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ANALYS I S OF GRAV ITAT IONAL -WAVE DATA

Research in this eld has grown considerably in recent years due to thecommissioning of a world-wide network of large-scale detectors. This net-work collects a very large amount of data that is currently being ana-lyzed and interpreted. This book introduces researchers entering the eld,and researchers currently analyzing the data, to gravitational-wave dataanalysis.

An ideal starting point for studying the issues related to currentgravitational-wave research, the book contains detailed derivations ofthe basic formulae related to the detectors responses and maximum-likelihood detection. These derivations are much more complete andmore pedagogical than those found in current research papers, and willenable readers to apply general statistical concepts to the analysis ofgravitational-wave signals. It also discusses new ideas on devising the e-cient algorithms needed to perform data analysis.

Piotr Jaranowski is an Associate Professor in the Faculty of Physicsat the University of Bialystok, Poland. He has been a visiting sci-entist at the Max Planck Institute for Gravitational Physics and theFriedrich Schiller University of Jena, both in Germany, and the Insti-tut des Hautes Etudes Scientiques, France. He currently works in theeld of gravitational-wave data analysis and general-relativistic problemof motion.

Andrzej Krolak is a Professor in the Institute of Mathematics atthe Polish Academy of Sciences, Poland. He has twice been awarded theSecond Prize by the Gravity Research Foundation (once with BernardSchutz). He has been a visiting scientist at the Max Planck Institutefor Gravitational Physics, Germany, and the Jet Propulsion Laboratory,USA. His eld of research is gravitational-wave data analysis and generaltheory of relativity, and the phenomena predicted by this theory such asblack holes and gravitational waves.

CAMBRIDGE MONOGRAPHSON PARTICLE PHYS ICS ,

NUCLEAR PHYS ICS AND COSMOLOGY

General Editors: T. Ericson, P. V. Landsho

1. K. Winter (ed.): Neutrino Physics2. J. F. Donoghue, E. Golowich and B. R. Holstein: Dynamics of the Standard Model3. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle

Physics, Volume 1: Electroweak Interactions, the New Particles and the Parton Model4. E. Leader and E. Predazzi: An Introduction to Gauge Theories and Modern Particle

Physics, Volume 2: CP-Violation, QCD and Hard Processes5. C. Grupen: Particle Detectors6. H. Grosse and A. Martin: Particle Physics and the Schrodinger Equation7. B. Anderson: The Lund Model8. R. K. Ellis, W. J. Stirling and B. R. Webber: QCD and Collider Physics9. I. I. Bigi and A. I. Sanda: CP Violation

10. A. V. Manohar and M. B. Wise: Heavy Quark Physics11. R. K. Bock, H. Grote, R. Fruhwirth and M. Regler: Data Analysis Techniques for

High-Energy Physics, Second edition12. D. Green: The Physics of Particle Detectors13. V. N. Gribov and J. Nyiri: Quantum Electrodynamics14. K. Winter (ed.): Neutrino Physics, Second edition15. E. Leader: Spin in Particle Physics16. J. D. Walecka: Electron Scattering for Nuclear and Nucleon Scattering17. S. Narison: QCD as a Theory of Hadrons18. J. F. Letessier and J. Rafelski: Hadrons and Quark-Gluon Plasma19. A. Donnachie, H. G. Dosch, P. V. Landsho and O. Nachtmann: Pomeron Physics

and QCD20. A. Homann: The Physics of Synchroton Radiation21. J. B. Kogut and M. A. Stephanov: The Phases of Quantum Chromodynamics22. D. Green: High PT Physics at Hadron Colliders23. K. Yagi, T. Hatsuda and Y. Miake: Quark-Gluon Plasma24. D. M. Brink and R. A. Broglia: Nuclear Superuidity25. F. E. Close, A. Donnachie and G. Shaw: Electromagnetic Interactions and

Hadronic Structure26. C. Grupen and B. A. Shwartz: Particle Detectors, Second edition27. V. Gribov: Strong Interactions of Hadrons at High Energies28. I. I. Bigi and A. I. Sanda: CP Violation, Second edition29. P. Jaranowski and A. Krolak: Analysis of Gravitational-Wave Data

