GRAVITATIONAL WAVES FROM COMPACT BINARY SYSTEMS
TABLE OF CONTENTS 1 INTRODUCTION 1 1 1 GRAVITATIONAL WAVE ASTRONOMY 1 1 1 1 GRAVITATIONAL WAVES 1 1 1 2 BENE TS OF THEORETICAL MODELING OF GRAVITATIONAL WAVE SOURCES 12 1 2 NEUTRON STARS 13 1 2 1 POTENTIAL GRAVITATIONAL WAVE MEASUREMENTS 19 1 3 EXTREME MASS RATIO INSPIRALS 24 1 3 1 MODELLING EXTREME MASS RATIO INSPIRALS 27 2 CONSTRAINING NEUTRON STAR TIDAL LOVE NUMBERS WITH GRAVITATIONAL WAVE DETECTORS 33 2 1 BACKGROUND AND MOTIVATION 33 2 2 TIDAL INTERACTIONS IN COMPACT BINARIES 35 2 3 TIDAL LOVE NUMBER 36 2 4 E ECT ON GRAVITATIONAL WAVE SIGNAL 37 2 5 ACCURACY OF MODEL 41 2 6 MEASURING THE LOVE NUMBER 43 3 TIDAL LOVE NUMBERS OF NEUTRON STARS 46 3 1 INTRODUCTION AND MOTIVATION 46 3 2 DE NITION OF THE LOVE NUMBER 49 3 3 CALCULATION OF THE LOVE NUMBER 52 3 3 1 EQUILIBRIUM CON GURATION 52 3 3 2 STATIC LINEARIZED PERTURBATIONS DUE TO AN EXTERNAL TIDAL ELD 52 3 3 3 NEWTONIAN LIMIT 55 3 4 RESULTS AND DISCUSSION 55 4 TWO TIMESCALE ANALYSIS OF EXTREME MASS RATIO INSPIRALS IN KERR I ORBITAL MOTION 60 4 1 INTRODUCTION AND SUMMARY 61 4 1 1 BACKGROUND AND MOTIVATION 61 4 1 2 METHODS OF COMPUTING ORBITAL MOTION AND WAVEFORMS 64 4 1 3 THE TWO TIMESCALE EXPANSION METHOD 69 4 1 4 ORBITAL MOTION 71 4 1 5 TWO TIMESCALE EXPANSION OF THE EINSTEIN EQUATIONS AND ADI ABATIC WAVEFORMS 75 4 1 6 ORGANIZATION OF THIS PAPER 79 4 1 7 NOTATION AND CONVENTIONS 79 4 2 EXTREME MASS RATIO INSPIRALS IN KERR FORMULATED USING ACTION ANGLE VARIABLES 80 4 2 1 REVIEW OF ACTION ANGLE VARIABLES IN GEOMETRIC HAMILTONIAN MECHANICS 81 4 2 2 GENERALIZED ACTION ANGLE VARIABLES FOR NON COMPACT LEVEL SETS 85 4 2 3 APPLICATION TO BOUND GEODESIC MOTION IN KERR 87 4 2 4 EXPLICIT EXPRESSIONS IN TERMS OF BOYER LINDQUIST COORDINATES 90 4 2 5 APPLICATION TO SLOW INSPIRAL MOTION IN KERR 94 4 2 6 RESCALED VARIABLES AND INCORPORATION OF BACKREACTION ON THE BLACK HOLE 97 4 2 7 CONSERVATIVE AND DISSIPATIVE PIECES OF THE FORCING TERMS 105 4 3 A GENERAL WEAKLY PERTURBED DYNAMICAL SYSTEM 109 4 4 SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 112 4 4 1 OVERVIEW 112 4 4 2 FOURIER EXPANSIONS OF THE PERTURBING FORCES 113 4 4 3 TWO TIMESCALE ANSATZ FOR THE SOLUTION 115 4 4 4 RESULTS OF THE TWO TIMESCALE ANALYSIS 117 4 4 5 DERIVATION 122 4 5 SYSTEMS WITH SEVERAL DEGREES OF FREEDOM SUBJECT TO NON RESONANT FORCING 