Analysis of gravitational-wave data

271 Pages·2009·1.42 MB·English

by  Piotr Jaranowski

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ANALYSIS OF GRAVITATIONAL-WAVE DATA
Research in this field has grown considerably in recent years due to the
commissioning of a world-wide network of large-scale detectors. This net-
work collects a very large amount of data that is currently being ana-
lyzedandinterpreted.Thisbookintroducesresearchersenteringthefield,
and researchers currently analyzing the data, to gravitational-wave data
analysis.
An ideal starting point for studying the issues related to current
gravitational-wave research, the book contains detailed derivations of
the basic formulae related to the detectors’ responses and maximum-
likelihood detection. These derivations are much more complete and
more pedagogical than those found in current research papers, and will
enable readers to apply general statistical concepts to the analysis of
gravitational-wave signals. It also discusses new ideas on devising the effi-
cient algorithms needed to perform data analysis.
Piotr Jaranowski is an Associate Professor in the Faculty of Physics
at the University of Bial(cid:2)ystok, Poland. He has been a visiting sci-
entist at the Max Planck Institute for Gravitational Physics and the
Friedrich Schiller University of Jena, both in Germany, and the Insti-
tut des Hautes E´tudes Scientifiques, France. He currently works in the
field of gravitational-wave data analysis and general-relativistic problem
of motion.
Andrzej Kro´lak is a Professor in the Institute of Mathematics at
the Polish Academy of Sciences, Poland. He has twice been awarded the
Second Prize by the Gravity Research Foundation (once with Bernard
Schutz). He has been a visiting scientist at the Max Planck Institute
for Gravitational Physics, Germany, and the Jet Propulsion Laboratory,
USA. His field of research is gravitational-wave data analysis and general
theory of relativity, and the phenomena predicted by this theory such as
black holes and gravitational waves.
CAMBRIDGE MONOGRAPHS
ON PARTICLE PHYSICS,
NUCLEAR PHYSICS AND COSMOLOGY
General Editors: T. Ericson, P. V. Landshoff
1. K.Winter (ed.):Neutrino Physics
2. J.F.Donoghue,E.Golowich and B.R.Holstein:Dynamics of the Standard Model
3. E.Leader and E.Predazzi:An Introduction to Gauge Theories and Modern Particle
Physics, Volume 1: Electroweak Interactions, the ‘New Particles’ and the Parton Model
4. E.Leader and E.Predazzi:An Introduction to Gauge Theories and Modern Particle
Physics, Volume 2: CP-Violation, QCD and Hard Processes
5. C.Grupen:Particle Detectors
6. H.Grosse and A.Martin:Particle Physics and the Schro¨dinger Equation
7. B.Anderson:The Lund Model
8. R.K.Ellis,W.J.Stirling and B.R.Webber:QCD and Collider Physics
9. I.I.Bigiand A.I.Sanda:CP Violation
10. A.V.Manohar and M.B.Wise:Heavy Quark Physics
11. R.K.Bock,H.Grote,R.Fru¨hwirth and M.Regler:Data Analysis Techniques for
High-Energy Physics, Second edition
12. D.Green:The Physics of Particle Detectors
13. V.N.Gribov and J.Nyiri:Quantum Electrodynamics
14. K.Winter (ed.):Neutrino Physics, Second edition
15. E.Leader:Spin in Particle Physics
16. J.D.Walecka:Electron Scattering for Nuclear and Nucleon Scattering
17. S.Narison:QCD as a Theory of Hadrons
18. J.F.Letessier and J.Rafelski: Hadrons and Quark-Gluon Plasma
19. A.Donnachie,H.G.Dosch,P.V.Landshoff and O.Nachtmann:Pomeron Physics
and QCD
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25. F.E.Close,A.Donnachie and G.Shaw:Electromagnetic Interactions and
Hadronic Structure
26. C.Grupen and B.A.Shwartz:Particle Detectors, Second edition
27. V.Gribov:Strong Interactions of Hadrons at High Energies
28. I.I.Bigiand A.I.Sanda:CP Violation, Second edition
29. P.Jaranowskiand A.Kro´lak:Analysis of Gravitational-Wave Data
ANALYSIS OF
GRAVITATIONAL-WAVE DATA
PIOTR JARANOWSKI
University of Bial(cid:1)ystok, Poland
ANDRZEJ KRO´ LAK
Polish Academy of Sciences, Poland
CAMBRIDGE UNIVERSITY PRESS
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© P. Jaranowski and A. Krolak 2009
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Contents
Preface page vii
Notation and conventions x
1 Overview of the theory of gravitational radiation 1
1.1 Linearized general relativity 2
1.2 Plane monochromatic gravitational waves 7
1.3 Description in the TT coordinate system 10
1.4 Description in the observer’s proper reference frame 13
1.5 Gravitational waves in the curved background 17
1.6 Energy–momentum tensor for gravitational waves 19
1.7 Generation of gravitational waves and
radiation reaction 20
2 Astrophysical sources of gravitational waves 26
2.1 Burst sources 27
2.2 Periodic sources 28
2.3 Stochastic sources 29
2.4 Case study: binary systems 30
2.5 Case study: a rotating triaxial ellipsoid 42
2.6 Case study: supernova explosion 45
2.7 Case study: stochastic background 47
3 Statistical theory of signal detection 51
3.1 Random variables 52
3.2 Stochastic processes 56
3.3 Hypothesis testing 62
3.4 The matched filter in Gaussian noise:
deterministic signal 71
3.5 Estimation of stochastic signals 76
v
vi Contents
3.6 Estimation of parameters 79
3.7 Non-stationary stochastic processes 90
4 Time series analysis 99
4.1 Sample mean and correlation function 99
4.2 Power spectrum estimation 101
4.3 Tests for periodicity 107
4.4 Goodness-of-fit tests 109
4.5 Higher-order spectra 111
5 Responses of detectors to gravitational waves 114
5.1 Detectors of gravitational waves 114
5.2 Doppler shift between freely falling observers 115
5.3 Long-wavelength approximation 122
5.4 Responses of the solar-system-based detectors 124
6 Maximum-likelihood detection in Gaussian noise 131