ANALYSIS OF

GRAVITATIONAL-WAVE DATA

P IOTR JARANOWSK IUniversity of Bialystok, Poland

ANDRZE J KR OLAKPolish Academy of Sciences, Poland

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-86459-6

ISBN-13 978-0-511-60518-5

P. Jaranowski and A. Krolak 2009

2009

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Contents

Preface page viiNotation and conventions x

1 Overview of the theory of gravitational radiation 11.1 Linearized general relativity 21.2 Plane monochromatic gravitational waves 71.3 Description in the TT coordinate system 101.4 Description in the observers proper reference frame 131.5 Gravitational waves in the curved background 171.6 Energymomentum tensor for gravitational waves 191.7 Generation of gravitational waves and

radiation reaction 20

2 Astrophysical sources of gravitational waves 262.1 Burst sources 272.2 Periodic sources 282.3 Stochastic sources 292.4 Case study: binary systems 302.5 Case study: a rotating triaxial ellipsoid 422.6 Case study: supernova explosion 452.7 Case study: stochastic background 47

3 Statistical theory of signal detection 513.1 Random variables 523.2 Stochastic processes 563.3 Hypothesis testing 623.4 The matched lter in Gaussian noise:

deterministic signal 713.5 Estimation of stochastic signals 76

v

vi Contents

3.6 Estimation of parameters 793.7 Non-stationary stochastic processes 90

4 Time series analysis 994.1 Sample mean and correlation function 994.2 Power spectrum estimation 1014.3 Tests for periodicity 1074.4 Goodness-of-t tests 1094.5 Higher-order spectra 111

5 Responses of detectors to gravitational waves 1145.1 Detectors of gravitational waves 1145.2 Doppler shift between freely falling observers 1155.3 Long-wavelength approximation 1225.4 Responses of the solar-system-based detectors 124

6 Maximum-likelihood detection in Gaussian noise 1316.1 Deterministic signals 1316.2 Case studies: deterministic signals 1506.3 Network of detectors 1676.4 Detection of stochastic signals 184

7 Data analysis tools 1927.1 Linear signal model 1927.2 Grid of templates in the parameter space 1977.3 Numerical algorithms to calculate the F-statistic 2017.4 Analysis of the candidates 208

Appendix A: The chirp waveform 212

Appendix B: Proof of the NeymanPearson lemma 218

Appendix C: Detectors beam-pattern functions 221C.1 LISA detector 222C.2 Earth-based detectors 225

Appendix D: Response of the LISA detectorto an almost monochromatic wave 229

Appendix E: Amplitude parameters of periodic waves 233

References 235Index 249

Preface

Gravitational waves are predicted by Einsteins general theory of rela-tivity. The only potentially detectable sources of gravitational waves areof astrophysical origin. So far the existence of gravitational waves hasonly been conrmed indirectly from radio observations of binary pulsars,notably the famous Hulse and Taylor pulsar PSR B1913+16 [1]. As gravi-tational waves are extremely weak, a very careful data analysis is requiredin order to detect them and extract useful astrophysical information. Anygravitational-wave signal present in the data will be buried in the noiseof a detector. Thus the data from a gravitational-wave detector are real-izations of a stochastic process. Consequently the problem of detectinggravitational-wave signals is a statistical one.

The purpose of this book is to introduce the reader to the eld ofgravitational-wave data analysis. This eld has grown considerably in thepast years as a result of commissioning a world-wide network of long arminterferometric detectors. This network together with an existing networkof resonant detectors collects a very large amount of data that is currentlybeing analyzed and interpreted. Plans exist to build more sensitive laserinterferometric detectors and plans to build interferometric gravitational-wave detectors in space.

