a single value of l and there is a degeneracy in m. For each assigned value of m, we deﬁne the vectors  ˆ |m| m   |m| m  H1 AZer in (σ)  |m|+1 m  |m|+1 m  Hˆ  A (σ)  1   Zer in   .   .   .   .          ˆ m   m   H ≡  |m| m  and A ≡  |m| m  (7.54)  Hˆ   A (σ)   2   RW in   ˆ |m|+1 m  |m|+1 m  H2  ARW in (σ)      .   .   .   .  with N(L − |m| + 1) and 2(L − |m| + 1) components respectively. Since the perturbed equations are linear, these vectors are related by the matrix equation Am(σ) = Mm(σ)Hˆ m (7.55) ˆ lm (the constants Hj do not depend on σ, as noted before Eq. (7.49)). The entries of the complex matrix Mm(σ) must be evaluated by integrating the perturbed equations. Equation (7.52), which identiﬁes the QNM eigenfrequencies, may be written as m m m M (σ0)Hˆ = 0 ∀Hˆ . (7.56) 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods 131

A discrete set of QNMs exists if Mm is a square matrix, i.e. if N = 2. Thus, such equation is equivalent to

m det (M (σ0)) = 0 . (7.57)

By counting the number of independent equations, in the cases of spherical and of slowly rotating stars one finds indeed N = 2. Let us now restrict the frequency to the real axis and normalize the constants Hˆ m m f lm lm so that the solutions Hi (r, θ) (and consequently ZZer(r, f), ZRW(r, f)) are real: this is always possible since the perturbed Einstein equations in the frequency domain have real coefficients, as long as f is real (if one assumes the fluid to be non dissipative, so that the equations are time-symmetric). Thus, in the wave zone we have

lm lm −ifr∗ lm ifr∗ ZZer(r, f) = AZer in(f)e + AZer out(f)e ∈ R lm lm −ifr∗ lm ifr∗ ZRW(r, f) = ARW in(f)e + ARW out(f)e ∈ R . (7.58) lm lm The ingoing wave amplitudes AZer in and ARW in may be found, as shown in [230], by evaluating ZZer and ZRW at different values of r∗ and ﬁtting both functions with a superposition ˆ lm lm of sin(fr∗) and cos(fr∗). It should be noted that, although H is real, AZer in and ARW in lm lm ∗ lm lm ∗ are complex (they satisfy the conditions AZer out = (AZer in) and ARW out = (ARW in) ) and m m m thus M is complex. For f ∈ R, the vectors Hˆ and A are related by Am(f) = Mm(f)Hˆ m . (7.59)

By expanding Eq. (7.57) about σ0 = f0 − i/T0 (with |1/T0| f0) as is done for spherical stars, one ﬁnds that if f ∼ f0,

m det M (f) = det M(f0)[1 − iT0(f − f0)] (7.60) so that

s 2 m 2 1 |det M (f)| ∝ (f − f0) + . (7.61) T0 Thus, the QNM frequencies are found by evaluating the (complex) matrix Mm for real values of the frequency f and by determining the minima of the modulus of its determinant. The standing wave approach has thus been generalised to rotating stars.

7.4.2 Spectral Methods for Stellar Oscillations Spectral methods are very powerful when solving partial differential equations and are particularly useful to implement boundary conditions. They have been used to solve the linearised Einstein perturbation equations for a slowly rotating star in [20]. As in the case of rapidly rotating stars (see Chapter8), this is a 2D problem. The r-dependence and the θ-dependence are treated with Chebyshev and associated Legendre polynomials respectively.

Chebyshev Polynomials Chebyshev polynomials

def Tn(x(θ)) = cos(nθ) n ∈ N (7.62) 132 7. Perturbation Theory, Quasi-Normal Modes and Instabilities satisfy the orthogonality relations Z 1 dx π Tm(x)Tn(x)√ = (1 + δm0) δmn (7.63) −1 1 − x2 2 on the interval x ∈ [−1, 1] and may be used to expand a function f(x) in the form ∞ X f(x) = anTn(x) . (7.64) n=0 Integrals involving Chebyshev polynomials may be performed numerically by exploiting the Gaussian quadrature method [183] in the case of Chebyshev weights and abscissas (or collocation points): N Z 1 dx π X g(x)√ = g(x ) (7.65) 2 N + 1 n −1 1 − x n=0 π(n + 1/2) x = cos n = 0, 1,...,N (7.66) n N + 1 where the expansion is truncated at n = N. The derivative operator matrix representation is: 0 X f (x) = (Dmnan)Tm , (7.67) n,m where

DNn = 0

Dk−1 n = Dk+1 n + 2kδkn k = 2,...,N (7.68) 1 D = [D + 2δ ] . 0n 2 2 n 1n The matrix representation of the multiplication of a function f(x), whose Chebyshev expansion is X f(x) = anTn(x) , (7.69) n by a function V (x) is X X X V (x)f(x) = bnTn(x) = Tn(x) Vnmam , (7.70) n n m where 2 − δm0 X V = V (x )T (x )T (x ) . (7.71) nm N + 1 k n k m k k If the independent variable is deﬁned in a different range, say y ∈ [a, b] , (7.72) in order to use the Chebyshev expansion, it may be rescaled to [−1, 1] with the transformation 2y − b − a x = ∈ [−1, 1] b − a (b − a)x + (b + a) y = (7.73) 2 2 ∂ = ∂ . y b − a x 7.4 Oscillations of Slowly Rotating Neutron Stars With Spectral Methods 133

Associated Legendre polynomials Functions of θ are expanded with the aid of the basis of the associated Legendre polynomials (see Appendix E.1) |m| def d P lm(x) = (−1)m(1 − x2)(|m|/2) P l(x) , (7.74) dx|m| where y = cos θ ∈ [−1, 1] (7.75) These polynomials are related to the spherical harmonics Y lm(θ, φ) by Eq. (E.3). Expanding a function in circular harmonics eimφ and in associated Legendre polynomials is thus equivalent to performing an expansion in spherical harmonics. As discussed in Eq .(E.4), associated Legendre polynomials are convenient in expanding functions of the polar angle θ because they are eigenfunctions of the Laplacian operator. Furthermore, they form a complete basis and automatically satisfy the asymptotic behaviour near the axis (see Eq. (7.74) and Eq. (E.7)) P lm ∼ (1 − x2)m/2 = (sin θ)m if θ ' 0, π (7.76) which is required for any angular expansion of a regular function near the axis sin θ = 0 (Section 6 of [115]). A function f(r, θ, φ) which is ﬁrst expanded in circular harmonics eimφ and is regular near the star centre and the polar axis thus yields the functions f m(r, cos θ) which may be expanded as X f m(r, x) = alm(r)P lm(x) (7.77) l for a given m. In order to apply the Gaussian quadrature method to the associated Legendre polynomials, one uses the fact that, for a given value of m, the polynomials P lm(x) P¯lm(x) ≡ (7.78) (1 − x2)m/2 form a complete basis with orthogonality relation Z 1 ¯lm ¯l0m 2 m 2 (l + m)! P (x)P (x)(1 − x ) dx = δll0 . (7.79) −1 2l + 1 (l − m)! ¯lm (α,β) The P are thus a particular case of Jacobi polynomials Ji (y) (see [183]) Z 1 α β Ji(y)Jj(y)(1 − y) (1 + y) ∝ δij , (7.80) −1 with α = β = m. Therefore, Gaussian integration takes the form 1 lm Z X f(yk)P (yk) f m(y)P lm(y)dy = w (7.81) k 1 − y2 −1 k k where the xk’s and the wk’s are the collocation points and the weights for the Jacobi polynomials with α = β = m. In particular, the coefﬁcients in Eq. (7.77) take the form lm 2l + 1 (l − m)! X f(yk)P (yk) alm = w . (7.82) 2 (l + m)! k 1 − y2 k k 134 7. Perturbation Theory, Quasi-Normal Modes and Instabilities

Differential Equations and Boundary Conditions Consider a one-dimensional ﬁrst order differential equation

Z0(x) + V (x)Z(x) = 0 (7.83) over the interval x ∈ [−1, 1] and the expansion in a polynomial basis Tn(x) truncated at n = N

N X Z(x) = anTn(x) . (7.84) n=0 The differential equation thus corresponds to the algebraic equation

N X (Dnm + Vnm)am = 0 (n = 0,...,N) , (7.85) m=0 where Dnm and Vnm are the matrix representations of the derivative operator and of the multiplication by the scalar function V (x). In order to solve this equation, the speciﬁcation of a boundary condition is required. In [20], the boundary condition is implemented by using the so-called τ-method. In the case of this example, this consists in cutting the last row of Eq. (7.85) and replacing it with the boundary condition, say Z(x0) = z0, which yields     D00 + V00 D01 + V01 ...D0N + V0N a0  D + V D + V ...D + V   a   10 10 11 11 1N 1N   1   . . . .   .   . . .. .   .   . . .   .      D(N−1)0 + V(N−1)0 D(N−1)1 + V(N−1)1 ...D(N−1)N + V(N−1)N  aN−1 T0(x0) T1(x0) ...TN (x0) aN  0   0     .  =  .  (7.86)  .     0  z0

The differential equation with the boundary condition Z(x0) = z0 hence reduces to a matrix equation which can be solved by LU decomposition [183]. This approach can be easily generalised to a higher order differential equations, to systems of coupled differential equations and to the case of partial differential equations in more than one dimension [254, 255]. Chapter 8

The Perturbed Einstein Equations for a Rapidly Rotating Star

In order to detect the gravitational signals emitted by NSs it is important to know their pulsation frequencies. As discussed in Section 7.3, non-rotating stars, whose QNMs can easily be computed within perturbation theory [193, 194, 195, 196, 197, 198, 199], are an ideal case since all NS rotate. It is therefore necessary to determine the QNM spectrum of rotating neutron stars. As a matter of fact, due to rotation, some modes may grow unstable through the CFS instability mechanism [71, 72, 73, 74] and these instabilities may have important effects on the subsequent evolution of the star; for example they may be associated with a further emission of GWs, the amount of which would depend on when and whether the growing modes are saturated by non-linear couplings or dissipative processes. In order to detect the gravitational signals emitted by compact stars it is therefore important to know their pulsation frequencies and to study under which conditions the corresponding modes become unstable. As discussed in Section 7.4, a perturbative approach for calculating the quasi-normal modes of slowly rotating neutron stars was developed in [20] and it allows one to avoid the simplifying assumptions discussed at the end of Section 7.3.2. We intend to start extending this method to rapidly rotating neutron stars, since they have been observed (see page3), they are promising gravitational wave sources and their QNM spectrum is far to be complete. We shall, therefore begin this chapter by discussing the background solution for rotating relativistic stars; a code, named RNS, for constructing models of rapidly rotating, relativistic, compact stars using tabulated equations of state was written by Nikolaos Stergioulas and is available at http://www.gravity.phys.uwm.edu/rns/. We will then be ready to perturb the rapidly rotating star solution of the Einstein equations; this will yield the perturbation equations that one eventually has to solve in order to calculate the QNM spectrum of rapidly rotating neutron stars.

8.1 Geometry of a Rotating Star

In order to build the metric of a stationary axisymmetric rotating ﬂuid, i.e. a rotating star, one requires that the spacetime admits two Killing vectors: a timelike one, tµ, which generates asymptotic time-translations and a second one, φµ, which generates rotations and thus corresponds to the axial symmetry. Moreover, one assumes the spacetime to be

135 136 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

µ µ µ asymptotically ﬂat, that is, t tµ = −1, φ φµ = +∞ and tµφ = 0 at spatial inﬁnity [209]. The required symmetries of the metric imply the vanishing of the Lie derivatives (see Appendix E.3) of the metric

Ltgµν = 0 Lφgµν = 0 (8.1) and of the commutator [t, φ] which is again a Lie derivative

µ Ltφ = 0 . (8.2)

This property then implies that one may choose the coordinates x0 = t and x3 = φ such that tµ and φµ are coordinate vector ﬁelds, i.e. there exists a family of 2-surfaces spanned by tµ and φµ and of scalars t, φ for which1

µ µ t ∇µt = φ ∇µφ = 1 (8.3) µ µ t ∇µφ = φ ∇µt = 0 . (8.4)

If one also assumes that there are no meridional convective currents (circular flow or circularity condition), i.e. that the 4-velocity of the fluid belongs to the hypersurface defined by the Killing vectors and thus takes the form

uµ = ut(tµ + Ωφµ) , (8.5)

t µ where u = u ∇µt, there exists a family of 2-surfaces orthogonal to the two Killing vectors which may be described by the two remaining coordinates x1 and x2. A common choice 1 2 2 for x and x are quasi-isotropic coordinates, i.e. grθ = 0 and gθθ = r grr (in spherical coordinates) or gz$ = 0 and gzz = g$$ (in cylindrical coordinates). Three of the metric functions appearing in gµν may be expressed as invariant combinations of the two Killing vectors through the relations

µ gtt = tµt (8.6) µ gφφ = φµφ (8.7) µ gtφ = tµφ , (8.8) while the fourth metric function determines the conformal factor that characterizes the µ µ µ µ geometry of the 2-surfaces orthogonal to t = δt and φ = δφ. With the two aforementioned quasi-isotropic coordinate choices, the rotating star metric (conventionally) takes the form

ds2 = −e2νdt2 + e2ψ(dφ − ωdt)2 + e2µ(dr2 + r2dθ2) (8.9) and

ds2 = −e2νdt2 + e2ψ(dφ − ωdt)2 + e2µ(d$2 + dz2) . (8.10)

A different, but equivalent, form of the metric is used in the formalism due to Hartle and 2 Sharp [256] in which gθθ = gφφ/ sin θ is required (in spherical coordinates):

ds2 = −e2νdt2 + e2λdr2 + e2µr2[dθ2 + sin2 θ(dφ − ωdt)2] , (8.11)

1The following are all Lie derivatives of scalars. 8.1 Geometry of a Rotating Star 137

which reduces to the usual non-rotating metric when ω → 0 and one has the gauge freedom to put µ = 0. The equivalence with Eq. (8.9), for example, is evident if one performs the formal replacement e2µr2 → e2ψ/ sin2 θ and then the coordinate change dθ → eλ−ψr sin θdθ, which is equivalent to replacing the quasi-isotropic spherical coordinate requirement with 2 Hartle and Sharp’s requirement gθθ = gφφ/ sin θ. In virtue of Eq. (8.1), the four potentials appearing in the metric — in whichever form one prefers to write it — are of course functions of r and θ. A peculiarity of relativistic rotating stars which has no Newtonian counterpart is the dragging of local inertial frames by the rotation of the gravitational ﬁeld source. In studying this relativistic effect, one is assisted by Zero Angular Momentum Observers (ZAMOs), also known as Eulerian observers. These observers have worldlines orthogonal to the t = const. hypersurfaces and uφ = 0 (since their angular momentum must vanish), therefore

φ t 0 = uφ = gφφu + gφtu (8.12)

from which

φ µ dφ u gφt t φµ = t = − = − ν = ω (8.13) dt u gφφ φ φν no matter which form of the metric we decide to use. This means that particles with zero angular momentum move along trajectories whose angular velocity with respect to an observer at rest at infinity is given by the metric function ω which is thus known as dragging potential. The other two metric functions which may be expressed through the invariant combinations of the two Killing vectors — Eqs. (8.6)-(8.8) — also have a physical interpretation: e−ν is the time dilation factor between the local ZAMO’s proper time and the coordinate time t (the proper time at infinity) along a radial coordinate line, whereas 2πeψ — 2πeµr sin θ in Hartle and Sharp’s formalism, see the comment following Eq. (8.11) — is the proper circumferential radius of a circle around the symmetry axis. We now return to the fluid 4-velocity field given in Eq. (8.5). ZAMOs provide us with a natural tetrad field with basis 1-forms in Hartle and Sharp’s formalism

e(0) = eνdt e(1) = eλdr e(2) = eµrdθ e(3) = eµr sin θ(dφ − ωdt) (8.14)

and basis vectors e−µ e−µ e = e−ν(∂ + ω∂ ) e = e−λ∂ e = ∂ e = ∂ . (0) t φ (1) r (2) r θ (3) r sin θ φ (8.15)

In terms of the ﬂuid 3-velocity v, we may thus write the non-zero components of the 4-velocity along these frame vectors: 1 v u(0) = √ u(3) = √ . (8.16) 1 − v2 1 − v2 Therefore e−ν ωe−ν + ve−µr−1 sin−1 θ ut = √ uφ = √ (8.17) 1 − v2 1 − v2 138 8. The Perturbed Einstein Equations for a Rapidly Rotating Star and one ﬁnds

uφ veν−µ Ω ≡ = ω + (8.18) ut r sin θ for the fluid angular velocity relative to infinity, i.e. the fluid angular velocity as seen by an observer at rest at infinity. The 3-velocity v is hence written in terms of Ω as

v = (Ω − ω)r sin θeµ−ν . (8.19)

In the other two formalisms one therefore has

v = (Ω − ω)eψ−ν . (8.20)

The formulas for v are explained by noticing that Ω − ω is the star angular velocity as measured by a local inertial observer, i.e. Ω is corrected by the dragging potential ω, e−ν is the time dilation factor of this observer and 2πeµr sin θ (or 2πeψ) is the circumference of a circle centred about the axis of symmetry.