128 4 5 1 OVERVIEW 128 4 5 2 FOURIER EXPANSIONS OF PERTURBING FORCES 129 4 5 3 THE NO RESONANCE ASSUMPTION 131 4 5 4 TWO TIMESCALE ANSATZ FOR THE SOLUTION 133 4 5 5 RESULTS OF THE TWO TIMESCALE ANALYSIS 135 4 5 6 DERIVATION 141 4 6 NUMERICAL INTEGRATION OF AN ILLUSTRATIVE EXAMPLE 150 4 7 DISCUSSION 153 4 7 1 CONSISTENCY AND UNIQUENESS OF APPROXIMATION SCHEME 153 4 7 2 E ECTS OF CONSERVATIVE AND DISSIPATIVE PIECES OF THE SELF FORCE 155 4 7 3 THE RADIATIVE APPROXIMATION 157 4 7 4 UTILITY OF ADIABATIC APPROXIMATION FOR DETECTION OF GRAVITA TIONAL WAVE SIGNALS 160 4 8 CONCLUSIONS 165 4 8 1 ACKNOWLEDGEMENTS 166 4 9 APPENDIX EXPLICIT EXPRESSIONS FOR THE COE CIENTS IN THE ACTION ANGLE EQUATIONS OF MOTION 166 4 10 APPENDIX COMPARISON WITH TREATMENT OF KEVORKIAN AND COLE 168 5 EVOLUTION OF THE CARTER CONSTANT FOR INSPIRALS INTO A BLACK HOLE E ECT OF THE BLACK HOLE QUADRUPOLE 169 5 1 INTRODUCTION AND SUMMARY 170 5 2 E ECT OF AN AXISYMMETRIC MASS QUADRUPOLE ON THE CONSERVATIVE OR BITAL DYNAMICS 177 5 2 1 CONSERVATIVE ORBITAL DYNAMICS IN A BOYER LINDQUIST LIKE CO ORDINATE SYSTEM 178 5 2 2 E ECTS LINEAR IN SPIN ON THE CONSERVATIVE ORBITAL DYNAMICS 181 5 3 E ECTS LINEAR IN QUADRUPOLE AND QUADRATIC IN SPIN ON THE EVOLUTION OF THE CONSTANTS OF MOTION 182 5 3 1 EVALUATION OF THE RADIATION REACTION FORCE 182 5 3 2 INSTANTANEOUS UXES 186 5 3 3 ALTERNATIVE SET OF CONSTANTS OF THE MOTION 189 5 3 4 TIME AVERAGED UXES 193 5 4 APPLICATION TO BLACK HOLES 196 5 4 1 QUALITATIVE DISCUSSION OF RESULTS 196 5 4 2 COMPARISON WITH PREVIOUS RESULTS 196 5 5 NON EXISTENCE OF A CARTER TYPE CONSTANT FOR HIGHER MULTIPOLES 199 5 5 1 SEPARABILITY ANALYSIS 200 5 5 2 DERIVATION OF NON EXISTENCE OF ADDITIONAL CONSTANTS OF THE MOTION 205 5 6 CONCLUSION 214 5 7 ACKNOWLEDGMENTS 215 5 8 APPENDIX TIME VARIATION OF QUADRUPOLE ORDER OF MAGNITUDE ESTI MATES 215 5 8 1 ESTIMATE OF THE SCALING FOR THE NONSPINNING CASE 216 5 8 2 ESTIMATE OF THE SCALING FOR THE SPINNING CASE 216 5 8 3 APPLICATION TO KERR INSPIRALS 217 5 9 APPENDIX COMPUTATION OF TIME AVERAGED UXES 218 5 9 1 AVERAGING METHOD THAT PARALLELS FULLY RELATIVISTIC AVERAGING 218 5 9 2 AVERAGING METHOD USING THE EXPLICIT PARAMETERIZATION OF NEWTONIAN ORBITS 222 6 CARTER CONSTANT EVOLUTION IN THE ADIABATIC REGIME 225 6 1 INTRODUCTION 225 6 2 THE KERR SPACETIME 228 6 2 1 TEUKOLSKY PERTURBATION FORMALISM 