Analysis of Gravitational-Wave Data

0521864593......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Notation and conventions......Page 12
1 Overview of the theory of gravitational radiation......Page 15
1.1.1 Linearized Einstein equations......Page 16
1.1.2 Global Poincaré transformations......Page 18
1.1.3 Gauge transformations......Page 20
1.2 Plane monochromatic gravitational waves......Page 21
1.3 Description in the TT coordinate system......Page 24
1.4 Description in the observer\'s proper reference frame......Page 27
1.4.1 Plus polarization......Page 29
1.5 Gravitational waves in the curved background......Page 31
1.6 Energy–momentum tensor for gravitational waves......Page 33
1.7 Generation of gravitational waves and radiation reaction......Page 34
2 Astrophysical sources of gravitational waves......Page 40
2.1.1 Coalescing compact binaries......Page 41
2.2 Periodic sources......Page 42
2.3 Stochastic sources......Page 43
2.4 Case study: binary systems......Page 44
2.4.1 Newtonian binary dynamics......Page 47
2.4.2 Post-Newtonian binary dynamics......Page 53
2.5 Case study: a rotating triaxial ellipsoid......Page 56
2.6 Case study: supernova explosion......Page 59
2.7 Case study: stochastic background......Page 61
3 Statistical theory of signal detection......Page 65
3.1 Random variables......Page 66
3.2 Stochastic processes......Page 70
3.3 Hypothesis testing......Page 76
3.3.1 Bayesian approach......Page 77
3.3.2 Minimax approach......Page 80
3.3.3 Neyman–Pearson approach......Page 82
3.4.1 Cameron–Martin formula......Page 85
3.4.2 Stationary noise......Page 87
3.4.3 Matched filtering......Page 89
3.5 Estimation of stochastic signals......Page 90
3.6.1 Uniformly minimum variance unbiased estimators......Page 93
3.6.2 Exponential families of probability distributions......Page 96
3.6.3 Fisher information......Page 97
3.6.4 Kullback–Leibler divergence......Page 98
3.6.5 L1-norm distance......Page 99
3.6.7 Bayesian estimation......Page 100
3.6.9 Maximum-likelihood estimation......Page 101
3.6.10 Lower bounds on the variance of estimators......Page 102
3.7.1 Karhunen–Loéve expansion......Page 104
3.7.2 Wiener processes......Page 105
3.7.3 Resampled stochastic processes......Page 107
3.7.4 Cyclostationary processes......Page 110
4.1 Sample mean and correlation function......Page 113
4.2 Power spectrum estimation......Page 115
4.2.1 Periodogram......Page 116
4.2.2 Averaging......Page 118
4.2.3 Windowing and smoothing......Page 119
4.2.5 Multitapering......Page 120
4.3 Tests for periodicity......Page 121
4.4.1 Pearson’s chi2 test......Page 123
4.4.2 Kolmogorov–Smirnov test......Page 124
4.5 Higher-order spectra......Page 125
5.1 Detectors of gravitational waves......Page 128
5.2 Doppler shift between freely falling observers......Page 129
5.3 Long-wavelength approximation......Page 136
5.4 Responses of the solar-system-based detectors......Page 138
5.4.1 LISA detector: time-delay interferometry......Page 140
5.4.2 Ground-based laser interferometric detector......Page 141
5.4.3 Ground-based resonant-bar detector......Page 143
6.1 Deterministic signals......Page 145
6.1.1 The F-statistic......Page 146
6.1.2 Signal-to-noise ratio and the Fisher matrix......Page 148
6.1.3 False alarm and detection probabilities......Page 152
6.1.4 Number of templates......Page 159
6.1.5 Suboptimal filtering......Page 161
6.2.1 Periodic signal from a spinning neutron star......Page 164
6.2.2 Chirp signal from an inspiraling compact binary......Page 172
6.2.3 Signal from a supernova explosion......Page 179
6.3 Network of detectors......Page 181
6.3.1 Chirp signal from an inspiraling compact binary......Page 183
6.3.2 Periodic signal from a spinning neutron star......Page 191
6.3.3 Detection of periodic signals by LISA......Page 194
6.4 Detection of stochastic signals......Page 198
7.1 Linear signal model......Page 206
7.1.1 Examples......Page 208
7.2.1 Covering problem......Page 211
7.2.2 The covering problem with constraints......Page 212
7.3.1 The FFT algorithm......Page 215
7.3.2 Resampling......Page 216
7.3.3 Fourier interpolation......Page 218
7.3.4 Optimization algorithms......Page 220
7.4.1 Coincidences......Page 222
7.4.2 Signal-to-noise gain......Page 223
7.4.3 Upper limits......Page 224
Appendix A The chirp waveform......Page 226
Appendix B Proof of the Neyman–Pearson lemma......Page 232
Appendix C Detector’s beam-pattern functions......Page 235
C.1 LISA detector......Page 236
C.2 Earth-based detectors......Page 239
C.2.1 Interferometric detector......Page 240
C.2.2 Resonant-bar detector......Page 241
Appendix D Response of the LISA detector to an almost monochromatic wave......Page 243
Appendix E Amplitude parameters of periodic waves......Page 247
References......Page 249
Index......Page 263