8.2 Perfect Fluids

On a sub-millimetre scale, superfluid neutrons and protons in the neutron star interior do not behave as a perfect fluid; on larger scales, though, a single averaged velocity field uµ accurately describes a NS. The approximation of uniform rotation is invalid below a centimetre, but the error in computing the stellar structure on larger scales is negligible [209]:

2 −11 δgµν ∼ (1cm/R) ∼ 10 . (8.21)

A perfect ﬂuid is described by stress-energy tensor

T µν = ( + P )uµuν + P gµν ≡ uµuν + P γµν , (8.22)

µ µ where the 4-velocity u is a timelike vector ﬁeld (uµu = −1), the scalars and P are the energy density and pressure measured by a comoving observer (an observer with 4-velocity uµ), and

def γµν = gµν + uµuν (8.23) is the projector orthogonal to u. A general EOS of the form = (n, s), P = P (n, s) expresses the energy density and the pressure as functionals of the baryon density n and the entropy per baryon s. Since neutron stars cool down quickly, one parameter equations of state (i.e. a zero temperature equations of state) are generally used; they have the form = (P ), or equivalently = (n) and P = P (n). A useful thermodynamical quantity linked to n is the baryon mass density ρ = mnn: the baryonic mass is a conserved quantity since all known interaction conserve the baryonic number. By projecting the equations of motion

µν T ;ν = 0 (8.24) 8.2 Perfect Fluids 139 along the 4-velocity and orthogonally to it with γµν, one obtains the energy conservation equation

µ µ u ∇µ = −( + P )∇µu (8.25) and Euler’s equation γµ∇ P uµ∇ u = − ν µ . (8.26) µ ν + P µ Equations (8.25) and (8.26) are ﬁve, but they are constrained by u uµ = −1 so that the original number of relations in Eq. (8.24) is recovered. Other two equations worth mentioning are the baryon number conservation law

µ ∇µ(nu ) = 0 (8.27) and the second law of thermodynamics in differential form which relates the energy density, the baryon density and the entropy per baryon (compare with Eq. (3.101))

d = hdn + T ds , (8.28) where

def + P h = (8.29) n is the comoving enthalpy per baryon. A stationary ﬂow is described by a spacetime with a timelike Killing vector tµ which generates time translations that leave the metric and ﬂuid variables unchanged, i.e.

µ Ltgµν = Ltu = Lt = LtP = 0 . (8.30) In the same way, an axisymmetric ﬂow has a Killing vector φµ which generates translations around an axis which leave the metric and ﬂuid variables unchanged. The two Killing vectors tµ and φµ — which rotating neutron star solutions possess — allow us to write the relativistic version of Bernoulli’s law

µ Lu(huµt ) = 0 , (8.31) which in Newtonian physics expresses the conservation of enthalpy for a stationary ﬂow, and the conservation of the angular momentum of a ﬂuid element

µ Lu(huµφ ) = 0 . (8.32) These conservation laws may be demonstrated using the hydrodynamics equations (8.25)- (8.26) and exploiting the properties of Killing vectors. Finally, the ﬂow of an isentropic ﬂuid conserves circulation; the differential conservation law takes the form

Luωµν = 0 , (8.33) where def ωµν = ∇µ(huν) − ∇ν(huµ) (8.34) is the relativistic vorticity. 140 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

8.3 Boundary Conditions

In building a rotating star model, the following boundary conditions are required [257]: 1. star centre (a) freedom from singularity in all physical variables (b) vanishing radial derivatives of the four metric potentials (c) finite pressure and energy density 2. star surface (a) vanishing pressure (b) continuity and differentiability of the four metric potentials (i.e. they must be limited) 3. symmetry axis (a) vanishing gradients of the four metric potentials orthogonal to the axis 2 (b) local flatness: gφφ ∼ sin θgrr 4. possible inner border surfaces (a) continuity in P (b) continuity and differentiability of the four metric potentials 5. radial infinity (a) asymptotic flatness, i.e. lim {ν, ω, µ} = 0 (or lim {ν, ω, λ} = 0) and ν ∼ r−1, r→∞ r→∞ −3 2 ω ∼ r , gφφ ∼ sin θgrr

8.4 Determining the Perturbation Equations

In discussing the perturbation equations, we shall first work with the background metric in the form given in Eq. (8.9) and perturb it following Priou [258], who determined the perturbation equations for the solid crust of a rapidly rotating star: it will be sufficient to set to zero all crust-related perturbative quantities introduced by Priou. However, in his formalism the scalar, vector and tensor behaviours under rotations on the 2-sphere (see Appendix E.1) of the perturbative variables are mixed up: this makes the formalism compact, but not very straightforward to use in extending the spectral method approach used by Ferrari, Gualtieri and Marassi for slowly rotating stars [20], which was synthesised in Section 7.4. We shall therefore use Priou’s formalism and the background metric in the form of Eq. (8.9) to check that the Maple files we have written are correct; we then shift to Hartle and Sharp’s form for the metric (Eq. (8.11)) and to perturbative quantities whose scalar, vector and tensor nature is kept explicit all the time. Since our goal is to make the extension of the approach of [20] to rapidly rotating stars possible, this second formalism works more naturally. There are several equations one may perturb: the Einstein equations, the fluid equations, µ the constraint u uµ = −1, which becomes

µ µ ν 2δuµu = u u hµν , (8.35) 8.5 Perturbation Equations in Priou’s Approach 141

µ where δu and hµν are, respectively, the Eulerian variation of the ﬂuid 4-velocity ﬁeld and of the background metric tensor (Section 7.1). Before proceeding, we provide the following universal expressions useful in calculating the perturbed equations in terms of Eulerian variations: 1 δR = hα − (h α + h ;α) (8.36) µν (µ;ν);α 2 α ;µ;ν µν;α [µ;α] αβ δR = −2hµ ;α − R(0)hαβ (8.37)

δTµν = (δ + δP )uµuν + 2( + P )u(µδuν) + δP gµν + P hµν ≡

≡ δP γµν + P δγµν + δuµuν + 2u(µδuν) (8.38) δT = 3δP − δ . (8.39)

The formula for δRµν follows from 1 δΓα = hα − h ;α . (8.40) βγ (β;γ) 2 βγ Other useful expressions are the ones which allow us to switch from the Eulerian perturbations δuµ to the Lagrangian displacement ξµ [259]: 1 δuµ = uµuαuβh − γµα[ξ, u] (8.41) 2 αβ α 1 δu = h uν + u uαuβh − γ [ξ, u]α (8.42) µ µν 2 µ αβ µα α β α δγµν = γµ γν hαβ − 2u(µγν)α[ξ, u] (8.43) µν µα βν α (µ ν)β (µ ν)α δγ = (−γ γ + 2u u γ )hαβ − 2u γ [ξ, u]α . (8.44) Whether one decides to work with δuµ or ξµ, the set of perturbation variables is made up of ten degrees of freedom for the metric, four for the fluid velocity/position, one for the pressure and one for the energy density perturbation, for a total of sixteen variables. The equations to perturb are ten (the Einstein equations or a part of them plus the conservation µν equations T ;ν = 0 in some convenient form) and the equation of state will yield an eleventh constraint. This leaves us with five uncovered degrees of freedom: if one uses δuµ a twelfth µ equation follows from the constraint u uµ = −1 (see Eq. (8.35)) and one is left with four degrees of freedom which are covered by gauge transformations of the first kind (see Section 7.1.1); if, instead, one uses the Lagrangian displacement ξ, there is the additional freedom to perform a gauge transformation of the second kind which covers the unphysical degree of freedom of the Lagrangian displacement vector ξ (again, see Section 7.1.1)

8.5 Perturbation Equations in Priou’s Approach

We start with the background metric of a rapidly rotating star in quasi-isotropic cylindrical coordinates, i.e. 2 2ν 2 2ψ 2 2µ 2 2 ds(0) = −e dt + e (dφ − ωdt) + e (d$ + dz ) , (8.45) or in matrix form −e2ν + e2ψω2 0 0 −e2ψω  2µ   0 e 0 0  gµν =  2 2µ  . (8.46)  0 0 r e 0  −e2ψω 0 0 e2ψ 142 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

In [258], Priou writes the perturbations in their most general form by working on the line element as follows:

ds2 = −e2ν(1 + 2h)dt2 + e2ψ(1 + 2w)[dφ − (ω + y)dt − adr − bdθ]2 + + e2µ[(1 + 2k)dr2 + r2(1 + 2p)dθ2] + 2qdrdθ + 2Ldtdr + 2Mdtdθ, (8.47) where the ten functions h, w, y, a, b, k, p, q, L, M depend on (t, r, θ, φ). In other words the components of the metric perturbation tensor hµν are

−2he2ν + 2(ωw + y)ωe2ψ L + ωae2ψ M + ωbe2ψ −(2ωw + y)e2ψ  2ψ 2µ 2ψ   L + ωae 2ke q −ae  hµν =  2ψ 2 2µ 2ψ  .  M + ωbe q 2r pe −be  −(2ωw + y)e2ψ −ae2ψ −be2ψ 2we2ψ

The form of the linearly perturbed Einstein equations Priou chooses is

1 1 δR = 8π δT − g δT − h T , (8.48) µν µν 2 µν 2 µν and in his paper all the equations are written out without performing any gauge choice. We carried out the same calculations and successfully cross-checked the results of our Maple ﬁles with [258], so that we were conﬁdent about the results we would eventually obtain when switching to the Hartle-Sharp formalism.

8.5.1 A Note About Gauge Choice

We will shortly discuss three possible gauge choices of the first kind (i.e. coordinate transformations) we have encountered when handling this formalism and looking into ways of simplifying hµν. The first gauge is the BCL (Battiston, Cazzola, Lucaroni) gauge. This gauge was first presented in [260] for non-rotating stars. In such case odd perturbations do not induce fluid motion except for a stationary rotation, so that, apart from how this influences the shape of the potential barrier associated with the spacetime curvature, the even perturbations are the most interesting ones to work on (Section 7.3.1). Battiston, Cazzolla and Lucaroni indeed deal with polar perturbations only and impose htt = hθθ = hθφ = hφφ = 0. The BCL gauge was then generalized to the slowly rotating case (using the Arnowitt-Deser-Misner formalism) by Ruoff, Stavridis and Kokkotas [237], who formulated it as

hθθ = hθφ = hφφ = 0

htt = −2ωhtφ , (8.49) i.e. the generalisation consists in setting to zero the metric tensorial perturbations (using three gauge degrees of freedom) and setting htt proportional to htφ (using the fourth and last gauge degree of freedom). In Priou’s notation this becomes

p = b = w = h = 0 (8.50) 8.5 Perturbation Equations in Priou’s Approach 143 and thus  2yωe2ψ L + ωae2ψ M −ye2ψ  2ψ 2µ 2ψ L + ωae 2ke q −ae  hµν =   . (8.51)  M q 0 0  −ye2ψ −ae2ψ 0 0

A second gauge choice we encountered in the literature is due to Friedman and Ster- gioulas who, after trying many different gauge choices, elected the following one as the most convenient gauge choice [250]:

b = q = y = 0 (8.52) w = p , (8.53) i.e. −2he2ν + 2ω2pe2ψ L + ωae2ψ M −2ωpe2ψ  2ψ 2µ 2ψ   L + ωae 2ke 0 −ae  hµν =  2 2µ  .  M 0 2r pe 0  −2ωpe2ψ −ae2ψ 0 2pe2ψ

They note that in the non-rotating limit this gauge coincides with the extensively used Regge-Wheeler gauge [212] (see Section 8.6.1) and that b = 0 and w = p are equivalent to setting the tensorial part of hµν proportional to the metric on the 2-sphere (just as in the Regge-Wheeler gauge). As a last option, guided by the naive idea of reducing the number of addends in δRµν by simplifying hµν as much as possible, we thought of setting

a = b = w = y = 0 , (8.54) i.e. −2he2ν LM 0  2µ   L 2ke q 0 hµν =  2 2µ  , (8.55)  M q 2pr e 0 0 0 0 0 since a, b, y, w are the only quantities appearing more than once in hµν. We noticed that with this gauge choice the order of the system of differential equations obtained by perturbing the Einstein equations is lower than with the other two. Since the quality of a gauge is not solely determined by how short it makes the set of equations, but mainly by how simple it makes their integration, in Table 8.5.1 we make a gross comparison among the three gauge choices just discussed by listing the highest r-derivatives appearing in each of the components of the perturbed Ricci tensor for every gauge choice. Such derivatives are, in fact, the only ones left if one assumes an eimφ+iσt dependence for the perturbations and treats the θ-dependence with an expansion in associated Legendre polynomials and then exploits the properties of such polynomials for the derivatives with respect to θ. This is not a detailed comparison of course: one would have to manipulate the equations ﬁrst and then see all three gauges at work with a speciﬁc numerical method to judge them. This crude parallel, however, shows that: 144 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

δRµν BCL FS New tt y00 h00, p00, k0 h00 tr M 0, y0 h0,M 0, p0 h0,M 0, p0 tθ M 00 M 00 M 00 tφ y00 p00 h0, k0,L0, p0, q0 rr a0, k0,L0, q0, y0 h00, p00 h00, p00 rθ M 0, y0 h0,M 0, p0 h0,M 0, p0 rφ y0 h0, p0 h0, p0 θθ k0, q0 p00 p00 θφ a0,L0,M 0, q0 a0,M 0,L0 L0,M 0, a0 φφ a0, k0, q0, y0 p00 h0, k0, p0, q0 Table 8.1. List of the highest r-derivatives appearing in each of the components of perturbed Ricci tensor for the three gauges discussed in the text (FS stands for Friedman-Stergioulas). A prime (0) indicates a derivative with respect to r.

1. M 00 is the only non-eliminable second derivative in all three gauges

2. with no algebraical manipulation, the BCL equations are one order lower than the other two gauges

When these simple analysis were completed, we thought that it would have been interesting to test the “new gauge” within some numerical approach. A short while later, we came to know about the work of Vincent and de Freitas Pacheco [261] who thought of the gauge choice described in Eq. (8.54), analysed it thoroughly and use it to calculate, within linear perturbation theory, the axisymmetric oscillations of a rapidly rotating neutron star by using a Runge-Kutta scheme. They also discuss the fact that the proposed gauge is ill-deﬁned in the non-rotating limit and it is not suitable in the non-axisymmetric case.

8.6 Perturbation Equations in Hartle and Sharp’s Formalism

We are now ready to switch to the Hartle and Sharp form of the background metric of a rapidly rotating star in spherical coordinates, i.e.

2 2ν 2 2λ 2 2µ 2 2 2 2 ds(0) = −e dt + e dr + e r [dθ + sin θ(dφ − ωdt) ] . (8.56) or in matrix form −e2ν + e2µω2r2 sin2 θ 0 0 −e2µωr2 sin2 θ  2λ   0 e 0 0  gµν =  2 2µ  . (8.57)  0 0 r e 0  −e2µωr2 sin2 θ 0 0 e2µr2 sin2 θ

To describe the fluid, we thus build the stress energy tensor of a perfect fluid (Eq. (8.22)) upon the 4-velocity field   µ 1 Ω u = q , 0, 0, q  (8.58) e2ν − (Ω − ω)2e2µr2 sin2 θ e2ν − (Ω − ω)2e2µr2 sin2 θ 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 145 which was derived at the end of Section 8.1. In the slow rotation regime, we may drop the ω2 terms and recover the problem in the form treated by Ferrari, Gualtieri and Marassi with spectral methods in [20] (see also Section 7.4). The non vanishing Einstein equations for the background metric take the form

e−2νr2 sin2 θ (ν + ν2 + λ ν + ν cot θ)e−2µ − (ω2 + e2µ−2λr2ω2) θθ θ θ θ θ θ r 2 ν + ν + ν2 − λ ν + 2ν µ + 2 r e−2λr2 rr r r r r r r " # e2ν + P e2µr2 sin2 θ(Ω − ω)2 − 3P = 8πr2 + (8.59) e2ν − e2µr2 sin2 θ(Ω − ω)2 2

4 ω + + 4µ − ν − λ ω e2µ−2λr2 + ω + (3 cot θ + 2µ + λ − ν )ω rr r r r r r θθ θ θ θ θ (ω − Ω)(P + )r2e2ν+2µ = 16π (8.60) e2ν − e2µr2 sin2 θ(Ω − ω)2

1 e2λ−2µ e2µ−2νr2 sin2 θω2 − (λ ν + λ cot θ + λ + λ2) − 2µ − ν − ν2 2 r θ θ θ θθ θ r2 rr rr r µ λ − 4 r + 2 r + 2λ µ − 2µ2 + λ ν r r r r r r r = 4πe2λ( − P ) (8.61)

ν λ 1 θ − ν − µ + θ − ν ν + λ ν + ν µ + λ µ + e2µ−2νr2ω sin2 θω = 0 r rθ rθ r r θ θ r θ r θ r 2 θ r (8.62)

1 e2µ−2νr2 sin2 θ e2µ−2λr2ω2 + ω2 − λ − ν − ν2 − λ2 + ν cot θ + λ cot θ 2 r θ θθ θθ θ θ θ θ + 2µθνθ + 2µθλθ (Ω − ω)2(P + ) = 8πr4 sin2 θe4µ (8.63) e2µr2 sin2 θ(Ω − ω)2 − e2ν

−2λ+2µ 2 2 2 2 2 1 − e (1 + 4rµr − λrr + νrr + r νrµr + 2r µr + r µrr − λrr µr) − µθθ 1 − µ ν − µ λ − cot θ(ν + λ + µ ) − sin2 θr2e−2ν+2µ(r2ω2e2λ+2µ + ω2) θ θ θ θ θ θ θ 2 r θ h P e2ν − e2µr2 sin2 θ(Ω − ω)2 1 i = 8π − r2e2µ + r2e2µ( − 3P ) (8.64) e2ν − e2µr2 sin2 θ(Ω − ω)2 2 where the r and θ subscripts of the metric potentials — which therefore cannot be mistaken for tensor indices — denote partial derivatives. The six equations have been obtained from 146 8. The Perturbed Einstein Equations for a Rapidly Rotating Star the Einstein equations