6.1 Deterministic signals 131
6.2 Case studies: deterministic signals 150
6.3 Network of detectors 167
6.4 Detection of stochastic signals 184
7 Data analysis tools 192
7.1 Linear signal model 192
7.2 Grid of templates in the parameter space 197
7.3 Numerical algorithms to calculate the F-statistic 201
7.4 Analysis of the candidates 208
Appendix A: The chirp waveform 212
Appendix B: Proof of the Neyman–Pearson lemma 218
Appendix C: Detector’s beam-pattern functions 221
C.1 LISA detector 222
C.2 Earth-based detectors 225
Appendix D: Response of the LISA detector
to an almost monochromatic wave 229
Appendix E: Amplitude parameters of periodic waves 233
References 235
Index 249
Preface
Gravitational waves are predicted by Einstein’s general theory of rela-
tivity. The only potentially detectable sources of gravitational waves are
of astrophysical origin. So far the existence of gravitational waves has
only been confirmed indirectly from radio observations of binary pulsars,
notably the famous Hulse and Taylor pulsar PSR B1913+16 [1]. As gravi-
tationalwavesareextremelyweak,averycarefuldataanalysisisrequired
in order to detect them and extract useful astrophysical information. Any
gravitational-wave signal present in the data will be buried in the noise
of a detector. Thus the data from a gravitational-wave detector are real-
izations of a stochastic process. Consequently the problem of detecting
gravitational-wave signals is a statistical one.
The purpose of this book is to introduce the reader to the field of
gravitational-wave data analysis. This field has grown considerably in the
past years as a result of commissioning a world-wide network of long arm
interferometric detectors. This network together with an existing network
ofresonantdetectorscollectsaverylargeamountofdatathatiscurrently
being analyzed and interpreted. Plans exist to build more sensitive laser
interferometric detectors and plans to build interferometric gravitational-
wave detectors in space.
This book is meant both for researchers entering the field of gravitatio-
nal-wave data analysis and the researchers currently analyzing the data.
In our book we describe the basis of the theory of time series analysis,
signal detection, and parameter estimation. We show how this theory
applies to various cases of gravitational-wave signals. In our applications
we usually assume that the noise in the detector is a Gaussian and sta-
tionary stochastic process. These assumptions will need to be verified
in practice. In our presentation we focus on one very powerful method
of detecting a signal in noise called the maximum-likelihood method.
This method is optimal by several criteria and in the case of Gaussian
vii
viii Preface
noise it consists of correlating the data with the template that is matched
to the expected signal. Robust methods of detecting signals in non-
Gaussian noise in the context of gravitational-wave data analysis are dis-
cussed in [2, 3]. In our book we do not discuss alternative data analysis
techniques such as time-frequency methods [4, 5], wavelets [6, 7, 8, 9],
Hough transform [10, 11], and suboptimal methods like those proposed
in [12, 13].
Early gravitational-wave data analysis was concerned with the detec-
tion of bursts originating from supernova explosions [14] and it consisted
mainly of analysis of coincidences among the detectors [15]. With the
growing interest in laser interferometric gravitational-wave detectors that
are broadband it was realized that sources other than supernovae can
also be detectable [16] and that they can provide a wealth of astrophys-
ical information [17, 18]. For example, the analytic form of the gravi-
tational-wave signal from a binary system is known to a good approxi-
mation in terms of a few parameters. Consequently, one can detect such
a signal by correlating the data with the waveform of the signal and
maximizing the correlation with respect to the parameters of the wave-
form. Using this method one can pick up a weak signal from the noise
by building a large signal-to-noise ratio over a wide bandwidth of the
detector [16]. This observation has led to a rapid development of the the-
ory of gravitational-wave data analysis. It became clear that detectability
of sources is determined by an optimal signal-to-noise ratio, which is the
power spectrum of the signal divided by power spectrum of the noise
integrated over the bandwidth of the detector.
An important landmark was a workshop entitled Gravitational Wave
Data Analysis held in Dyffryn House and Gardens, St. Nicholas near
Cardiff, in July 1987 [19]. The meeting acquainted physicists interested in
analyzing gravitational-wave data with the basics of the statistical theory
of signal detection and its application to the detection of gravitational-
wave sources. As a result of subsequent studies the Fisher information
matrix was introduced to the theory of the analysis of gravitational-
wave data [20, 21]. The diagonal elements of the Fisher matrix give lower
bounds on the variances of the estimators of the parameters of the sig-
nal and can be used to assess the quality of the astrophysical informa-
tion that can be obtained from detections of gravitational-wave signals
[22, 23, 24, 25]. It was also realized that the application of matched-
filtering to some signals, notably to periodic signals originating from neu-
tron stars, will require extraordinarily large computing resources. This
gave a further stimulus to the development of optimal and efficient algo-
rithms and data analysis methods [26].
Averyimportantdevelopmentwasthepaper[27],whereitwasrealized
that for the case of coalescing binaries matched-filtering was sensitive to