Table of Contents
Preface to the Second Edition xix
Preface xxi
1 The Search for Gravitational Waves 1
1.1 The Importance of the Search 2
1.2 A Bit of History 3
1.3 The Practice of Gravitational Wave Detection 5
1.4 A Guide for the Reader 6
2 The Nature of Gravitational Waves 9
2.1 Waves in General Relativity 9
2.2 The Michelson-Morley Experiment 12
2.3 A Schematic Detector of Gravitational Waves 18
2.4 Description of Gravitational Waves in Terms of Force 25
3 Sources of Gravitational Waves 27
3.1 Physics of Gravitational Wave Generation 27
3.2 In the Footsteps of Heinrich Hertz? 31
3.3 Observation of Gravitational Wave Emission 34
3.4 Astronomical Sources of Gravitational Waves 37
3.4.1 Neutron star binaries 38
3.4.2 Supernovae 39
3.4.3 Pulsars 42
3.4.4 "Wagoner stars" 43
3.4.5 Black holes 44
3.4.6 Stochastic backgrounds 45
3.4.7 Discussion 46
4 Linear Systems, Signals and Noise 49
4.1 Characterizing a Time Series 49
4.1.1 The Fourier transform 50
4.1.2 Cross-correlation and autocorrelation 50
4.1.3 Convolution 52
4.1.4 The power spectrum 52
4.1.5 The Periodogram 53
4.1.6 Interpretation of power spectra 54
4.1.7 The amplitude spectral density 55
4.2 Linear Systems 55
4.2.1 Bode plots 61
4.2.2 Frequency response example 61
4.3 The Signal-to-Noise Ratio 62
4.3.1 Noise statistics 63
4.3.2 Matched templates and matched niters 65
4.3.3 SNR rules of thumb 67
4.3.4 The characteristic amplitude 68
5 Optical Readout Noise 71
5.1 Photon Shot Noise 72
5.2 Radiation Pressure Noise 75
5.3 Shot Noise in Classical and Quantum Mechanics 80
5.4 The Remarkable Precision of Interferometry 83
6 Folded Interferometer Arms 85
6.1 Herriott Delay Line 86
6.2 Beam Diameter and Mirror Diameter 88
6.3 Fabry-Perot Cavities 91
6.4 A Long Fabry-Perot Cavity 97
6.5 Hermite-Gaussian Beams 97
6.6 Scattered Light in Interferometers 99
6.7 Comparison of Fabry-Perot Cavities with Delay Lines 101
6.8 Optical Readout Noise in Folded Interferometers 101
6.9 Transfer Function of a Folded Interferometer 103
6.10 To Fold, or Not to Fold? 105
7 Thermal Noise 107
7.1 Brownian Motion 107
7.2 Brownian Motion of a Macroscopic Mass Suspended in a Dilute Gas 108
7.3 The Fluctuation-Dissipation Theorem 110
7.4 Remarks on the Fluctuation-Dissipation Theorem 112
7.5 The Quality Factor, Q 114
7.6 Thermal Noise in a Gas-Damped Pendulum 115
7.7 Dissipation from Internal Friction in Materials 117
7.8 Special Features of the Pendulum 121
7.9 Thermal Noise of the Pendulum's Internal Modes 123
8 Seismic Noise and Vibration Isolation 127
8.1 Ambient Seismic Spectrum 127
8.2 Seismometers 129
8.3 Vibration Isolators 130
8.4 Myths About Vibration Isolation 131
8.5 Isolation in an Interferometer 132
8.6 Stacks and Multiple Pendulums 135
8.7 Q: High or Low? 138
8.8 A Gravitational "Short Circuit" Around Vibration Isolators 140
8.9 Beyond Passive Isolation 141
9 Design Features of Large Interferometers 143
9.1 How Small Can We Make a Gravitational Wave Inteferometer? 143
9.2 Noise from Residual Gas 146
9.2.1 Simple model 147
9.2.2 Exact result 148
9.2.3 Implications for Interferometer Design 149
9.3 The Space-Borne Alternative 149
10 Null Instruments 151
10.1 Some Virtues of Nullity 152
10.1.1 Null hypotheses 152
10.1.2 Null experiments 152
10.1.3 Null instruments 155
10.1.4 Null features of a gravitational wave interferometer 157
10.1.5 Active null instruments 158
10.2 The Advantages of Chopping 161
10.3 The Necessity to Operate a Gravitational Wave Interferometer as an Active Null Instrument 162