1 e2νr2Rtt = 8πe2νr2 T tt − gttT 2 −2 2ν t −2 2ν t −2 sin θe R φ = −16π sin θe T φ 1 R = 8π T − g T rr rr 2 rr Rrθ = 8πTrθ R T R − φφ = 8π T − φφ θθ sin2 θ θθ sin2 θ 1 R sin−2 θ = 8π sin−2 θ T − g T φφ φφ 2 φφ and can be used to simplify or manipulate the perturbation equations, for example by eliminating second derivatives with respect to r of the background quantities. We are now ready to introduce the metric and ﬂuid perturbations using the same perturbative formalism as in [20]. All perturbations are expanded in circular harmonics eimφ, and Fourier-transformed in time. Perturbations belonging to different polar indices m and different frequencies σ do not couple, so that m and σ may be treated as ﬁxed and the perturbed quantities may be decomposed as follows

mσ imφ+iσt hµν(t, r, θ, φ) = hµν (r, θ)e (8.65) µ µ imφ+iσt δu (t, r, θ, φ) = δumσ(r, θ)e (8.66) δ(t, r, θ, φ) = δ(r, θ)eimφ+iσt (8.67) δP (t, r, θ, φ) = δP (r, θ)eimφ+iσt , (8.68) where the frequency σ is of course complex in general. The metric perturbation may be written as

 2ν mσ mσ mσ im mσ mσ mσ  −2e H0 H1 ∂θh0,pol − sin θ h0,ax imh0,pol + sin θ∂θh0,ax  ∗ 2e2λHmσ ∂ hmσ − im hmσ imhmσ + sin θ∂ hmσ  hmσ =  2 θ 1,pol sin θ 1,ax 1,pol θ 1,ax µν  2µ mσ 2 L mσ   ∗ ∗ 2e {K r Ωab + [∇a∇b − 2 Ωab]G −  1 cd cd mσ − 2 [∇abcΩ ∇d + ∇bacΩ ∇d]hAX }

(8.69)

where asterisks stand for symmetric components, the indices a, b run over {θ, φ}, Ωab = 2 diag(1, sin θ) is the metric on the 2-sphere, ab is the Levi-Civita tensor on the 2-sphere 1/2 (θφ = −φθ = Ω = sin θ), ∇a denotes the covariant derivative on the 2-sphere and L is Laplacian operator on the sphere (Eq. (E.5)). In this expression, we have used a shorthand mσ notation for the expansion of the θ-dependence of hµν so that a multipole expansion of the 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 147 tensor is actually performed (compare with Appendix E.1); it works as follows:

mσ X lmσ lm H0 (r, θ) ≡ H0 (r)Y (θ, 0) (8.70) l≥|m| mσ X lmσ lm H1 (r, θ) ≡ H2 (r)Y (θ, 0) (8.71) l≥|m| mσ X lmσ lm H2 (r, θ) ≡ H2 (r)Y (θ, 0) (8.72) l≥|m| mσ X lmσ lm h0,ax(r, θ) ≡ h0,ax (r)Y (θ, 0) (8.73) l≥|m| mσ X lmσ lm h0,pol(r, θ) ≡ h0,pol(r)Y (θ, 0) (8.74) l≥|m| mσ X lmσ lm h1,ax(r, θ) ≡ h1,ax (r)Y (θ, 0) (8.75) l≥|m| mσ X lmσ lm h1,pol(r, θ) ≡ h1,pol(r)Y (θ, 0) (8.76) l≥|m| X Kmσ(r, θ) ≡ Klmσ(r)Y lm(θ, 0) (8.77) l≥|m| X Gmσ(r, θ) ≡ Glmσ(r)Y lm(θ, 0) (8.78) l≥|m| mσ X lmσ lm hAX (r, θ) ≡ hAX (r)Y (θ, 0) (8.79) l≥|m| where the Y lm(θ, φ) ’s are scalar spherical harmonics2. We have explicitly indicated the dependence on r and θ to show that with this kind of expansion the metric perturbations are ready to be treated with spectral methods: each 2D function appearing in Eqs. (8.70)- (8.79) may, in fact, be treated with the double expansion in Chebyshev polynomials for the r-dependence and associated Legendre polynomials — which are related to spherical harmonics by Eq. (E.3) — for the θ-dependence (see Section 7.4.2). Notice that all quantities on the left hand side of Eqs. (8.70)-(8.79) are scalars since they are sums of products of scalars under rotations. For the ﬂuid perturbations we may proceed in a similar way, by expanding the energy density and pressure scalar perturbations as

X δmσ(r, θ) ≡ δlmσ(r)Y lm(θ, 0) (8.80) l≥|m| X δP mσ(r, θ) ≡ δP lmσ(r)Y lm(θ, 0) (8.81) l≥|m|

2See Eq. (E.3) 148 8. The Perturbed Einstein Equations for a Rapidly Rotating Star and the 4-velocity perturbation in vector spherical harmonics as eν δur = e−2λRmσ(r, θ) (8.82) mσ 4π( + P ) eν ∂V mσ(r, θ) imU mσ(r, θ) δuθ = − (8.83) mσ 4πr2( + P ) ∂θ sin θ eν ∂U mσ(r, θ) δuφ = imV mσ(r, θ) + sin θ , (8.84) mσ 4πr2 sin2 θ( + P ) ∂θ where X Rmσ(r, θ) ≡ Rlmσ(r)Y lm(θ, 0) (8.85) l≥|m| X U mσ(r, θ) ≡ U lmσ(r)Y lm(θ, 0) (8.86) l≥|m| X V mσ(r, θ) ≡ V lmσ(r)Y lm(θ, 0) . (8.87) l≥|m|

t We have not written δumσ since it is determined by Eq. (8.35).

8.6.1 The Regge-Wheeler Gauge In this formalism we shall work in the Regge-Wheeler gauge [212] for two reasons: 1. to maintain the gauge choice of the work we intend extending [20] and 2. because we have tried to adopt the BCL gauge (Section 8.5.1) in this formalism and to solve the perturbations equation with spectral methods in the non-rotating limit, and it proved to be very unstable, thus discouraging its use in the rotating case. The Regge-Wheeler gauge choice consists in 1. setting to zero the polar terms in the vector harmonics of the metric perturbation and

2. imposing hab ∝ Ωab on the tensorial harmonics of the metric perturbation. lm lm lm lm By setting h0,pol = h1,pol = G = hAX = 0 as required, the metric perturbation in Eq. (8.69) thus reduces to  2ν mσ mσ im mσ mσ  −2e H0 H1 − sin θ h0,ax sin θ∂θh0,ax  ∗ 2e2λHmσ − im hmσ sin θ∂ hmσ  mσ,RW  2 sin θ 1,ax θ 1,ax  hµν =  2µ mσ 2  . (8.88)  ∗ ∗ 2e K r 0  ∗ ∗ ∗ 2e2µKmσr2 sin2 θ Hereafter we shall simplify the notation by dropping m, σ, “RW” and “ax” since this will not generate any confusion; therefore:  2ν im  −2e H0 H1 − sin θ h0 sin θ∂θh0  ∗ 2e2λH − im h sin θ∂ h   2 sin θ 1 θ 1  hµν(r, θ) =  2µ 2  . (8.89)  ∗ ∗ 2e Kr 0  ∗ ∗ ∗ 2e2µKr2 sin2 θ Once again, the (six) perturbations are functions deﬁned on the 2D domain r × θ, and their multipole expansion shown in Eqs. (8.70)-(8.79) is implicit in this notation. 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 149

8.6.2 The Perturbed Einstein Equations Putting together the expressions for the perturbations we gave at the beginning of this chapter, the formalism of Section 8.6 and the Regge-Wheeler gauge, we calculated the linearly perturbed Einstein equations. The ten less lengthy ones follow from 1 1 r2e4ν δRtt − 8π δT tt − httT − gttδT = 0 (8.90) 2 2 1 2r2e2ν δRt − 8π δT t − ht T = 0 (8.91) r r 2 r 1 2r2e2ν δRt − 8π δT t − ht T = 0 (8.92) θ θ 2 θ 2e2ν 1 δRt − 8π δT t − ht T = 0 (8.93) sin2 θ φ φ 2 φ 1 1 2e2ν δR − 8π δT − h T − g δT = 0 (8.94) rr rr 2 rr 2 rr 1 2e2ν δR − 8π δT − h T = 0 (8.95) rθ rθ 2 rθ 1 2e2ν δR − 8π δT − h T = 0 (8.96) rφ rφ 2 rφ 1 δR − 8π δT − h T = 0 (8.97) θφ θφ 2 θφ 1 1 δR − δR − 8π δT − δT = 0 (8.98) θθ sin2 θ φφ θθ sin2 θ φφ 1 1 δR + δR − 8π δT + δT − g δT − h T = 0 . (8.99) θθ sin2 θ φφ θθ sin2 θ φφ θθ θθ Before writing the equations down explicitly, we shall explain the notation we adopt. We have six spacetime perturbation variables and five fluid perturbation variables (we have not used any EOS constraint to link δP and δ yet since this may be done as a simple final substitution). The eleven quantities (and their derivatives) will be multiplied by functions of the background solution that will be abbreviated by using eleven coefficient symbols; these are associated with the perturbation variables with the following convention:

Perturbation: H0 H1 H2 h0 h1 K R U V δP δ Coefficient: ABCDEF ABCDE so that calligraphic letters are related to the fluid perturbations and not to the spacetime perturbations. The coefficients A, B, C, D, E, F, A, B, C, D and E have superscripts between parenthesis to indicate the perturbation equation they belong to, with equations (8.98) and (8.98) denoted by (−) and (+) respectively. Furthermore, subscripts will identify the derivative operator a coefficient multiplies; the use of such subscripts is clarified by this example extracted from Eq. (8.92): ∂2H ∂2H (function of the background variables) 1 ≡ B(tθ) 1 . (8.100) ∂r∂θ rθ ∂r∂θ If no subscripts appear, the perturbation variable is not derived; notice that, because of the form of the Einstein equations and of the expansions we make of the perturbation variables, A, D and E can never present any subscript. 150 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

THE SET OF EQUATIONS (tt) (tt) (tt) (tt) (tt) (Arr ∂rr + Ar ∂r + Aθθ ∂θθ + Aθ ∂θ + A )H0 (tt) (tt) + (Br ∂r + B )H1 (tt) (tt) (tt) + (Cr ∂r + Cθ ∂θ + C )H2 (tt) (tt) (tt) (tt) (tt) (tt) + (Drrθ ∂rrθ + Drθ ∂rθ + Dθθθ ∂θθθ + Dθθ ∂θθ + Dθ ∂θ + D )h0 (tt) (tt) (tt) (tt) + (Er ∂r + Erθ ∂rθ + E ∂θ + E )h1 (tt) (tt) (tt) (tt) (tt) + (Frr ∂rr + Fr ∂r + Fθθ ∂θθ + Fθ ∂θ + F )K (tt) (tt) (tt) (tt) + Bθ ∂θU + C V + D δP + E δ = 0 (8.101)

(tr) A H0 (tr) (tr) (tr) + (Bθθ ∂θθ + Bθ ∂θ + B )H1 (tr) + C H2 (tr) (tr) (tr) + (Dr ∂r + Dθ ∂θ + D )h0 (tr) (tr) (tr) (tr) + (Eθθθ ∂θθθ + Eθθ ∂θθ + Eθ ∂θ + E )h1 (tr) (tr) + (Fr ∂r + F )K + A(tr)R = 0 (8.102)

(tθ) A H0 (tθ) (tθ) (tθ) (tθ) + (Br ∂r + Brθ ∂rθ + Bθ ∂θ + B )H1 (tθ) (tθ) + (Cθ ∂θ + C )H2 (tθ) (tθ) (tθ) (tθ) (tθ) + (Drr ∂rr + Dr ∂r + Dθθ ∂θθ + Dθ ∂θ + D )h0 (tθ) (tθ) (tθ) (tθ) (tθ) (tθ) + (Er ∂r + Erθθ ∂rθθ + Erθ ∂rθ + Eθθ ∂θθ + Eθ ∂θ + E )h1 (tθ) (tθ) + (Fθ ∂θ + F )K (tθ) (tθ) + B U + Cθ ∂θV = 0 (8.103)

(tφ) (tφ) (tφ) (Ar ∂r + Aθ ∂θ + A )H0 (tφ) (tφ) + (Br ∂r + B )H1 (tφ) (tφ) (tφ) + (Cr ∂r + Cθ ∂θ + C )H2 (tφ) (tφ) (tφ) (tφ) (tφ) (tφ) + (Drrθ ∂rrθ + Drθ ∂rθ + Dθθθ ∂θθθ + Dθθ ∂θθ + Dθ ∂θ + D )h0 (tφ) (tφ) (tφ) (tφ) + (Er ∂r + Erθ ∂rθ + Eθ ∂θ + E )h1 (tφ) (tφ) (tφ) (tφ) (tφ) + (Frr ∂rr + Fr ∂r + Fθθ ∂θθ + Fθ ∂θ + F )K (tφ) (tφ) (tφ) (tφ) + Bθ ∂θU + C V + D δP + E δ = 0 (8.104) 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 151

(rr) (rr) (rr) (rr) (Arr ∂rr + Ar ∂r + Aθ ∂θ + A )H0 (rr) (rr) + (Br ∂r + B )H1 (rr) (rr) (rr) (rr) + (Cr ∂r + Cθθ ∂θθ + Cθ ∂θ + C )H2 (rr) (rr) (rr) (rr) (rr) + (Drrθ ∂rrθ + Drθ ∂rθ + Dθθ ∂θθ + Dθ ∂θ + D )h0 (rr) (rr) (rr) (rr) + (Er ∂r + Erθ ∂rθ + Eθ ∂θ + E )h1 (rr) (rr) (rr) (rr) + (Frr ∂rr + Fr ∂r + Fθ ∂θ + F )K + D(rr)δP + E(rr)δ = 0 (8.105)

(rθ) (rθ) (rθ) (rθ) (Ar ∂r + Arθ ∂rθ + Aθ ∂θ + A )H0 (rθ) (rθ) + (Bθ ∂θ + B )H1 (rθ) + Cθ H2,θ (rθ) (rθ) (rθ) (rθ) (rθ) (rθ) + (Dr ∂r + Drθθ ∂rθθ + Drθ ∂rθ + Dθθ ∂θθ + Dθ ∂θ + D )h0 (rθ) (rθ) (rθ) + (Eθθ ∂θθ + Eθ ∂θ + E )h1 (rθ) (rθ) (rθ) (rθ) + (Fr ∂r + Frθ ∂rθ + Fθ ∂θ + F )K = 0 (8.106)

(rφ) (rφ) (Ar ∂r + A )H0 (rφ) (rφ) + (Bθ ∂θ + B )H1 (rφ) + C H2 (rφ) (rφ) (rφ) + (Drθ ∂rθ + Dθ ∂θ + D )h0 (rφ) (rφ) (rφ) (rφ) + (Eθθθ ∂θθθ + Eθθ ∂θθ + Eθ ∂θ + E )h1 (rφ) (rφ) + (Fr ∂r + F )K + A(rφ)R = 0 (8.107)

(θφ) (θφ) (Aθ ∂θ + A )H0 (θφ) (θφ) + (Br ∂r + B )H1 (θφ) (θφ) + (Cθ ∂θ + C )H2 (θφ) (θφ) (θφ) (θφ) + (Dr ∂r + Dθθ ∂θθ + Dθ ∂θ + D )h0 (θφ) (θφ) (θφ) (θφ) (θφ) (θφ) + (Er ∂r + Erθθ ∂rθθ + Erθ ∂rθ + Eθθ ∂θθ + Eθ ∂θ + E )h1 (θφ) (θφ) + (Fθ ∂θ + F )K (θφ) (θφ) + B U + Cθ ∂θV = 0 (8.108) 152 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

(−) (−) (−) (Aθθ ∂θθ + Aθ ∂θ + A )H0 (−) (−) (−) + (Cθθ ∂θθ + Cθ ∂θ + C )H2 (−) (−) (−) (−) (−) + (Drθ ∂rθ + Dθθθ ∂θθθ + Dθθ ∂θθ + Dθ ∂θ + D )h0 (−) (−) (−) (−) + (Er ∂r + Erθ ∂rθ + Eθ ∂θ + E )h1 (−) (−) (−) (−) + (Fr ∂r + Fθθ ∂θθ + Fθ ∂θ + F )K (−) (−) (−) (−) + Bθ ∂θU + C V + D δP + E δ = 0 (8.109)

(+) (+) (+) (+) (Ar ∂r + Aθθ ∂θθ + Aθ ∂θ + A )H0 (+) + B H1 (+) (+) (+) (+) + (Cr ∂r + Cθθ ∂θθ + Cθ ∂θ + C )H2 (+) (+) (+) (+) + (Drθ ∂rθ + Dθθθ ∂θθθ + Dθθ ∂θθ + Dθ ∂θ)h0 (+) (+) + (Eθ ∂θ + E )h1 (+) (+) (+) (+) (+) + (Frr ∂rr + Fr ∂r + Fθθ ∂θθ + Fθ ∂θ + F )K (+) (+) (+) (+) + Bθ ∂θU + C V + D δP + E δ = 0 (8.110) 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 153