This book is meant both for researchers entering the eld 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 theoryapplies to various cases of gravitational-wave signals. In our applicationswe usually assume that the noise in the detector is a Gaussian and sta-tionary stochastic process. These assumptions will need to be veriedin practice. In our presentation we focus on one very powerful methodof 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 matchedto 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 analysistechniques such as time-frequency methods [4, 5], wavelets [6, 7, 8, 9],Hough transform [10, 11], and suboptimal methods like those proposedin [12, 13].

Early gravitational-wave data analysis was concerned with the detec-tion of bursts originating from supernova explosions [14] and it consistedmainly of analysis of coincidences among the detectors [15]. With thegrowing interest in laser interferometric gravitational-wave detectors thatare broadband it was realized that sources other than supernovae canalso 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 sucha signal by correlating the data with the waveform of the signal andmaximizing the correlation with respect to the parameters of the wave-form. Using this method one can pick up a weak signal from the noiseby building a large signal-to-noise ratio over a wide bandwidth of thedetector [16]. This observation has led to a rapid development of the the-ory of gravitational-wave data analysis. It became clear that detectabilityof sources is determined by an optimal signal-to-noise ratio, which is thepower spectrum of the signal divided by power spectrum of the noiseintegrated over the bandwidth of the detector.

An important landmark was a workshop entitled Gravitational WaveData Analysis held in Dyryn House and Gardens, St. Nicholas nearCardi, in July 1987 [19]. The meeting acquainted physicists interested inanalyzing gravitational-wave data with the basics of the statistical theoryof signal detection and its application to the detection of gravitational-wave sources. As a result of subsequent studies the Fisher informationmatrix was introduced to the theory of the analysis of gravitational-wave data [20, 21]. The diagonal elements of the Fisher matrix give lowerbounds 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-ltering to some signals, notably to periodic signals originating from neu-tron stars, will require extraordinarily large computing resources. Thisgave a further stimulus to the development of optimal and ecient algo-rithms and data analysis methods [26].

A very important development was the paper [27], where it was realizedthat for the case of coalescing binaries matched-ltering was sensitive to

Preface ix

very small post-Newtonian eects of the waveform. Thus, these eectscan be detected. This leads to a much better verication of Einsteinstheory of relativity and provides a wealth of astrophysical informa-tion that would make a laser interferometric gravitational-wave detectora true astronomical observatory complementary to those utilizing the elec-tromagnetic spectrum. Further theoretical methods were introduced: tocalculate the quality of suboptimal lters [28], to calculate the numberof templates to do a search using matched-ltering [29], to determine theaccuracy of templates required [30], to calculate the false alarm prob-ability and thresholds [31]. An important point is the reduction of thenumber of parameters that one needs to search for in order to detect asignal. Namely, estimators of a certain type of parameters, called extrinsicparameters, can be found in a closed analytic form and consequently elimi-nated from the search. Thus a computationally intensive search needs onlybe performed over a reduced set of intrinsic parameters [21, 31, 32, 33].

The book is organized as follows. Chapter 1 introduces linearized Ein-stein equations and demonstrates how the gravitational waves arise inthis approximation. Chapter 2 reviews the most important astrophysicalsources of gravitational waves and presents in more detail calculations oftwo independent polarizations of waves from a compact binary systemand emitted by a rotating triaxial ellipsoid. Also the main features ofa gravitational-wave burst from a supernova explosion and of stochasticgravitational-wave background are considered. Chapter 3 is an introduc-tion to the statistical theory of the detection of signals in noise and estima-tion of the signals parameters. We discuss the basic properties of not onlystationary stochastic processes but also certain aspects of non-stationaryprocesses with applications to gravitational-wave data analysis. Chap-ter 4 is an introduction to the time series analysis. Chapter 5 studiesthe responses of detectors to gravitational waves for both ground-basedand space-borne detectors. Chapter 6 introduces the maximum-likelihooddetection method with an application to detection and estimation ofparameters of gravitational-wave signals buried in the noise of the detec-tor. The noise in the detector is assumed to be Gaussian and stationary.We consider both deterministic and stochastic signals. We also considerdetection by networks of detectors. Chapter 7 presents an in detail appli-cation of the methods presented in Chapter 6 to the case of the periodicgravitational-wave signal from a rotating neutron star. The methodol-ogy explained in Chapter 7 can be used in the analysis of many othergravitational-wave signals such as signals originating from the white-dwarfbinaries.