10.3.1 The need to chop 163
10.3.2 The need to actively null the output 164
11 Feedback Control Systems 167
11.1 The Loop Transfer Function 169
11.2 The Closed Loop Transfer Function 170
11.3 Designing the Loop Transfer Function 172
11.4 Instability 174
11.4.1 Causes of instability 175
11.4.2 Stability tests 176
11.5 The Compensation Filter 177
11.6 Active Damping: A Servo Design Example 179
11.7 Feedback to Reduce Seismic Noise Over a Broad Band 186
11.7.1 Suspension point interferometer 186
11.7.2 Active isolation 187
12 An Interferometer as an Active Null Instrument 189
12.1 Fringe-Lock in a Non-Resonant Interferometer 189
12.2 Shot Noise in a Modulated Interferometer 195
12.3 Rejection of Laser Output Power Noise 197
12.4 Locking the Fringe 198
12.5 Fringe Lock for a Fabry-Perot Cavity 200
12.6 A Simple Interferometer with Fabry-Perot Arms 204
12.7 Beyond the Basic Interferometer 206
12.7.1 Power recycling 206
12.7.2 Signal recycling 208
12.7.3 Resonant sideband extraction 208
13 Resonant Mass Gravitational Wave Detectors 211
13.1 Does Form Follow Function? 211
13.2 The Idea of Resonant Mass Detectors 212
13.3 A Bar's Impulse Response and Transfer Function 213
13.4 Resonant Transducers 216
13.5 Thermal Noise in a Bar 219
13.6 Bandwidth of Resonant Mass Detectors 222
13.6.1 When are narrow bandwidths optimum? 222
13.6.2 Interpreting narrow-band observations 224
13.7 A Real Bar 226
13.8 Quantum Mechanical Sensitivity "Limit" 227
13.9 Beyond the Quantum "Limit"? 230
14 Detecting Gravitational Wave Signals 233
14.1 The Signal Detection Problem 233
14.2 Probability Distribution of Time Series 234
14.3 Coincidence Detection 239
14.4 Optimum Orientation 241
14.5 Local Coincidences 242
14.6 Searching for Periodic Gravitational Waves 243
14.6.1 When is a spectral peak improbably strong? 243
14.6.2 Signatures of periodic gravitational waves 244
14.6.3 Frequency noise in the source and elsewhere 248
14.7 Searching for a Stochastic Background 248
15 Gravitational Wave Astronomy 253
15.1 Gravitational Wave Source Positions 253
15.1.1 Network figure of merit 255
15.1.2 Why measure positions? 257
15.1.3 Inferences from precise positions 260
15.1.4 Temporal coincidence with non-gravitational observations 261
15.2 Interpretation of Gravitational Waveforms 262
15.2.1 Core collapse 263
15.2.2 Binary coalescences 264
15.2.3 A gravitational standard candle 265
15.2.4 Recognizing signals from black holes 266
15.3 Previous Gravitational Wave Searches 267
15.3.1 Room temperature bars 267
15.3.2 Cryogenic bars 268
15.3.3 The Strange case of Supernova 1987A 270
15.3.4 Gravitational wave searches with interferometers 272
15.3.5 Other observational upper limits 273
16 Prospects 277
16.1 A Prototype Interferometer 277
16.2 LIGO 278
16.3 Proposed Features of 4 km Interferometers 279
17 Epilogue 283
17.1 Introduction 283
17.2 Physics/Engineering Background (Chapters 4, 10, 11) 285
17.3 Prehistory of Gravitational Wave Detection (Chapter 1) 286
17.4 Gravitational Waves and their Interactions with Detectors (Chapter 2) 287
17.5 Sources of Gravitational Waves (Chapter 3) 287
17.6 Quantum Measurement Noise (Chapter 5) 288
17.7 Interferometer Configurations (Chapters 6 and 12) 289
17.8 Thermal Noise (Chapter 7) 289
17.9 Seismic Noise (Chapter 8 and Section 11.7.2) 291
17.10 Resonant Mass Detectors (Chapter 13) 294
17.11 Large Interferometers (Chapters 9 and 16) 295
17.12 Data Analysis (Chapter 14) 296
17.13 Gravitational Wave Astronomy (Chapter 15) 297
References 301