(tt) COEFFICIENTS

(tt) 2 −2λ+2ν Arr = r e (8.111) (tt) −2λ+2ν Ar = (2 − rλr + 2rµr + 2rνr)re (8.112) (tt) 2ν−2µ Aθθ = e (8.113) (tt) 2ν−2µ Aθ = e (λθ + 2νθ + cot θ) (8.114) (tt) h 2 2 2 2 2 2 2 4 2 2 2 A = 2 r sin θωθ (r λrνr − 2rνr − 2r νrµr − r νr + r sin θωr − r νrr) m2 i × e−2λ+2ν − ν2 + + λ ν + ν cot θ + ν e2ν−2µ θ 2 sin2 θ θ θ θ θθ + 32πe2νr2 e2ν + e2µr2 sin2 θP (Ω − ω)[(Ω − 2ω) + e2µr2 sin2 θ(Ω − ω)2] × H (8.115) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] 0 (tt) 2 −2λ Br = ir e (mω + σ) (8.116) (tt) −2λ B = ire [(mω + σ)(2rµr + 2 − rλr) + rmωr] (8.117) (tt) 2 −2λ+2ν Cr = −r νre (8.118) (tt) 2ν−2µ Cθ = νθe (8.119) (tt) 4 2 2 2 2 2 2 2 2ν−2λ C = (r sin θωr − 2r νr − 2r νrr − 4r νrµr − 4rνr + 2r λrνr)e + (mω + σ)2r2 (8.120) (tt) 2 −2λ Drrθ = −r sin θe ω (8.121) (tt) −2λ Drθ = r sin θe (2rωνr + rωλr − 2ω − rωr − 2rµrω) (8.122) (tt) −2µ Dθθθ = − sin θe ω (8.123) (tt) −2µ Dθθ = sin θe (2ωνθ − ωθ − ωλθ − 3ω cot θ) (8.124) nh m2 D(tt) = 1 + − λ cot θ + 4ν + 4λ ν − cot2 θ + 2ν2 + 6 cot θν ω θ sin2 θ θ θθ θ θ θ θ i −2µ + (2νθ − λθ − 4 cot θ − 2µθ)ωθ − ωθθ sin θe 2 2 2 2 2 + [2ω(2r νrr + 4rνr + r νr − 2r λrνr + 4r νrµr) 2 2 2 2 −2λ + ωr(r λr − 4r + 2r νr − 4r µr) − r ωrr] sin θe 2 −2ν 3 2 2 2µ−2λ 2 o − 2r e sin θω(ωθ + r e ωr ) n 2 − 16πr2 sinθ ω( − 3P ) + e2ν − e2µr2 sin2 θ(Ω − ω)2 e2νΩ − e2µr2 sin2 θP ω(Ω − ω)2 o + 2e2ν (8.125) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θ(Ω − ω)ω] me−2µ D(tt) = (mωλ + mω + σλ ) (8.126) sin θ θ θ θ ime−2µ+2ν−2λν E(tt) = θ (8.127) r sin θ (tt) 2 −2λ Erθ = ir sin θe ω(σ + mω) (8.128) (tt) −2λ Eθ = ir sin θe ω[(mω + σ)(2 + 2rµr − rλr) + rmωr] (8.129) 154 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

ime2ν−2µ−2λ E(tt) = 2ν ν − λ ν + 2ν − λ ν − r2 sin2 θe2ν−2µω ω (8.130) sin θ θ r r θ rθ θ r r θ (tt) 4 2 2µ−2λ 2 Frr = −r sin θe ω (8.131) (tt) 2 2 −2λ+2µ Fr = 2r sin θe e2ν−2µν × − 3r2µ ω2 + r2ω2ν − 3rω2 − r2ωω + r + r2ω2λ (8.132) r r r sin2 θ r (tt) 2 2 2 Fθθ = −r sin θω (8.133) (tt) 2 2 Fθ = r sin θω(2ωνθ − 5ω cot θ − 4ωµθ − ωλθ − 2ωθ) (8.134)

h1 F (tt) = 2 r2 ω2m2 + σ2 + ωσm + (ων − µ ω − 2µ2ω + 2ν ω − ω 2 θ θθ θ θθ θθ 2 + ω − 4ω cot θ + 2λθνθω − λθµθω − 4µθωθ + 2νθωθ + 4νθω cot θ 2 i − λθω cot θ − 5 cot θωθ − λθωθ − 5µθω cot θ + 2νθµθω) sin θω

n 2 2 + [(2rνr + rλr − 6rµr − 6)rωr + (6rνr − 8rµr − r µrr + 2r νrr 2 2 2 2 2 2 2 2 + rλr − 4r µr − 3 + r λrµr − 2r λrνr + r νr + 6r νrµr)ω − r ωrr]ω 1 o − r2ω2 r2 sin2 θe−2λ+2µ − r6e−2λ−2ν+4µ sin4 θω2ω2 2 r r ! 2 4 4 2 2 −2ν+2µ − (νθθ + νθ + λθνθ + νθ cot θ + 2r sin θωθ ω )e

n − 16πr4 sin2 θ ω2( − 3P ) e2µP r2 sin2 θ(Ω − ω)[e2νω2(Ω − 2ω) + e2µr2 sin2 θω3(Ω − ω)2] + 2 [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] e4ν[Ω2 + P (Ω2 − ω2)] o + 2 (8.135) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] 5ν+2µ 2 (tt) 8e r sin θ(Ω − ω) B = − q (8.136) [e2ν + e2µr2 sin2 θ(Ω − ω)ω] e2ν − e2µr2 sin2 θ(Ω − ω)2 5ν+2µ 2 (tt) 8ime r (Ω − ω) C = − q (8.137) [e2ν + e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 h e2µr2 sin2 θ(Ω − ω)2 3i D(tt) = −16πr2e2ν + (8.138) e2ν − e2µr2 sin2 θ(Ω − ω)2 2 h e2ν 1i E(tt) = −16πr2e2ν − (8.139) e2ν − e2µr2 sin2 θ(Ω − ω)2 2 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 155

(tr) COEFFICIENTS

(tr) 2 A = ir mωr (8.140) (tr) −2µ Bθθ = e (8.141) (tr) −2µ Bθ = e (cot θ − νθ − λθ) (8.142) h m2 B(tr) = e−2µ − 2λ cot θ + 2λ ν − 2λ2 − − 2λ θ θ θ θ sin2 θ θθ ν µ λ i + 4r2e−2λ µ λ + ν µ + r − 2 r − µ2 + r − µ r r r r r r r r rr n2r2[e2µP r2 sin θ2(Ω − ω)2 − e2ν] o − 8π + r2(3P − ) (8.143) e2ν − e2µr2 sin2 θ(Ω − ω)2 (tr) C = ir(4mω + 4rmµrω + 4rσµr + rmωr + 4σ) (8.144) ie−2µm D(tr) = (ν − λ ) (8.145) r sin θ θ θ (tr) 2 −2ν Dθ = −ir e sin θωωrm (8.146) 2ime−2µ λ ν D(tr) = − θ − θ − λ µ + µ − ν µ (8.147) sin θ r r θ r rθ θ r (tr) −2µ Eθθθ = sin θe ω (8.148) (tr) −2µ Eθθ = sin θe (3ω cot θ − ωλθ + ωθ − ωνθ) (8.149) h ωm2 E(tr) = e−2µ(cot θω + cot2 θω − − ω − 3ωλ cot θ − 2λ ω θ θ sin2 θ θ θ θ 2 −2λ 2 − 2ωλθθ − ω cot θνθ − 2ωλθ + 2λθνθω) + e (r νrωr − 8rωµr 2 2 2 2 2 + 4rνrω − r ωrr − 4r ωµr + 4r νrµrω + r λrωr + 4rωλr − 4rωr 2 2 2 i + 4r ωλrµr − 4r µrωr − 4r ωµr,r) sin θ he2ν(P Ω − P ω + Ω) + e2µP ωr2 sin2 θ(Ω − ω)2 − 16πr2 sin θ e2ν − e2µr2 sin2 θ(Ω − ω)2 1 i + ω(3P − ) (8.150) 2 me−2µ E(tr) = [(σ + mω)(ν − λ ) − mω ] (8.151) sin θ θ θ θ (tr) 2 Fr = 2r i(mω + 2σ) (8.152) (tr) 2 2 4 −2ν+2µ 2 2 F = 4ir σµr + 4irσ − 4ir σνr − ir e sin θmω ωr (8.153) 2 3ν (tr) 4r e A = −q (8.154) e2ν − e2µr2 sin2 θ(Ω − ω)2 156 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

(tθ) COEFFICIENTS

(tθ) 2 A = ir mωθH0 (8.155) (tθ) 2 −2λ Br = 2r e νθ (8.156) (tθ) 2 −2λ Brθ = −r e (8.157) (tθ) 2 −2λ Bθ = r e (λr − νr) (8.158) ν λ B(tθ) = 2r2e2λ θ + θ − µ + ν µ + λ ν − λ ν + λ µ (8.159) r r rθ θ r θ r r θ θ r (tθ) 2 Cθ = 2ir (mω + σ) (8.160) (tθ) 2 C = ir [mωθ + 2(σ + mω)(λθ − νθ)] (8.161) ir2e−2λm D(tθ) = − (8.162) rr sin θ ir2e−2λm D(tθ) = (ν + λ ) (8.163) r sin θ r r ie−2µm D(tθ) = − (8.164) rθθ sin θ im D(tθ) = (2e−2µν − r2e−2ν sin2 θωω − e−2µ cot θ) (8.165) θ sin θ θ θ 2im h D(tθ) = e−2µ λ2 − λ ν − ν cot θ + µ − µ ν − µ λ + cot θµ − 1 sin θ θ θ θ θ θθ θ θ θ θ θ m2 + λ + θθ 2 sin2 θ −2λ 2 2 2 2 2 i + e (4rµr − r λrµr + 2r µr − r νrµr + r µrr − rλr + 1 − rνr) h e2µr2 sin2 θP m(Ω − ω)2 im(3P + )i + 16πir2 sin θ + (8.166) e2ν − e2µr2 sin2 θ(Ω − ω)2 2 sin2 θ r2e−2λm E(tθ) = − (σ + ωm) (8.167) r sin θ (tθ) 2 −2λ Erθθ = −r sin θe ω (8.168) (tθ) 2 −2λ Erθ = r sin θe (2ωνθ − ωθ − ω cot θ) (8.169) (tθ) 2 −2λ Eθθ = r sin θe ω(λr − νr) (8.170) (tθ) 2 −2λ Eθ = 2r e sin θ (8.171) ωλ ων ω × ων µ + θ + θ − rθ − ω cot θ − ωλ ν + ωλ µ + ω ν θ r r r 2 r r θ θ r r θ 1 1 1 1 + ωλ ν − ωµ + ωλ cot θ + λ ω − µ ω − ω ν − ω cot θν θ r rθ 2 r 2 r θ θ r 2 θ r 2 r mr2e−2λ 2 E(tθ) = (σ + mω) ν + λ − 2µ − − mω (8.172) sin θ r r r r r (tθ) 2 Fθ = 2ir σ (8.173) (tθ) 2 F = 2ir (2mωνθ − 2mω cot θ − λθσ − σνθ − 2mµθω − ωλθm 1 − mω − r2e2µ−2ν sin2 θmω2ω ) (8.174) θ 2 θ 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 157

2 3ν+2µ (tθ) 4imr e B = q (8.175) sin θ e2ν − e2µr2 sin2 θ(Ω − ω)2 2 3ν+2µ (tθ) 4r e Cθ = −q (8.176) e2ν − e2µr2 sin2 θ(Ω − ω)2

(tφ) COEFFICIENTS

(tφ) 2 2µ−2λ Ar = r e ωr (8.177) (tφ) Aθ = ωθ (8.178) (tφ) n A = 2 ωθθ + (3 cot θ + 2µθ − νθ + λθ)ωθ

−2λ+2µo + [rωrr + (4 − rνr − rλr + 4rµr)ωr]re 16πe2µ+4νr2(P + )(Ω − 2ω) + (8.179) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] ime−2λ B(tφ) = − (8.180) r sin2 θ ime−2λ B(tφ) = (λ − ν ) (8.181) sin2 θ r r (tφ) 2 −2λ+2µ Cr = r ωre (8.182) (tφ) Cθ = −ωθ (8.183) (tφ) n 2 −2λ+2µ C = 2 [r ωrr + (4 + 4rµr − rλr − rνr)rωr]e m o − (σ + mω) (8.184) sin2 θ e−2λ D(tφ) = (8.185) rrθ sin θ e−2λ D(tφ) = − (ν + λ + r2 sin2 θe2µ−2νωω ) (8.186) rθ sin θ r r r e−2µ D(tφ) = (8.187) θθθ r2 sin θ e−2µ D(tφ) = (λ − r2 sin2 θe2µ−2νωω − 2µ − ν + cot θ) (8.188) θθ r2 sin θ θ θ θ θ (tφ) n 2 2µ−2ν−2λ Dθ = [2rωωrr + rωr − (2rλr − 8rµr + 4rνr − 8)ωωr]r sin θe e−2µ − 1 − λ cot θ − 2µ λ + ν cot θ − 2µ − 4µ cot θ r2 sin θ θ θ θ θ θθ θ m2 e−2λ − cot2 θ + 2ν µ − − 2 λ µ − µ θ θ sin2 θ sin θ r r rr 1 µ ν − 2µ2 + λ + ν µ − − 4 r + r r r r r r2 sin2 θ r sin2 θ r sin2 θ 2 −2νo + [2ωωθθ + ωθ − (4νθ − 2λθ − 4µθ − 7 cot θ)ωωθ] sin θe 16π n3P − − (8.189) sin θ 2 158 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

e4ν + e2µ+2νr2 sin2 θ[(Ω2 − Ωω − ω2) + P (2Ω2 − 4Ωω + ω2)] + [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] e4µr4 sin4 θ(Ω − ω)3ω o + (8.190) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] m2e−2µ D(tφ) = (2µ + 2 cot θ − λ − ν ) (8.191) r2 sin3 θ θ θ θ ime−2λω E(tφ) = − θ (8.192) r sin θ ie−2λ E(tφ) = − (σ + 2mω) (8.193) rθ sin θ ie−2λ E(tφ) = θ sin θ 2σ 2mω × σλ + 2mωλ − − 2σµ − + σν − 2mµ ω − mω (8.194) r r r r r r r r ime−2λ E(tφ) = (ν ω − 2 cot θω − 2µ ω + λ ω − 2ω − 2µ ω sin θ θ r r r θ θ r θ θ r + ωθνr + λrωθ − 2ωrθ) (8.195) (tφ) 2 −2λ+2µ Frr = 2r e ω (8.196) (tφ) 2ν−2λ Fr = 2e 1 × − rων + 4r2ωµ + 4rω − 2r2ωλ − e2µ−2ν sin2 θr4ω2ω (8.197) r r r 2 r (tφ) Fθθ = 2ω (8.198) 1 F (tφ) = 2 2µ ω − ων + ω + ωλ + 3 cot θω − r2 sin2 θe2µ−2νω2ω (8.199) θ θ θ θ θ 2 θ (tφ) n 2 3 −2λ−2ν+4µ F = − rωωrr − (2rνr − 5 − 5rµr + rλr)ωωr + rωr ]r sin θe ω 2 2 2 2µ−2ν + [ωωθθ + (3µθ + λθ + 4 cot θ − 2νθ)ωωθ + ωθ ]r sin θe ω m o − ω − (3 cot θ + λ − ν + 2µ )ω + (σ + mω) θθ θ θ θ θ sin2 θ n − 16πe2νr2 e2νω(3P − ) 2e4µ−2νP r4 sin4 θω(Ω − ω)3 + [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] e2µr2 sin2 θ[Ω(2ω2 − Ω2) + P (Ω3 + 2ωΩ2 − 8Ωω2 + 4ω3)] − [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] 2e2ν[Ω + P (Ω − ω)] o − (8.200) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 159

3ν+2µ 2ν 2µ 2 2 2 (tφ) 4e [e + e r sin θ(Ω − ω) ] Bθ = − q (8.201) sin θ[e2ν + e2µr2 sin2 θ(Ω − ω)ω] e2ν − e2µr2 sin2 θ(Ω − ω)2 3ν+2µ 2ν 2µ 2 2 2 (tφ) 4ime [e + e r sin θ(Ω − ω) ] C = − q (8.202) sin2 θ[e2ν + e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 16πe2ν+2µr2(Ω − ω) D(tφ) = − (8.203) e2ν − e2µr2 sin2 θ(Ω − ω)2 16πe2ν+2µr2(Ω − ω) E(tφ) = (8.204) e2ν − e2µr2 sin2 θ(Ω − ω)2

(rr) COEFFICIENTS

(rr) 2ν Arr = −2e (8.205) (rr) 2ν Ar = 2e (λr − 2νr) (8.206) e2ν−2µ+2λλ A(rr) = −2 θ (8.207) θ r2 (rr) 2 2 2 2µ A = −2r sin θωr e (8.208) (rr) Br = −2i(σ + ωm) (8.209) (rr) B = 2iλr(ωm + σ) (8.210) 2 C(rr) = 2e2ν ν + + 2µ (8.211) r r r r e2ν−2µ+2λ C(rr) = −2 (8.212) θθ r2 e2ν−2µ+2λ C(rr) = 2 (cot θ + 2λ + ν ) (8.213) θ r2 θ θ h e2ν−2µ+2λ m2 C(rr) = 4 − λ2 − λ − λ cot θ − λ ν r2 2 sin2 θ θ θθ θ θ θ i − 2e2λ(σ + mω)2 − 16πe2ν+2λ( − P ) (8.214) (rr) Drrθ = 2ω sin θ (8.215) (rr) Drθ = 2 sin θ(−2ωνr + ωr − λrω) (8.216) sin θωλ e−2µ+2λ D(rr) = 2 θ (8.217) θθ r2 h ω D(rr) = 2 sin θ 2 r − λ ω − 2ν ω + ω − 2ων + 2ω µ + 2λ ων θ r r r r r rr rr r r r r e−2µ+2λ i + e−2ν+2µr2 sin2 θωω2 + (ωλ cot θ − 2λ ων + λ ω ) (8.218) r r2 θ θ θ θ θ λ me−2µ+2λ D(rr) = −2 θ (ωm + σ) (8.219) r2 sin θ 2imν e2ν−2µ E(rr) = − θ (8.220) r r2 sin θ 160 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