228 6 2 2 BOYER LINDQUIST COORDINATES 234 6 3 VACUUM EQUATIONS 240 6 3 1 SEPARATION OF VARIABLES 240 6 3 2 BASIS OF MODES 243 6 3 3 IN UP OUT AND DOWN MODES 245 6 3 4 RELATIONS BETWEEN THE SCATTERING AND TRANSMISSION COE CIENTS249 6 3 5 MODE EXPANSION OF THE POTENTIAL FOR THE METRIC PERTURBATION 254 6 4 CONSTRUCTION OF THE GREEN S FUNCTIONS FOR THE TEUKOLSKY VARIABLES 257 6 4 1 FORMULA FOR THE RETARDED GREEN S FUNCTION 257 6 4 2 DERIVATION 258 6 4 3 CONSTRUCTION OF THE RADIATIVE GREEN S FUNCTION FOR THE TEUKOLSKY VARIABLES 264 6 4 4 THE INHOMOGENEOUS POTENTIALS 268 6 4 5 HARMONIC DECOMPOSITION OF THE AMPLITUDES 270 6 4 6 DERIVATION 274 6 4 7 EXPRESSIONS FOR THE TIME DERIVATIVES OF THE CONSTANTS OF MOTION277 6 5 COMPARISON OF THE NOTATION TO OTHER CONVENTIONS 287 6 6 TWO TIMESCALE METHOD 288 6 6 1 ANALYSIS OF THE O EINSTEIN EQUATION 288 6 7 APPENDIX SKETCH OF THE DERIVATION OF THE TEUKOLSKY STAROBINSKY IDENTITIES 306 CHAPTER 1 INTRODUCTION 1 1 GRAVITATIONAL WAVE ASTRONOMY TO INTRODUCE THE WORK IN THIS THESIS ON THEORETICAL TOOLS FOR ANALYZING SOURCES OF GRAVITATIONAL WAVES WE RST GIVE SOME WELL KNOWN BACKGROUND MATERIAL THAT CAN BE FOUND IN TEXTBOOKS SUCH AS 1 2 1 1 1 GRAVITATIONAL WAVES ALMOST A CENTURY AGO EINSTEIN S THEORY OF GENERAL RELATIVITY RADICALLY CHANGED THE NOTION OF SPACE AND TIME THEY ARE NOT JUST THE STAGE UPON WHICH EVENTS OCCUR INSTEAD SPACETIME IS A DYNAMIC ENTITY WHICH CURVES EXPANDS AND CONTRACTS AROUND MATTER AND ENERGY THE THEORY OF GENERAL RELATIVITY PREDICTS THE EXISTENCE OF TRANSVERSE DISTORTIONS OF SPACETIME CURVATURE CALLED GRAVITATIONAL WAVES WHICH AS A CONSEQUENCE OF CAUSALITY PROPAGATE AT THE SPEED OF LIGHT SINCE INFORMATION ABOUT THE CHANGING GRAVITATIONAL ELD CANNOT REACH DISTANT OBSERVERS FASTER THAN LIGHT HOWEVER SCIENTISTS AT THE TIME CONCLUDED THAT GRAVITATIONAL RADIATION WOULD NOT BE OBSERVABLE BECAUSE IT IS PRODUCED ONLY IN EXTREMELY SMALL QUANTITIES IN EVERYDAY AND ATOMIC PROCESSES FOR EXAMPLE THE PROBABILITY FOR AN ELECTRON TRANSITION OF ENERGY E 1EV BETWEEN TWO ATOMIC STATES BY GRAVITATIONAL RADIATION RATHER THAN ELECTROMAGNETIC RADIATION IS OF ORDER THE RATIO OF THE SQUARE OF THE DIMENSIONLESS COUPLING CONSTANTS FOR THE GRAVITATIONAL AND ELECTROMAGNETIC INTERACTIONS 1 G C5 E2 E2 C 10 54 WHICH RE ECTS HOW WEAKLY GRAVITATIONAL WAVES INTERACT WITH MATTER ELDS 1