(rr) 2 Erθ = −2iω sin θ(ω m + σ) (8.221) (rr) 2 Eθ = 2i sin θ(ω mλr + ωσλr) (8.222) 2ie2ν−2µm E(rr) = (λ ν − λ ν ) (8.223) r2 sin θ r θ θ r (rr) 2µ 2 2 2 2ν Frr = 2(e r sin θω − 2e ) (8.224) h 4 ω F (rr) = 2ω2r2 sin2 θe2µ + 2 r − λ + 4µ − 2ν r r ω r r r 1 i + 4e2ν(λ − 2 − 2µ ) (8.225) r r r (rr) 2 2 2λ Fθ = 2 sin θω λθe (8.226) h µ ω2 1 ω2λ ω2ν F (rr) = 4 sin2 θ e2µr2 ωω − ω2ν + 4 r + ω2 − r − 2 r rr rr r 2 r r r 4ωω + 2µ2ω2 + ω2 + r + ω2µ − ωλ ω r r rr r r 2 2 2 + ω λrνr − 2ωνrωr − ω λrµr − 2ω νrµr + 4µrωωr 1 + e2λλ ω(ω + ωµ + ω cot θ − ων ) + e−2ν+4µr4 sin2 θω2ω2 θ θ θ θ 2 r e2ν−2µ+2λ i + (λ ν + λ cot θ + λ + λ2) (8.227) r2 sin2 θ θ θ θ θθ θ D(rr) = 8πe2ν+2λ (8.228) E(rr) = −8πe2ν+2λ (8.229)

(rθ) COEFFICIENTS

(rθ) 2ν Ar = 2e (λθ − νθ) (8.230) (rθ) 2ν Arθ = −2e (8.231) 1 A(rθ) = 2e2ν µ + − ν (8.232) θ r r r (rθ) 2 2 2µ A = −2 sin θr ωθωre (8.233) (rθ) Bθ = −i(mω + σ) (8.234) (rθ) B = 2iλθ(σ + ωm) (8.235) 1 C(rθ) = 2e2ν µ + + ν (8.236) θ r r r m D(rθ) = − (σ + mω) (8.237) r sin θ (rθ) Drθθ = 2 sin θω (8.238) (rθ) Drθ = sin θ(−2ωλθ + ωθ + 2ω cot θ − 2ωνθ) (8.239) ω D(rθ) = sin θ − 2ων − 2 − 2ωµ + ω (8.240) θθ r r r r 2ων D(rθ) = 2 sin θ θ − ω λ − ω cot θµ + 2ωλ ν + 2ωµ ν (8.241) θ r r θ r θ r r θ 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 161

2µ−2ν 2 2 + ωrµθ − ω cot θνr − 2ωνrθ + e sin θr ωθωωr − νθωr 3 ω cot θ − ω ν + ω + ω cot θ − (8.242) θ r rθ 2 r r 2m D(rθ) = (rσµ + ωm + rωmµ + 2σ) (8.243) r sin θ r r e2ν−2µm E(rθ) = −i sin θ mω2 + ωσ − (8.244) θθ r2 sin2 θ (rθ) h Eθ = i sin θ 2ωλθ(mω + σ) − ω cot θ(mω + σ) me2ν−2µ i + (cot θ − 2λ ) (8.245) r2 sin2 θ θ (rθ) h −2λ+2ν 2 2 2 2 2 E = 2e (1 + r µrr + rνr + r µrνr − r λrµr − rλr + 2r µr + 4rµr) im × + r2(σ2 + ω2m2 + 2ωσm) r2 sin θ m2 i + e2ν−2µ 2λ cot θ + 2λ ν + 2λ2 + 2λ − θ θ θ θ θθ sin2 θ 8πime2ν(P − ) − (8.246) sin θ (rθ) 2µ 2 2 2 2 2 2ν−2µ Fr = 2e sin θ(2µθr ω − r ω λθ + λθe 2 2 2 2 2ν−2µ 2 − r ω νθ + 2r cot θω + νθe + r θωωθ) (rθ) 2ν 2µ 2 2 2 Frθ = 2(−e + e r sin θω ) (8.247) (rθ) 2µ 2 Fθ = 2e r sin θω(rωr + µrrω − rωνr + ω) (8.248) h1 1 1 F (rθ) = 4r2 sin2 θe2µ cot θ + µ + cot θµ − λ − cot θν − µ ν − ν r r θ r r θ r θ r rθ 1 1 + µ µ + µ − λ µ + λ ν + e2µ−2ν sin2 θr2ω ω ω2 + ω ω (8.249) θ r rθ θ r θ r 2 θ r 2 θ r 1 i + ω − λ ω + ω µ + 2ω cot θ + ω + ω µ − ν ω − ω ν ω r θ θ r r θ r rθ θ r θ r θ r 162 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

(rφ) COEFFICIENTS

(rφ) 2ν Ar = −2me (8.250) h 1 i A(rφ) = i 2e2νm − ν + µ − (σ + mω)r2 sin2 θω e2µ (8.251) r r r r (rφ) 2 Bθ = sin θωθ (8.252) (rφ) 2 2 2 2 2 2µ−2λ B = [(−4µrr ωr + r λrωr + r ωrνr − 4rωr − r ωrr) sin θe 2 2 + ωm + σm − 2 sin θλθωθ] 16πe2ν+2µr2 sin2 θ(P + )(Ω − ω) − (8.253) e2ν − e2µr2 sin2 θ(Ω − ω)2 h 2 i C(rφ) = i 2µ + 2ν + me2ν − (σ + mω)r2 sin2 θω e2µ (8.254) r r r r (rφ) Drθ = i sin θ(mω − σ) (8.255) (rφ) −2ν+2µ 2 2 2 −2ν+2µ 2 2 Dθ = i sin θ e r sin θmω ωr + e r sin θωσωr + 2σµr 2σ − 2mων + mω + (8.256) r r r (rφ) D = im sin θ(2ωrµθ + 2ωr cot θ − λθωr + ωrθ − ωrνθ) (8.257) e2ν−2µ sin θ E(rφ) = − (8.258) θθθ r2 e2ν−2µ sin θ E(rφ) + cot θ + λ − ν + 2µ + e−2ν+2µr2 sin2 θωω (8.259) θθ r2 θ θ θ θ h µ ν λ E(rφ) = sin θ 2e−2λ+2ν − 4 r − µ − 2µ2 − r2 − µ ν − r + r + λ µ θ r rr r r r r r r r 2 − (mω − σ)σ + sin θωωθ(cot θ − 2λθ) e2ν−2µ + cot θ(2µ − ν − 3λ + cot θ) r2 θ θ θ m2 + + 1 − 4λ µ + r sin2 θe−2λ+2µ sin2 θ θ θ 2 2 i × rνrωωr − 4ωωr − 4µrωωr − rωr + rλrωωr − r ωωrr n P e2ν + e2µr2 sin2 θ(ωP + Ω)(Ω − ω) o − 8πe2ν sin θ 2 + − 3P (8.260) e2ν − e2µr2 sin2 θ(Ω − ω)2 h m2e2ν−2µ i E(rφ) = (ν + λ − 2 cot θ − 2µ ) − (mω + σ)m sin θω (8.261) θ θ θ r2 sin θ θ (rφ) 2 2 2µ 2ν Fr = −2i(r sin θωσe − me ) (8.262) (rφ) 2 2µ 4 2 −2ν+2µ 2 2 F = i sin θe {r sin θe ω ωr(mω + σ) + 2r[rω m(1 − νr + µr)

+ (mω + σ)rωr]} (8.263) q A(rφ) = −frac4e3ν+2µr2 sin2 θ(Ω − ω) e2ν − e2µr2 sin2 θ(Ω − ω) (8.264)

(θφ) COEFFICIENTS

(θφ) Aθ = −im (8.265) 1 A(θφ) = i[m(µ − ν + cot θ) − r2 sin2 θe2µ−2νω (mωσ)] (8.266) θ θ 2 θ 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 163

1 B(θφ) = − sin2 θr2ω e−2ν+2µ−2λ (8.267) r 2 θ B(θφ) = r2 sin2 θe2µ−2ν−2λ 1 ω 1 1 × ω ν − µ ω − θ − ω + λ ω (8.268) 2 θ r r θ r 2 rθ 2 r θ (θφ) Cθ = −im (8.269) 1 C(θφ) = i(mµ − λ m + m cot θ + sin2 θσr2ω e−2ν+2µ θ θ 2 θ 1 + sin2 θmr2ωω e−2ν+2µ) (8.270) 2 θ 1 D(θφ) − i sin θmr2ω e−2ν+2µ−2λ (8.271) r 2 r 1 D(θφ) = i sin θe−2ν(mω − σ) (8.272) θθ 2 1 D(θφ) = i sin θe−2ν θ 2 2 2 × [sin θωr ωθ(mω + σ) − 2mωνθ + 2σµθ + (mω + σ) cot θ] (8.273) h 1 σm 1 3 D(θφ) = im sin θe−2ν µ ω − ω ν − + λ ω + cot θω θ θ 2 θ θ 2 sin2 θ 2 θ θ 2 θ 1 ωm2 i + ω − + rω e2µ−2λ(1 + rµ ) 2 θθ 2 sin2 θ r r 8πie2µr2 sin θ(P + )(Ω − ω) + (8.274) e2ν − e2µr2 sin2 θ(Ω − ω)2 e−2λm2 E(θφ) = (8.275) r 2 sin θ 1 E(θφ) = sinθe−2λ (8.276) rθθ 2 1 1 E(θφ) − sin θe−2λ µ − sin2 θr2ωω e−2ν+2µ − cot θ (8.277) rθ θ 2 θ 2 1 E(θφ) = sin θe−2λ(ν − λ ) (8.278) θθ 2 r r n 1 1 E(θφ) = sin θe−2λ λ µ + λ cot θ − µ + r2 sin2 θe2µ−2ν θ r θ 2 r rθ 2 h 1 i o × (ν + λ − 2µ − )ωω − ω ω − ωω − cot θν − µ ν (8.279) r r r r θ r θ rθ r θ r 1 E(θφ) = − m sin θe−2λ 2 mν λ m × (mr2ωω e−2ν+2µ + σr2ω e−2ν+2µ − r + r ) (8.280) r r sin2 θ sin2 θ (θφ) 2 2 −2ν+2µ Fθ = −i sin θωr σe (8.281) (θφ) h 2 2 −2ν+2µ 2 F = i m(νθ + λθ) + r sin θe ω (cot θ + mµθ − mνθ) 1 i + r4 sin4 θe4µ−4νω ω2(σ + mω) (8.282) 2 θ 2 ν+4µ (θφ) 2imr sin θe (Ω − ω) B = q (8.283) e2ν − e2µr2 sin2 θ(Ω − ω)2 2 2 ν+4µ (θφ) 2r sin θe (Ω − ω) Cθ = −q (8.284) e2ν − e2µr2 sin2 θ(Ω − ω)2 164 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

(−) COEFFICIENTS

(−) Aθθ = −1 (8.285) (−) Aθ = cot θ + 2µθ − 2νθ (8.286) m2 A(−) = − sin2 θr4ω2e−2λ−2ν+4µ + 2 sin2 θe−2ν+2µr2ω2 + ) r θ sin2 θ 16πe2ν+4µr4 sin2 θω(Ω − ω)( + P ) + (8.287) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] (−) Cθθ = −1 (8.288) (−) Cθ = 2µθ + cot θ − 2λθ (8.289) m2 C(−) = − sin2 θr4ω2e−2λ−2ν+4µ (8.290) sin2 θ r (−) 2 −2ν+2µ−2λ Drθ = − sin θr ωre (8.291) (−) −2ν Dθθθ = sin θe ω (8.292) (−) −2ν −2ν Dθθ = − sin θe (2µθω + 2ωνθ − e ω cot θ) (8.293) m2ω ω mσ D(−) = sin θe−2ν 4ων µ − − − 2 − 2ων − 2ν ω θ θ θ sin2 θ sin2 θ sin2 θ θθ θ θ 2 4 2 4µ−2ν−2λ + 2ωθµθ − 2µθω cot θ + 3ωθ cot θ + sin θr ωωr e 2 2 2 −2ν+2µ 2µ−2λ + 2r sin θωωθ e + 2rωre (rµr + 1) + ωθθ 16πr2(e2ν − e2µr2 sin2 θω2)(P + )(Ω − ω) + (8.294) sin θ[e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] 2me−2ν D(−) = (σ + mω)(cot θ + µ ) (8.295) sin θ θ 2ime−2λ E(−) = (cot θ + µ ) (8.296) r sin θ θ 2ime−2λ E(−) = − (8.297) rθ sin θ h 2m i E(−) = ie−2λ sin θr2ω e−2ν+2µ(σ + mω) + (λ − ν ) (8.298) θ r sin θ r r 2ime−2λ E(−) = (8.299) sin θ 2 2 −2ν+2µ × (r sin θωθωre − cot θλr + νr cot θ + νrµθ + µrθ − µθλr) (−) 2 4 −2λ−2ν+4µ Fr = −2 sin θr ωωre (8.300) (−) 2 2 2 −2ν+2µ Fθθ = r sin θω e (8.301) h 3 i F (−) = 2 λ + ν + ( cot θ − ν + µ )r2 sin2 θe2µ−2νω2 (8.302) θ θ θ 2 θ θ (−) 2 2 2µ−2ν F = 2r sin θe ω[(µθθω + 2µθωθ − ω − ωνθθ + ωθθ − 2νθωθ) mσ ωm2 + cot θ(3ω + µ ω − ων ) − − θ θ θ sin2 θ 2 sin2 θ 1 + r2 sin2 θωe2µ−2ν(ω2 + r2ω2e2µ−2λ)] θ 2 r 8.6 Perturbation Equations in Hartle and Sharp’s Formalism 165

e2µr2 sin2 θ(Ω − ω) − 16πe2µr2 3P − [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] × {e2ν[P (Ω − ω) − 2Ω] + e2µr2 sin2 θ[ωΩ(Ω − 2ω) − P ω3]} (8.303) 4µ+3ν 2 (−) 4e r (Ω − ω) Bθ = q (8.304) sin2 θ[e2ν − e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 4µ+3ν 2 (−) 4ime r (Ω − ω) C = q (8.305) [e2ν + e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 8πe4µr4 sin2 θ(Ω − ω)2 D(−) = (8.306) e2ν − e2µr2 sin2 θ(Ω − ω)2 h e2µ i E(−) = −8πr2e2µ 1 − (8.307) e2ν − e2µr2 sin2 θ(Ω − ω)2

(+) COEFFICIENTS

(+) −2λ+2µ Ar = −2re (rµr + 1) (8.308) (+) Aθθ = −1 (8.309) (+) Aθ = 2νθ − cot θ (8.310) m2 A(+) = + sin2 θr4ω2e4µ−2λ−2ν sin2 θ r 16πe2ν+4µr4 sin2 θω(Ω − ω)( + P ) − (8.311) [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] (+) 2µ−2ν+2µ−2λ B = −2ire (1 + µr)(σ + mω) (8.312) (+) −2λ+2µ Cr = 2re (1 + rµr) (8.313) (+) Cθθ = −1 (8.314) (+) Cθ = cot θ − 2λθ (8.315) m2 C(+) = e−2λ+2µ 16rµ − 4rλ + e2λ−2µ + 4r2µ + 4 + 4r2ν µ r r sin2 θ rr r r 2 4 2 2 −2ν+2µ 2 2 − 4λrr µr + r sin θωr e + 8r µr + 4νrr (8.316) (+) 2µ−2ν−2λ Drθ = 2r sin θe (rµrω + ω + rωr) (8.317) (+) −2ν Dθθθ = sin θe ω (8.318) (+) −2ν Dθθ = sin θe (2ωθ − 2ωνθ + 3ω cot θ) (8.319) (+) −2ν 2 2µ−2λ 2 2µ−2λ Dθ = sin θe ω cot θ − ω − 4rωνre + 3ωθ cot θ − 4r ωνrµre ωm2 − 4ων cot θ − 2ν ω − sin2 θr4ωω2e4µ−2ν−2λ − − 2ων + ω θ θ θ r sin2 θ θθ θθ 16πr2(e2ν − e2µr2 sin2 θω2)(P + )(Ω − ω) − (8.320) sin θ[e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] (+) −2ν+2µ−2λ Eθ = −ir sin θe 2 2 × (rmωωr + 2σω + 2rmµrω + 2mω + 2rσµrω + rσωr) (8.321) 166 8. The Perturbed Einstein Equations for a Rapidly Rotating Star

ime−2λ ν λ 1 E(+) = λ µ − µ ν − µ − θ + θ − r2 sin2 θω ω e−2ν+2µ (8.322) sin θ θ r r θ rθ r r 2 θ r (+) 2 −2λ+2µ Frr = −2r e (8.323) (+) −2λ+2µ Fr = 2e 2 2 2 3 2 −2ν+2µ × [r λr − 4r µr − r νr − 4r + (rωµr + 2ω + rωr)ωr sin θe ](8.324) (+) 2 2 2 −2ν+2µ Fθθ = r sin θω e − 2 (8.325) (+) 2 2 −2ν+2µ Fθ = (5ω cot θ − 2ωνθ + 4ωθ + 4µθω)ωr sin θe − 2 cot θ (8.326) (+) 2µn −2λ 2 2 2 2 2 F = 2e 2e (λrr − 1 − 4rµr − 2r µrr − 2r µr − r νrµr + λrr µr − νrr) h 1 i m2 − (mω + σ)σ + ω2m2 + 2 sin2 θν ωω r2e−2ν + e−2µ 2 θ θ sin2 θ 2 2 −2νh 2 2 2 2ν−2µ + r sin θe ωθ − ω + µθθω + 4µθωωθ + 5ω cot θωθe 2 2 2 2 2 2 + 5µθω cot θ − ω νθθ + ωωθθ − 3ω νθ cot θ + 2ω cot θ − 2ω νθµθ 2 2 2 2 2 2 2 2 2 2 + 2µθω + 2(ω + 2rµrω + 2r µrω − rω νr + rωωr − 2r ω νrµr 1 io + r2µ ωω )e−2λ+2µ − sin2 θr4ω2ω2e4µ−2ν−2λ r r 2 r e2µr2 sin2 θ(Ω − ω) − 16πe2µr2 [e2ν − e2µr2 sin2 θ(Ω − ω)2][e2ν + e2µr2 sin2 θω(Ω − ω)] × {e2ν[P (Ω − ω) − 2Ω] + e2µr2 sin2 θ[ωΩ(Ω − 2ω) − P ω3]} + − 2P (8.327) 4µ+3ν 2 (+) 4e r (Ω − ω) Bθ = − q (8.328) sin2 θ[e2ν − e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 4µ+3ν 2 (+) 4ime r (Ω − ω) C = − q (8.329) [e2ν + e2µr2 sin2 θω(Ω − ω)] e2ν − e2µr2 sin2 θ(Ω − ω)2 h e2ν i D(+) = −8πe2µr2 − 2 (8.330) e2ν − e2µr2 sin2 θ(Ω − ω)2 8πr2e2µ+2ν E(+) = − (8.331) e2ν − e2µr2 sin2 θ(Ω − ω)2

8.7 Future Developments

Once the set of equations governing an oscillating rapidly rotating neutron star has been determined, they may be integrated numerically. As we have pointed out several times, the calculations have been carried out in order to favour spectral methods as an integration technique: all perturbation quantities in the gauge we have used behave as scalars under rotation and this is ideal to generalise the method of [20] described in Section 7.4. Obviously, solving numerically the equations found in the previous section is a task that will require a lot of coding and debugging effort. We also point out that the equations, in the form we have written them, still need some simple manipulation to lower the number of derivatives with respect to θ to take 2 3 care of numerically. For example, operators like ∂θθ and even ∂θθθ still appear in the 8.7 Future Developments 167 equations: since the angular dependence of the perturbations will be expanded in the basis of associated Legendre polynomials, derivatives of this kind may be eliminated by exploiting the properties of such polynomials (Eqs. (E.4) and (E.5)). Furthermore, it may happen that a rearrangement of some coefﬁcients with the aid of the background equations (8.59)-(8.64) may be preferable for the numerical integration. We expect two main issues to arise in implementing the numerical integration of these equations.

1. It will not be possible to impose the boundary condition ∆P = 0 on the surface of the star with a single analytic formula as in the non-rotating or slowly rotating case; this is due to the fact that with rapid rotation the surface of the star is not determined only by the value of the stellar radius, but it also depends on the polar angle. Thus, one does not have a simple equation ∆P (RNS) = 0 where all that is needed is the NS radius RNS — as for non-rotating or slowly rotating stars — but rather a 2D constraint of the form

∆P (r, θ) = 0 (8.332) surface where the surface is given by the numerically provided background solution.

2. A Gibbs phenomenon is expected to appear at the stellar surface, where there is a pressure discontinuity. This phenomenon arises when a function (or its derivative) with a discontinuity is approximated by a spectral series truncated at its N-th term; the sum of the truncated series overshoots the true value of the functions at the edge of the discontinuity by a ﬁxed amount (approximately 0.089 times the height of the jump, if one uses the Fourier series as a spectral series), independent of N. The Gibbs phenomenon carries along Gibbs oscillations: the sum of the truncated series oscillates with a wavelength which is O(1/N). The cure to this problem is developed in [262] and consists in reverting to multi-domain spectral methods.

Both problems will require an important upgrade of the codes written by the Rome I Gravitational Wave group for slowly rotating stars in order to be solved. The group plans on implementing the equations developed in this thesis and on unravelling the two issues discussed.

Conclusions and Final Remarks

In this thesis we have reported on our work on the modelling of two kinds of compact gravitational wave sources:

1. black hole-neutron star binaries, in Part I, and

2. rapidly rotating neutron stars, in Part II.

In the case of black hole-neutron star binaries, we improved an existing approximate model, the afﬁne model, by (see Chapter4 and [21])

• replacing, in the original formulation of the model, the Newtonian self-gravity potential of the neutron star with a pseudo-relativistic potential derived from the relativistic stellar structure equations and by

• extending it to the use of any barotropic equation of state.

The equations of the model in this new extended formulation were coded and have the advantage of allowing a quick computation of the orbital tidal disruption radius rtide — a quantity which is fundamental to discriminate possible short gamma-ray burst progenitors — and the exploration of a parameter space consisting of the binary mass ratio, the black hole spin, the neutron star mass and its equation of state. A comparison of the results yielded by our approach and recent relativistic quasi- equilibrium results was possible in the case of a Schwarzschild black hole and a polytropic neutron star; we obtained an excellent agreement up to big values of the mass ratio MNS/MBH (which depend on the neutron star compactness MNS/RNS), beyond which our model starts to underestimate rtide (Section 5.3). This happens because we assume that the centre of mass of the star moves on along a BH geodesic, a condition which is better satisfied by binaries with larger mass ratios. Our improved model agrees with relativistic calculations much better than the affine model in its original formulation with Newtonian self-gravity. Given the very good results of this test, in the rest of Chapter5 we focused on determining, in the quasi-equilibrium approximation, the orbital separation at which a NS is tidally disrupted (rtide) by a black hole tidal field for several BH-NS binaries in order to evaluate with our model the role played by (1) the binary mass ratio, (2) the black hole spin, (3) the equation of state of the NS fluid and (4) the neutron star mass. We found that the scenario of the formation of a massive accretion disk leading to a short gamma-ray burst is favoured by (keeping everything else fixed):

• high values of the mass ratio MNS/MBH

• high values of the BH spin

169 170 Conclusions and Final Remarks

• less compact neutron star models and

• low mass neutron stars. Moreover, we observed that coincident measurements of the masses and spins involved in a BH-NS coalescence leading to a short gamma-ray burst and of the burst itself could be useful in pinning down the neutron star equation of state. In Section 5.4.3 we showed that it may be possible to infer the neutron star radius, and thus to constrain the neutron star equation of state, from measurements of the gravitational radiation emitted by a BH-NS binary in which the neutron star is tidally disrupted. We estimated the accuracy up to which the neutron star radius may be measured to be ∼ 1 km if the binary mass ratio is 0.2. In Chapter6 we abandoned the quasi-equilibrium approximation and compared the waveform emitted by a BH-NS binary with a ﬁnite size neutron star with the waveform emitted by the same binary in the approximation of point-mass constituents. We observed that • the phase difference between the two waveforms grows as the mass ratio grows, i.e. as one move towards the equal mass case;

• the amplitude difference grows with the mass ratio;

• the phase and amplitude differences are greater if the black hole spin is larger. In all four cases the differences in the phase and the amplitude between the two waveforms develop in the last few cycles. To draw quantitative conclusions in this highly non-linear regime one will need fully-relativistic approaches; in the meantime the qualitative behaviours we observed indicate that the problem is well worth further investigation. All our conclusions in Part I of the thesis agree with the insight provided by the ﬁrst preliminary results obtained by the numerical relativity community [126, 139, 143, 158, 145, 141] and shed light on the role of the neutron star equation of state and on the chance of constraining it with gravitational wave observations.

——— ♦ ———

In the case of rapidly rotating stars, in Chapter8 we developed the perturbation equations which will eventually allow the calculation of their quasi-normal mode spectrum, a problem which has not been solved yet and which is important to solve since rapidly rotating neutron stars have been observed and are promising gravitational wave sources. The calculation of the perturbed equations we performed is intended as a ﬁrst step in extending the perturbative approach of [20] for slowly rotating stars which is based on: • studying the oscillations in the frequency domain

• expanding the perturbations in circular harmonics, so that for an assigned value of the complex frequency σ and of the azimuthal index m, the perturbed equations to solve are a 2D-system of linear, partial differential equations in the radial distance r and in the polar angle θ

• using the standing wave boundary condition

• integrating the perturbed equations with spectral methods. Conclusions and Final Remarks 171

The choice of extending this approach to the rapidly rotating case follows from the fact that, to our knowledge, all other existing perturbative or fully relativistic approaches for rotating neutron stars make use of simplifying approximations or are limited to restricted cases (see discussion in Section 7.3.2). The rather lengthy set of linearly perturbed Einstein equations we obtained for the rapidly rotating neutron star problem is an original result of this thesis and is reported in Section 8.6.2. The Rome I Gravitational Wave group is planning on implementing numerically these equations, dedicating special attention to numerical problems which are expected to arise at the stellar surface.

Appendix A

Noise Power Spectral Densities of Interferometric GW Detectors

−1 Detector fs/Hz f0/Hz S0/Hz Sh(x)/S0

−46 −30 −1 1−x2+0.5x4 GEO 40 150 1.0 · 10 (3.4x) + 34x + 20 1+0.5x2 LIGO 40 150 9.0 · 10−46 (4.49x)−56 + 0.16x−4.52 + 0.52 + 0.32x2 TAMA 75 400 7.5 · 10−46 x−5 + 13x−1 + 9(1 + x2) Virgo 20 500 3.2 · 10−46 (7.8x)−5 + 2x−1 + 0.63 + x2

−49 −4.14 −2 1−x2+0.5x4 Adv LIGO 20 215 1.0 · 10 x − 5x + 111 1+0.5x2 −52 α β ET 10 200 1.5 · 10 x + a0x + 2 3 4 5 6 1+b1x+b2x +b3x +b4x +b5x +b6x +b0 2 3 4 1+c1x+c2x +c3x +c4x

Table A.1. Noise power spectral densities Sh(f) of various terrestrial interferometers in operation and under construction: GEO600, (Initial) LIGO, TAMA, Virgo, Advanced LIGO (Adv LIGO), and Einstein Telescope (ET). For each detector the noise power spectral density is given in terms of a dimensionless frequency x = f/f0 and rises steeply above a lower cutoff frequency fs. The parameters in the ET design sensitivity curve are α = −4.1, β = −0.69, a0 = 186, b0 = 233, b1 = 31, b2 = −65, b3 = 52, b4 = −42, b5 = 10, b6 = 12, c1 = 14, c2 = −37, c3 = 19, c4 = 27 p [5]. The noise amplitude spectra Sh(f) are shown in the top panel of Figure A.1.

173 174 A. Noise Power Spectral Densities of Interferometric GW Detectors

Terrestrial GW Detectors

1e-21 TAMA GEO LIGO 1e-22

] Virgo -1/2 Adv LIGO 1e-23 (f) [Hz h S

√ ET 1e-24

1e-25 10 100 1000 10000 f [Hz]

LISA 1e-15

1e-16

] 1e-17 -1/2

1e-18 (f) [Hz h S

√ 1e-19

1e-20

1e-21 0.0001 0.001 0.01 0.1 1 f [Hz]

√ Figure A.1. Top panel: noise amplitude spectrum ( Sh, see caption in Table A.1) of ground- based GW detectors in operation (continuous lines) and under construction or design (dashed lines). Bottom panel: noise amplitude spectrum of LISA; the data appearing is available at http://www.srl.caltech.edu/ shane/sensitivity/. Appendix B

Relativistic Stellar Structure

B.1 The TOV Equations

The metric for a static and spherically symmetric matter distribution may be written as follows:

2 2ν(r) 2 2λ(r) 2 2 2 2 2 ds(0) = −e dt + e dr + r (dθ + sin θdφ ) . (B.1)

Inside the star, the two potentials ν and λ may be determined by solving the Einstein equations coupled to the equations of hydrostatic equilibrium which follow from Eq. (7.3). Assuming the star is composed of a perfect ﬂuid, the energy-momentum tensor may be written as usual

T µν = ( + P )uµuν + P gµν , (B.2) where uµ is the 4-velocity ﬁeld and the energy density and pressure P are assumed to be isotropic. Having deﬁned

r¯ def Z m(¯r) = 4πr2dr (B.3) 0 as the gravitational mass contained in a sphere of radius r¯, the relevant Einstein equations are: dν 1 dP = − (B.4) dr + P dr dP ( + P )(m(r) + 4πr3P ) = (B.5) dr r(r − 2M) 2m(r) e2λ = 1 − . (B.6) r These equations go under the name of Tolman-Oppenheimer-Volkoff (TOV) equations and they determine the hydrostatic equilibrium of a relativistic non-rotating spherical star [37, 36]. The system formed by the two ordinary differential equations (B.4) and (B.5), supplied with Eq. (B.3), is closed and therefore integrable once one speciﬁes an equation of state for the stellar ﬂuid which relates the pressure to the energy density. In order to obtain a stellar model, one has to choose a value for the pressure at the centre of the star; the star

175 176 B. Relativistic Stellar Structure radius R is found by exploiting the boundary condition P (R) = 0 and it then yields the total gravitational mass which is M = m(R). Finally, the boundary conditions for ν and λ may be determined with the aid of the Birkhoff theorem which states that the exterior solution of any spherically symmetric and static matter distribution is given by the Schwarzschild metric; therefore one obtains 2M e2ν| ≡ e−2λ| ≡ 1 − . (B.7) r=R r=R R

B.2 Calculating ΦTOV and VcTOV

One of the two limits in the integral definition of ΦTOV given in Eq. (4.6) is infinite. In performing this integral numerically with r as the independent variable, one would like to avoid this infinity, since an alternative to dealing with infinities numerically, if possible, is usually welcome. In other words we would like to be able to calculate ΦTOV(r) as

Z r 03 ( + P )(mTOV + 4πr P ) 0 ¯ ΦTOV(r) = 0 0 dr + ΦTOV(0) (B.8) 0 ρr (r − 2mTOV) ¯ where ΦTOV(0) is an integration constant we have to determine. In order to do so, we consider the deﬁnition of ΦTOV (Eq. (4.6)) and proceed as follows:

Z r 03 Z R 03 0 ( + P )(mTOV + 4πr P ) 0 ( + P )(mTOV + 4πr P ) ΦTOV(r) = dr 0 0 + dr 0 0 R ρr (r − 2mTOV) ∞ ρr (r − 2mTOV) Z r 03 Z R 0 ( + P )(mTOV + 4πr P ) 0 MTOV = dr 0 0 + dr 0 0 R ρr (r − 2mTOV) ∞ r (r − 2MTOV) Z r 03 Z R 0 ( + P )(mTOV + 4πr P ) 1 0 1 1 = dr 0 0 + dr − R ρr (r − 2mTOV) 2 ∞ r − 2MTOV r Z r 03 0 ( + P )(mTOV + 4πr P ) 1 2MTOV = dr 0 0 + ln 1 − , (B.9) R ρr (r − 2mTOV) 2 R where R is the star radius and MTOV = mTOV(R) is the gravitational mass. In the second step we use the fact that physically /ρ → 1 and P/ρ → 0 as P → 0. This therefore yields 1 2M Φ (R) = ln 1 − TOV , (B.10) TOV 2 R which matches the usual TOV boundary condition for ν shown in Eq. (B.7), as one may have expected from the relationship between ΦTOV and ν (see page 63). It follows that the boundary condition is given by 1 2M Φ¯ (0) = −Φ˜ (R) + ln 1 − TOV , (B.11) TOV TOV 2 R ˜ where ΦTOV(R) is the integral Z R ( + P )(m + 4πr3P ) dr TOV (B.12) 0 ρr(r − 2mTOV) performed numerically with the boundary condition Φ(0)˜ = 0. B.2 Calculating ΦTOV and VbTOV 177

We must therefore integrate

Φ ( + P )(m + 4πr3P ) TOV = TOV (B.13) dr ρr(r − 2mTOV) once from the star centre to the star surface1 with vanishing boundary condition at the star centre; this determines the true boundary condition (Eq. (B.11)) which we may use as explained in Eq. (B.8) to calculate ΦTOV at any r and VbTOV from

Z R Z R 3 3 dΦTOV 3 ( + P )(mTOV + 4πr P ) VbTOV = −4π drr = 4π drr . (B.14) 0 dr 0 ρr(r − 2mTOV) The integration routines may be tested in the Newtonian limit and with the use of a polytropic EOS; in this case we must obtain numerically the same value yielded by the formula given in Eq. (3.119), i.e.

3 M 2 Vb = . (B.15) n − 5 R Of course, in this limit, instead of the logarithm in the boundary condition we must use its Newtonian version −M/R.

1 One actually integrates from r0 1 to R.

Appendix C

Equations of Motion for BH-NS Coalescing Binaries

We provide the explicit expressions for the Equations (6.31)-(6.40) which determine the dynamical evolution of BH-NS binaries in the approach described in Chapter6. In order to recover a post-Newtonian point-mass evolution to order 2 PN with 2.5 PN gravitational wave dissipative terms, it is sufﬁcient to consider only the four dynamical equations (C.1)-(C.4) and to set ai = 0 in Eq. (C.3)

dr P = r dt µ ! 1 3ν − 1 P 2 1 GM [2 (3 + ν) P + 2νP ] + P 2 + φ P − Tot r r 2 c2µ3 r r2 r 2 c2µr !2 3 1 − 5ν + 5ν2 P 2 + P 2 + φ P 8 c4µ5 r r2 r " ! ! # 1 GM P 2 P 2 + Tot 4 5 − 20ν − 3ν2 P 2 + φ P − 4ν2P P 2 + φ − 16ν2P 3 8 c4µ3r r r2 r r r r2 r 1 G2M 2 + Tot [2 (5 + 8ν) P + 6νP ] 2 c4µr2 r r G Pr 3 MNS 3 MBH + 2 2 r 2 + JBH + 2 + JNS c r P 2 2 MBH 2 MNS 2 φ Pr + r2 ! 8 G2 P 2 − 2P 2 + 6 φ (C.1) 15 c5νr2 r r2 dφ P = φ dt r2µ ! 1 3ν − 1 P 2 GM (3 + ν) P + P 2 + φ P − Tot φ 2 c2µ3r2 r r2 φ c2µr3

179 180 C. Equations of Motion for BH-NS Coalescing Binaries

!2 3 1 − 5ν + 5ν2 P 2 + P 2 + φ P 8 c4µ5r2 r r2 φ " ! # 1 GM 5 − 20ν − 3ν2 P 2 4ν2P 2P + Tot 4 P 2 + φ P − r φ 8 c4µ3r r2 r r2 φ r2 G2M 2 (5 + 8ν) P + Tot φ c4µr4 G Pφ 3 MNS 3 MBH + 2 4 r 2 + JBH + 2 + JNS c r P 2 2 MBH 2 MNS 2 φ Pr + r2 8 G2P P − r φ (C.2) 3 c5νr4 dP P 2 GµM r = φ − Tot dt r3µ r2 ! " ! # 1 3ν − 1 P 2 1 GM P 2 + P 2 + φ P 2 − Tot (3 + ν) P 2 + φ + νP 2 2 c2µ3r3 r r2 φ 2 c2µr2 r r2 r 2 2 2 GMTot (3 + ν) P G µM − φ + Tot c2µr4 c2r3 !2 3 1 − 5ν + 5ν2 P 2 + P 2 + φ P 2 8 c4µ5r3 r r2 φ

 2 !2 2 !  1 GMTot 2 2 Pφ 2 2 2 Pφ 2 4 + +  5 − 20ν − 3ν P + − 2ν P P + − 3ν P  8 c4µ3r2 r r2 r r r2 r " ! # 1 GM P 2 + Tot 4 5 − 20ν − 3ν2 P 2 + φ P 2 − 4ν2P 2P 2 8 c4µ3r4 r r2 φ r φ " ! # G2M 2 P 2 + Tot (5 + 8ν) P 2 + φ + 3νP 2 c4µr3 r r2 r G2M 2 (5 + 8ν) P 2 3 G3 (1 + 3ν) µM 3 + Tot φ − Tot c4µr5 4 c4r4 s P 2 2G 2 φ 3 MNS 3 MBH + 2 3 Pr + 2 2 + JBH + 2 + JNS c r r 2 MBH 2 MNS 2 G 3 MNS 3 MBH Pφ + 2 5 2 + JBH + 2 + JNS r c r 2 MBH 2 MNS P 2 2 φ Pr + r2 GJ J − 6 BH NS c2r4 3 2KMBH 2KMBH + Mc c − cos(Ψ − φ)2 a2 + c − sin(Ψ − φ)2 a2 2r 11 r5 1 22 r5 2 2KM + c + BH a2 33 r5 3 ! 8 G2P P 2 1 − r φ − GνM 3 (C.3) 3 c5r4 νr 5 Tot 181

! dP 8 G2P Gν2M 3 P 2 φ = − φ 2 Tot + 2 φ − P 2 (C.4) dt 5 c5νr3 r r2 r

da1 Pa1 = RNS (C.5) dt Mc

da2 Pa2 = RNS (C.6) dt Mc

da3 Pa3 = RNS (C.7) dt Mc " # dP 1 V R2 Π a1 2 2 b 3 ˜ NS = Mc a1(Λ + Ω ) − 2a2ΛΩ + RNSa1A1 + − c11a1 (C.8) dt 2 Mc Mc a1 " # dP 1 V R2 Π a2 2 2 b 3 ˜ NS = Mc a2(Λ + Ω ) − 2a1ΛΩ + RNSa2A2 + − c22a2 (C.9) dt 2 Mc Mc a2 " # dP 1 V R2 Π a3 b 3 ˜ NS = Mc RNSa3A3 + − c33a3 (C.10) dt 2 Mc Mc a3 s dΨ M = BH (C.11) dt r3 dλ = Λ (C.12) dt dP λ ≡ C˙ = 0 (C.13) dt dϕ = Ω (C.14) dt dPϕ ˙ Mc 2 2 ≡ JNS = 2 c12 a1 − a2 (C.15) dt RNS

The tidal tensor components c11, c22, c33 and c12 are given in Eqs. (3.88)-(3.91), while A˜1, A˜2 and A˜3 are deﬁned by Eq. (3.114).

Appendix D

Symmetries of the Afﬁne Model Lagrangian and Conserved Quantities

The internal Lagrangian of the afﬁne model Eq. (3.83) is invariant under the action of two groups [14]: 1. the O(3) group corresponding to rotations of the Euclidean background space in which the isolated stellar ellipsoid model is situated and

2. the O(3) group corresponding to rotations of the reference state. In the language of the afﬁne model prescription — see Eq. (3.64) — this means that one may freely rotate the centre of mass coordinates ξi as well as the reference state centre of ˆi mass coordinates ξ . Altogether this gives the internal Lagrangian LI an invariance under an

O(4) ≡ O(3) ⊗ O(3) (D.1) group. The two O(3) transformations are of the form

i i j ξ → Ojξ (D.2) ˆi i ˆj ξ → Objξ , (D.3) where O = O−T and Ob = Ob −T are two orthogonal matrices. Their combined effect on the deformation matrix q is thus of form

qia → OijOˆabqjb (D.4) and the corresponding inﬁnitesimal transformation is given by

δqia = (oijqja − qiboˆba) δλ , (D.5)

T T where λ is a continuous variation parameter, and o = −o and ob = −ob are independent antisymmetric matrices generating the 6-dimensional algebra of O(4). Given this premise, we may apply Noether’s theorem which states that if a system has a continuous symmetry property — i.e. if a Lagrangian is left unchanged by a generalised

183 184 D. Symmetries of the Afﬁne Model Lagrangian and Conserved Quantities coordinate transformation — then there are corresponding conserved quantities [263]. In the present case, the components of the generator k are given by

i i k i l kj = okqj − ql oˆj , (D.6) and the corresponding conserved quantity is

i i q˙jkj . (D.7)

This yields d d oa (q ˙aqc) − (q ˙aqa) o c = 0 . (D.8) c dt b b dt b c bb

Since this expression holds for any antisymmetric matrices o and ob, the antisymmetric parts of the coefﬁcients must vanish and therefore d (q ˙aqc − q˙cqa) = 0 (D.9) dt b b b b d (q ˙aqa − q˙aqa) = 0 . (D.10) dt b c c b These expressions are connected to the spin angular momentum and the vorticity vectors (see [14] for extensive details). Thus, the spin angular momentum and the vorticity are conserved quantities for the Lagrangian LI. However, in the presence of a black hole, the dynamics of the stellar ellipsoid is also determined by the Lagrangian tidal term LT which changes the symmetry properties we have just discussed. The O(3) symmetry group acting on the ξba coordinates is preserved, while the other rotation group is not. In the case of tidal interactions, therefore,

1. the spin angular momentum is no longer a conserved quantity, so that the tidal ﬁeld rotates the stellar ellipsoid (also see Section 3.1.1), and

2. the vorticity is still a conserved quantity.

The conservation of the vorticity vector would cease if we were to add internal viscous dissipation terms to the star ﬂuid Lagrangian LT + LI. Appendix E

Essential Mathematical Toolkit

E.1 Multipole Expansion of the Metric Perturbation

A spherical symmetric background metric in standard coordinates takes the form 1 ds2 = −f(r)dt2 + dr2 + r2dΩ2 . (E.1) f(r) Because of the spherical symmetry of this background, it is convenient to think of the full spacetime as the product of a Lorentzian two-dimensional manifold M 2 associated with the coordinates (t, r), and the 2-sphere of unit radius S2 associated with the polar and azimuthal angles (θ, φ):

2 A B 2 a b ds = gABdx dx + r Ωabdx dx , (E.2)

2 2 where Ωab ≡ diag(1, sin θ) is the metric on S , and where we are using upper case indices (A, B, ...) to represent the coordinates in M 2 and lower case indices (a, b, ...) for coordinates in S2. We shall show how to perform a multipole expansion of the angular part of a perturbation hµν of this background; this kind of expansions are based on spherical harmonics and are typical of perturbation theory in General Relativity. Scalar spherical harmonics are denoted with Ylm; the polar index l is a non-negative integer, while the azimuthal index m is an integer which obeys the constraint |m| ≤ l and the spherical harmonics are deﬁned as s (2l + 1) (l − |m|)! Y lm(θ, φ) def= P lm(cos θ)eimφ m ≥ 0 4π (l + |m|)! def Y lm(θ, φ) = (−1)mY¯ l|m| m < 0 (E.3) where the bar indicates complex conjugation and P lm is the associated Legendre polynomial which solves the equation

LPlm(cos θ) = −l(l + 1)Plm(cos θ) , (E.4) with the Laplacian operator on the sphere given by

∂2 ∂ ∂2 ∂2 ∂ m2 ≡ + cot θ + φ ≡ + cot θ − . (E.5) L ∂θ2 ∂θ sin2 θ ∂θ2 ∂θ sin2 θ

185 186 E. Essential Mathematical Toolkit

Associated Legendre polynomials are deﬁned as

|m| def d P lm(x) = (−1)m(1 − x2)(|m|/2) P l(x) , (E.6) dx|m| where P l(x)’s denote Legendre polynomials, a set of orthogonal1 polynomials deﬁned on the interval x ∈ [−1, 1] by

l l def (−1) d P l(x) = (1 − x2)l . (E.7) 2ll! dxl The parity operator, which reverses the three Cartesian spatial axes, i.e. in spherical coordinates

(t, r, θ, φ) −→ (t, r, π − θ, π + φ) , (E.8) acts on the spherical harmonics in the following way:

Y lm(π − θ, π + φ) = (−)lY lm(θ, φ) , (E.9) so that spherical harmonics are said to have parity (−)l. The behaviour of the components of a symmetric (0, 2) tensor — as is the case of hµν — under rotations on the 2-sphere depends on what component one considers. Working in (t, r, θ, φ) coordinates, the nature of the components is the following:

SSVV    ∗ SVV    (E.10) ∗ ∗ TT  ∗ ∗ ∗ T where S, V and T stand for “scalar”, “vector” and “tensor” components. To expand the components of a tensor, we thus need 1. scalar harmonics for the S components

2. vector harmonics on the 2-sphere for the V components

3. tensor harmonics on the 2-sphere for the T components. Let us therefore introduce scalar, vector and tensor harmonics (see for example [264]). Scalar harmonics are the usual functions Y lm. Vector harmonics, on the other hand, come in two different types: even parity vector harmonics, which transform as (−)l under the action of the parity operator, and odd parity vector harmonics, which instead transform as (−)l+1. The even vector harmonics are simply deﬁned as the gradient of the scalar harmonics on the sphere

lm def lm lm Ya = ∇aY ≡ Y,a , (E.11) where ∇a denotes the covariant derivative on the 2-sphere, while the odd vector harmonics are constructed as curls on the sphere:

lm def b lm cb lm Sa = −a ∇bY = −acΩ ∇bY , (E.12)

1 1With respect to the inner product R f(x)g(x)dx. −1 E.1 Multipole Expansion of the Metric Perturbation 187

1/2 where ab is the Levi-Civita tensor on the 2-sphere (θφ = −φθ = Ω = sin θ)[264]. Similarly, one may deﬁne tensor harmonics of even and odd type [264]. Even tensor harmonics can be constructed in two ways: by multiplying the scalar harmonics with the lm angular metric Ωab, or by taking a second covariant derivative of the Y . This was the choice of Regge and Wheeler [212]; however, these functions do not form an orthonormal lm set so that instead of ∇a∇bY it is better to use the “Zerilli-Mathews” tensor harmonics which are deﬁned as

def 1 Zlm = ∇ ∇ Y lm + l(l + 1)Ω Y lm . (E.13) ab a b 2 ab Odd parity tensor harmonics, on the other hand, may be constructed only in one way:

def 1 Slm = ∇ Slm + ∇ Slm . (E.14) ab 2 a b b a The tensor harmonics built this way satisfy the following orthogonality relations:

ab Ω Ωab = 2δll0 δmm0 (E.15) ab l(l + 1)(l − 1)(l + 2) Z Z = δ 0 δ 0 (E.16) ab 2 ll mm ab l(l + 1)(l − 1)(l + 2) S S = − δ 0 δ 0 (E.17) ab 2 ll mm ab Ω Zab = 0 (E.18) ab Ω Sab = 0 (E.19) ab Z Sab = 0 (E.20) where the contraction of the indices is intended as, for example, Z ab ab ¯ lm lm Ω Zab ≡ Ω Y Zab d sin θdφ . (E.21)

Having deﬁned the vector and tensor harmonics, the perturbed metric is expanded in multipoles, and separated into its even sector, given by

lm lm lm h = HABY (E.22) even AB lm lm lm h = HA Y,b (E.23) even Ab lm 2 lm lm lm lm h = r K ΩabY + G Zab , (E.24) even ab and its odd sector, given by hlm = 0 (E.25) odd AB lm lm lm h = hA Sb (E.26) odd Ab lm lm lm h = h Sab , (E.27) odd ad

lm lm lm lm lm lm where the coefficients HAB,HA ,K ,G , hA and h are in general functions of r and t. The decomposition of hµν in multipoles thus naturally separates the metric perturbations into two classes with respect to parity: even parity/polar/spheroidal perturbations and odd 188 E. Essential Mathematical Toolkit parity/axial/toroidal perturbations (see Sections 7.2 and 7.3). Quantities belonging to the former class transform as (−1)l under the action of the parity operator, whereas for the latter class they transform as (−)l+1. Also notice that, since Y 00 is constant, both vector and tensor harmonics vanish for l = 0. On the other hand, for l = 1 vector harmonics do not vanish, but the tensor harmonics do. This means vector harmonics are non-zero only for l ≥ 1, while tensor harmonics are non-zero only if l ≥ 2. The scalar mode with l = 0 may be interpreted as a variation in the mass of the Schwarzschild spacetime, while the even mode with l = 1 is just gauge freedom and can be removed under a suitable transformation. On the other hand, the odd l = 1 mode can be interpreted as an infinitesimal angular momentum contribution, i.e. a “Kerr” mode, or, more precisely, a first order “Hartle-Thorne” [265] mode. When dealing with gravitational waves one is interested in modes with l ≥ 2. We may also picture the scalar, vector and tensor harmonics needed to expand symmetric tensors differently (see for example [213]): we may build a basis for tensors on M 2 × S2 by acting on Y lm with different combinations of the 1-form basis of M 2, i.e. µ et = (1, 0, 0, 0) (E.28) µ er = (0, 1, 0, 0) , (E.29) and of the covariant derivative operator and angular momentum operator on the 2-sphere, whose covariant components are

∇µ = (0, 0, ∇a) (E.30) a bc Lµ = (0, 0, irab∇ ) ≡= (0, 0, irabΩ ∇c) , (E.31) which act as a basis on S2 and which have opposite parity (1 and −1 respectively). A ﬁrst set of tensors we may build this way is the following:

lm lm lm Scalar: [etetY ][eterY ][ererY ], (E.32) lm lm lm lm Vector: [et∇Y ][etLY ][er∇Y ][erLY ], (E.33) Tensor: [∇∇Y lm][L∇Y lm][LLY lm]. (E.34) where we explicitly indicated the behaviour of each subset under rotations on the 2-sphere. The tensors may then be combined in a convenient manner to form the basis:

(0) lm alm = [etetY ] (E.35) lm alm = [ererY ] (E.36) (1) 1 2 lm alm = 2 [eterY ] (E.37) (0) 1 2 lm blm = 2 C1r[et∇Y ] (E.38) (0) 1 2 lm clm = 2 C1[etLY ] (E.39) 1 lm blm = 2 2 C1r[er∇Y ] (E.40) 1 lm clm = 2 2 C1[erLY ] (E.41) 1 lm 1 lm d = 2 2 C r [L∇Y ] + [e LY ] (E.42) lm 2 r r − 1 flm = 2 2 C2(elm + hlm) (E.43) − 1 2 glm = −2 2 C1 (elm − hlm) (E.44) (E.45) E.1 Multipole Expansion of the Metric Perturbation 189 where 2 e = r2 [∇∇Y lm] + [e ∇Y lm] (E.46) lm r r lm lm hlm = [LLY ] + r[er∇Y ] (E.47) and

def − 1 C1 = [l(l + 1)] 2 def − 1 C2 = [l(l + 1)(l − 1)(l + 2)] 2 . (E.48)

The ﬁrst three tensors are orthogonal to the 2-sphere (they expand the scalar components of a rank (0, 2) tensor), while the last three are tangent to it (they expand the tensorial components of a rank (0, 2) tensor); the remaining tensors expand the vector components of a rank (0, 2) tensor. The tensors we just deﬁned form an orthonormal set with respect to the inner product Z Z def µν∗ ∗ µλ νρ (R, P) = R PµνdΩ = RλρPµνη η dΩ , (E.49)

2 2 2 where ηµν = diag(−1, 1, r , r sin θ) is the ﬂat metric in spherical coordinates. Notice that with this inner product normality is deﬁned to be −1 and not 1. A generic symmetric tensor T may thus be expanded as

T = Tax + Tpol , (E.50) where (given the parity of et, er, ∇ and L)

ax Xh (0) (0) i T = Qlm clm + Qlmclm + Dlmdlm , (E.51) lm pol Xh (0) (0) (1) (1) T = Alm alm + Alm alm + Almalm + lm (0) (0) i + Blm blm + Blmblm + Glmglm + Flmflm (E.52) are the axial/odd and polar/even part of the tensor.

E.1.1 Explicit Expressions of the Tensor Basis

2 2 2 2 The tensor spherical harmonics normalized with ηµν = diag(−1, 1, r , r sin θ) are :

Y lm 0 0 0   (0)  0 0 0 0 alm =   (E.53)  0 0 0 0 0 0 0 0

2All the expressions that follow coincide with the ﬁrst (more implicit) description we gave of the multipole expansion of a symmetric tensor. 190 E. Essential Mathematical Toolkit

 0 Y lm 0 0 1  lm  (1) Y 0 0 0 alm = √   (E.54) 2  0 0 0 0 0 0 0 0

0 0 0 0  lm  0 Y 0 0 alm =   (E.55) 0 0 0 0 0 0 0 0

 ∂Y lm ∂Y lm  0 0 ∂θ ∂φ   (0) C1r  0 0 0 0  b = √  lm  (E.56) lm 2  ∂Y 0 0 0   ∂θ  ∂Y lm ∂φ 0 0 0

0 0 0 0   ∂Y lm ∂Y lm  C1r 0 0 ∂θ ∂φ  blm = √  lm  (E.57) 2 0 ∂Y 0 0   ∂θ  ∂Y lm 0 ∂φ 0 0

 1 ∂Y lm ∂Y lm  0 0 − sin θ ∂φ sin θ ∂θ   (0) iC1r  0 0 0 0  c = √  lm  (E.58) lm 2 − 1 ∂Y 0 0 0   sin θ ∂φ  ∂Y lm sin θ ∂θ 0 0 0

0 0 0 0   1 ∂Y lm ∂Y lm  iC1r 0 0 − sin θ ∂φ sin θ ∂θ  clm = √  lm  (E.59) 2 0 − 1 ∂Y 0 0   sin θ ∂φ  ∂Y lm 0 sin θ ∂θ 0 0

0 0 0 0  iC r2   2 0 0 0 0  dlm = √  1 lm lm (E.60) 2 0 0 sin θ X − sin θW  0 0 − sin θW lm − sin θXlm

0 0 0 0  C r2   2 0 0 0 0  flm = √  lm lm  (E.61) 2 0 0 W X  0 0 Xlm − sin2 θW lm

0 0 0 0  r2   0 0 0 0  glm = √  lm  (E.62) 2 0 0 Y 0  0 0 0 sin2 θY lm E.2 Pullback and Pushforward 191

where C1 ed C2 are given by Eq. (E.48) and ∂ ∂ Xlm(θ, φ) def= 2 − cot θ Y lm(θ, φ) ∂φ ∂θ " 2 2 # def ∂ ∂ 1 ∂ W lm(θ, φ) = − cot θ − Y lm(θ, φ) (E.63) ∂2θ ∂θ sin2 θ ∂2φ " # ∂2 = l(l + 1) + 2 . ∂θ2

E.1.2 Multipole Expansion of a Vector Working with multipole expansions, it is useful to be able to expand a generic 4-vector ﬁeld ξ. This is straightforward to do with the vector harmonics we previously introduced in Eqs. (E.11)-(E.12):

ξ = ξpol + ξax (E.64) where ∂ M (t, r) ∂ ξµ ≡ M (t, r)Y ,M (t, r)Y ,M (t, r) Y , 2 Y (E.65) pol 0 lm 1 lm 2 ∂θ lm sin2 θ ∂φ lm 1 ∂ M (r, t) ∂ ξµ ≡ 0, 0, −M (r, t) Y , 3 Y . (E.66) ax 3 sin θ ∂φ lm sin θ ∂θ lm and therefore ∂ ∂ ξ ≡ M (t, r)Y ,M (t, r)Y ,M (t, r) Y ,M (t, r) Y (E.67) µ,pol 0 lm 1 lm 2 ∂θ lm 2 ∂φ lm 1 ∂ ∂ ξ ≡ 0, 0, −M (r, t) Y ,M (r, t) sin θ Y . (E.68) µ,ax 3 sin θ ∂φ lm 3 ∂θ lm Notice that if ξ is the generator of a gauge transformation, one has one arbitrary function (M3) to impose a condition on the axial part of the metric perturbation and three arbitrary functions (M0,M1,M2) to impose three conditions on the polar part of the metric perturbation (Section 7.1.1).

E.2 Pullback and Pushforward

Let φ : M → N be a map from a manifold M to a manifold N (possibly of different dimensions) and f : N → R a function; the pullback φ∗f of f by φ is deﬁned as def φ∗f = (f ◦ φ) (E.69) and acts as

φ∗f : M → R . (E.70) If V (p) is a vector at a point p on M, the pushforward vector (φ∗V ) at the point φ(p) on N is instead given by

∗ def (φ V )(f) = V (φ∗f) , (E.71) 192 E. Essential Mathematical Toolkit that is, it is the action of V on the pullback of f by φ. If we call yµ(xν) the coordinates that n m map N(M) to R (R ) we have ∂yν ∂yν (φ∗V )µ∂ f = V µ∂ (φ f) = V µ∂ (f ◦ φ) = V µ ∂ f ⇒ (φ∗)ν = . µ µ ∗ µ ∂xµ ν µ ∂xµ (E.72)

In a similar manner, given the 1-form ω on N, the pullback φ∗ω is deﬁned by ∗ (φ∗ω)(V ) = ω(φ V ) (E.73) and we have ∂yν (φ ) ν = . (E.74) ∗ µ ∂xµ Therefore the pullback of a (0, l) tensor is given by

(1) (l) ∗ (1) ∗ (l) (φ∗T )(V , ..., V ) = T (φ V , ..., φ V ) (E.75) and the pushforward of a (k, 0) tensor is given by

∗ (1) (l) (1) (l) (φ T )(ω , ..., ω ) = T (φ∗ω , ..., φ∗ω ) . (E.76)

∗ The matrix representations of φ∗ and φ extend to higher rank as ∂yν1 ∂yνl (φ∗T )µ1...µ = ... Tν1...ν (E.77) l ∂xµ1 ∂xµl l ∂yµ1 ∂yµl (φ∗T )µ1...µl = ... T ν1...νl . (E.78) ∂xν1 ∂xνl If φ : M → N is invertible (and φ and φ−1 are both smooth), then it defines a diffeomorphism between M and N (which in this case are the same manifold). We can then use φ and φ−1 to move tensors from M to N and therefore define the pushforward (k, l) T µ1...µk M and pullback of arbitrary tensors. For a tensor field ν1...νl on we define the pushforward

∗ (1) (k) (1) (l) (1) (k) −1 ∗ (1) −1 ∗ (l) (φ T )(ω , ..., ω ,V , ..., V ) = T (φ∗ω , ..., φ∗ω , [φ ] V , ..., [φ ] V ) .

(E.79) In components this becomes

∂yµ1 ∂yµk ∂xβ1 ∂xβk ∗ µ1...µk µ1...µk (φ T )ν ...ν = ...... Tν ...ν . (E.80) 1 l ∂xα1 ∂xαk ∂yν1 ∂yνl 1 l This shows that diffeomorphisms and coordinate transformations are two different ways of doing exactly the same thing. Diffeomorphisms are “active coordinate transformations”, while traditional coordinate transformations are “passive”. Given an m-dimensional man- µ m ifold M with coordinate functions x : M → R , to change coordinates we can either µ m simply introduce new functions y : M → R (“keep the manifold ﬁxed and change the coordinate maps”), or we can introduce a diffeomorphism φ : M → M after which µ m the coordinates would just be the pullbacks (φ∗x) : M → R (“move the points on the manifold and then evaluate the coordinates of the new points”). In this sense Eq. (E.80) is the tensor transformation law. E.3 Lie Derivative 193

E.3 Lie Derivative

Since a diffeomorphism allows us to pullback and pushforward arbitrary tensors, it provides a way of comparing tensors at different points on a manifold [266]. Given a family of diffeomorphisms from M to M parametrized by t, φ t represents the flow down the integral curves3 and the associated vector field is the generator of the diffeomorphism. Given a vector field ξµ(x) then, we have a family of diffeomorphisms parametrized by t and for each t we may define the change of a tensor field along an integral curve as

def ∆ T µ1...µk (p) = φ [T µ1...µk (φ (p))] − T µ1...µk (p),, t ν1...νl t∗ ν1...νl t ν1...νl (E.81) i.e. the difference between the value of the tensor at some point p and its value at φ(p) pulled back to p. The Lie derivative of a tensor along a vector ﬁeld ξµ is then deﬁned as

µ ...µ ! ∆tT 1 k µ1...µk def ν1...νl LξTν ...ν (p) = lim . (E.82) 1 l t→0 t

Therefore the Lie derivative is a map from (k, l) tensor fields to (k, l) tensor fields which is manifestly coordinate independent. It is also defined to be linear and to obey the Leibniz rule (just as the conventional derivative):

Lξ(S ⊗ T ) = (LξS) ⊗ T + S ⊗ (LξT ) . (E.83)

It follows that the Lie derivative of a scalar is given by

µ Lξf = ξ ∂µf . (E.84)

Denoting the inner product with ( , ), this may be restated with differential forms as

∂f ∂ L f = (df(P ), ξ(P )) = ( dxµ, ξν ) = ξµ∂ f . (E.85) ξ ∂xµ ∂xν µ Using the commutator [ , ], for vectors one has instead

µ µ ν µ ν µ Lξv = [ξ, v] ≡ ξ ∂νv − v ∂νξ , (E.86) or equivalently

∂ ∂vµ ∂ξµ ∂ L v = [ξ, v] = (ξ(vµ) − v(ξµ)) = ξν − vν , (E.87) ξ ∂xµ ∂xν ∂xν ∂xµ whereas for 1-forms we have4

ν ν Lξvµ = ξ ∂νvµ + v ∂νξµ . (E.88)

3Given a vector field ξµ(x), its integral curves xµ(t) are defined as the solutions of dxµ/dt = ξµ 4The definition of Lie derivatives used should always be checked. For example the definitions we just gave are adopted by Friedman and Schutz in [73] and [74]. Chandrasekhar and Lebovitz use a different definition of the Lie derivative for vectors and 1-forms [267] which coincides with the one adopted by Lynden-Bell and Ostriker [268]. 194 E. Essential Mathematical Toolkit

We can also deﬁne the Lie derivative of an arbitrary tensor ﬁeld

L T µ1...µk = ξγ∂ T µ1...µk ξ ν1...νl γ ν1...νl − ∂ (ξµ1 )T γ...µk − ... − ∂ (ξµk )T µ1...γ γ ν1...νl γ ν1...νl + ∂ (ξγ)T µ1...µk + ... + ∂ (ξγ)T µ1...µk . ν1 γ...νl νl ν1...γ (E.89) Finally, Eqs. (E.86), (E.88) and (E.89) are completely covariant and it turns out that we can write in coordinate invariant fashion

L T µ1...µk = ξγ∇ T µ1...µk ξ ν1...νl γ ν1...νl − ∇ (ξµ1 )T γ...µk − ... − ∇ (ξµk )T µ1...γ γ ν1...νl γ ν1...νl + ∇ (ξγ)T µ1...µk + ... + ∇ (ξγ)T µ1...µk ν1 γ...νl νl ν1...γ (E.90) where ∇µ represents any symmetric (torsion-free) covariant derivative (including, of course, one derived from a metric). All terms involving connection coefﬁcients in Eq. (E.90) cancel out leaving only Eq. (E.89) which is hence already covariant. Appendix F

Quasi-Normal Mode Classiﬁcation

The terminology for the classiﬁcation of the QNMs follows the one used by Cowling in his classic paper [269] where the modes are labelled according to the prevailing force restoring a displaced ﬂuid element to its equilibrium position. We give a short description of the mode families; with ν we indicate values of the frequency of the oscillation which are typical for neutron stars and with τ values of its decay time which are typical for neutron stars [270, 225].

1. Polar (or spheroidal) ﬂuid modes (slowly damped and analogous to Newtonian ﬂuid pulsations)

• p(ressure)-modes: the radial component of the fluid displacement vector is significantly larger than the tangential component and the oscillations are thus almost radial; the pressure is the restoring force and it undergoes significant fluctuations when these modes are excited; there is an infinite set of these modes (ν > 4 kHz, τ > 1 s). Detailed data for the frequencies and damping times (due to gravitational radiation) of the p1-mode for various equations of state can be found in [43, 271]. • f(undamental)-modes: these modes are associated with bulk motions of the star; f-modes eigenfunctions have no nodes inside the star, and they grow towards the surface; they can thus be considered as the 0-th p-mode (ν ∼ 2 kHz, τ < 1 s). Detailed data for the frequencies and damping times (due to gravitational wave emission) of the f-mode for various equations of state are found in [227, 43, 271], whereas estimates for the damping times due to viscosity are available in [272, 273]. • g(ravity)-modes: they are associated with fluid displacement vectors whose tangential components are dominant in the fluid motion and hence with convective fluid motions; gravity is the restoring force since it tends to smooth out material inhomogeneities along equipotential level-surfaces; these modes exist only in non-isentropic stars or in stars with a composition gradient or a first order phase transition (ν < 0.5 kHz, τ > 5 s); in the barotropic case they are degenerate at zero frequency since the star is isentropic and therefore does not allow convective motions. For an extensive discussion about g-modes in relativistic stars see [274, 275], for a study on g-mode instability to gravitational radiation reaction in rotating stars refer to [276] and for very recent results on

195 196 F. Quasi-Normal Mode Classiﬁcation

g-modes of rapidly rotating stratiﬁed neutron stars in the general relativistic Cowling approximation see [277].

Note that

. . . < νg2 < νg1 < νf < νp1 < νp2 < . . . (F.1)

and that, once again, the fundamental mode (which is quasi radial) may be viewed as the lowest pressure mode.

2. Axial (or toroidal) and hybrid ﬂuid modes

• r(otation)-modes: these are present only with rotation and are degenerate at zero frequency in non-rotating stars for which ﬂuid modes are only polar; these modes are located mainly at the surface of the star are quasi-toroidal because rotation modiﬁes slightly the toroidal displacements; for these modes the frequency in the frame corotating with the star and in the inertial frame is given by

2mΩ ν(r) = r l(l + 1) 2 ν(i) = mΩ − 1 + O(Ω2) r l(l + 1)

so that all l ≥ 2 r-modes are generically unstable to the emission of gravitational radiation, due to the CFS mechanism1 [71, 72, 73, 74]; more precisely r-modes are a class of inertial modes (modes whose main restoring force is the Coriolis force and which are primarily velocity perturbations) that reduce to the classical axial r-modes in the Newtonian limit and are known in geophysics as Rossby waves; the frequencies range from zero to a thousand Hertz.

3. Polar and axial spacetime modes

• w(ave)-modes, which are analogous to black hole QNMs and consequently have no Newtonian analogue; they are decoupled from ﬂuid modes but can be trapped by the gravitational potential well and gradually decay; they are subject to gravitational redshift (ν > 6 kHz, τ ∼ 0.1 ms).

Note that, because of their different nature, ﬂuid perturbations propagate at a different speed from spacetime perturbations: the former travel at the speed of sound cs, whereas as the later move at the speed of light c. The family of spacetime modes (w-modes) may be additionally subdivided into three distinct categories that are similar both for polar and axial oscillations:

• trapped modes, which exist only for very compact, probably unrealistic, stars with R ≤ 3M; their frequencies can be of the order of a few hundred Hz to a few kHz and their damping times are slower than those of the other w-modes [202];

1The instability is active as long as its growth-time is shorter than the damping-time due to the viscosity of neutron star matter 197

• curvature modes which were identiﬁed after the discovery of trapped-modes [278]; these modes, usually considered to be the standard w-modes, are associated with the spacetime curvature and exist for all relativistic stars; their main characteristic is the very short damping time (∼ 0.1 ms); the ﬁrst w-modes have frequencies of about 6 ÷ 13 kHz which increase with the order of the mode;

• interface modes, or wII -modes, that are extremely rapidly damped modes (< 0.1 ms) whose frequency may vary from 2 kHz to 15 kHz [279].

Additionally other types of oscillations can be possible if one improves the description of the star interior: for example, considering a NS model with a fluid core, a solid crust and a surface fluid ocean, due to the non-zero shear, modes with a non-trivial behaviour appear in the crust (s-modes) or at the solid-fluid interfaces (i-modes). The efficiency of a mode in the GW emission depends on many factors, for example on the temperature of the star [231] and hence on the stage of evolution it is in. In the case of old NSs, the most important modes (i.e. efficient) are the fluid l = 2 fundamental and first pressure modes; higher order modes, (g, p, s and i) do not involve large mass motions and thus are not expected to be strong sources of gravitational radiation. The most relevant information carried by the GWs about the astrophysical properties of these sources is concentrated in the high frequency band, above ∼ 1 kHz. This means that the f-mode frequency lies in the region of the spectrum where the sensitivity of GW interferometric detectors like Virgo and LIGO begins to be strongly limited by shot noise. The construction of GW detectors with good sensitivity in the kHz region, such as the Einstein Telescope2, is hence crucial in order to observe NS oscillations and to start the era of GW astronomy of compact objects.

2http://www.et-gw.eu.

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