We would like to thank Gerhard Schafer for reading parts of the ma-nuscript of the book and for his useful remarks and comments.

Notation and conventions

General relativity. We adopt notation and conventions of the textbook byMisner, Thorne, and Wheeler [34]. Greek indices , , . . . run from 0 to 3and Latin indices i, j, . . . run from 1 to 3. We employ the Einstein sum-mation convention. We use the spacetime metric of signature (1, 1, 1, 1),so the line element of the Minkowski spacetime in Cartesian (inertial)coordinates (x0 = c t, x1 = x, x2 = y, x3 = z) reads

ds2 = dx dx = c2 dt2 + dx2 + dy2 + dz2. (1)

Numbers. N = {1, 2, . . .} is the set of natural numbers (i.e. the positiveintegers); Z = {. . . ,2,1, 0, 1, 2, . . .} denotes the set of all integers; R isthe set of real numbers. C denotes the set of complex numbers; for thecomplex number z = a+ i b (a, b R, i = 1) the complex conjugate ofz is denoted by z := a i b and |z| := a2 + b2 is the modulus of z; thereal and imaginary parts of z are denoted by (z) := a and (z) := b.3-vectors. For any 3-vectors a = (a1, a2, a3) and b = (b1, b2, b3) we denetheir usual Euclidean scalar product a b, and |a| denotes Euclideanlength of a 3-vector a:

a b :=3

i=1

aibi, |a| := a a = 3

i=1

(ai)2. (2)

Matrices are written in a sans serif font, e.g. M, N, . . . . Matrix multipli-cation is denoted by a dot, e.g. M N, and the superscript T stands forthe matrix transposition, e.g. MT.

Fourier transform. Dierent conventions are used for dening Fouriertransform. In the general case the one-dimensional Fourier transform

x

Notation and conventions xi

s = s() of a function s = s(t) is dened as

s() :=

|b|

(2)1a

+

eibt s(t) dt, (3)

where a and b (b = 0) are some real numbers. Then the solution of theabove equation with respect to the function s [i.e. the inverse Fouriertransform] reads

s(t) =

|b|

(2)1+a

+

eibt s() d. (4)

Throughout the book we employ two conventions. With the Fourier vari-able being the frequency f (measured in Hz) we use a = 0 and b = 2,i.e.

s(f) := +

e2if t s(t) dt, s(t) = +

e2if t s(f) df. (5)

When angular frequency := 2f is the Fourier variable we take a = 1and b = 1, i.e.

s() :=12

+

eit s(t) dt, s(t) = +

eit s() d. (6)

Spectral densities. In Section 3.2 we introduce the two-sided spectral den-sity S of a stationary stochastic process as a function of angular frequency, S = S(). The function S is the Fourier transform of the autocorrela-tion function R of the stochastic process, i.e.

S() = 12

+

eit R(t) dt, < < +. (7)

In Section 3.4 the two-sided spectral density S as a function of frequencyf is used, S = S(f). The function S is also the Fourier transform of thefunction R,

S(f) = +

e2if t R(t) dt, < f < +. (8)

Comparing Eqs. (7) and (8) we see that the relation between the functionsS and S is the following:

S() = 12S( 2

), S(f) = 2 S(2f). (9)

The relation between one-sided and two-sided spectral densities isexplained in Eqs. (3.93)(3.97).

xii Notation and conventions

Dirac delta function, . We employ integral representation of function:

(t) = +

e2if t df =12

+

eit d, (10)

as well as its periodic representation of the form:

n=(x 2n) = 1

2

n=

einx. (11)

1Overview of the theory ofgravitational radiation

In this chapter we very briey review the theory of gravitational radiation.A detailed exposition of the theory can be found in many textbooks ongeneral relativity, e.g. in Chapters 3537 of [34], Chapter 9 of [35], orChapter 7 of [36]. A detailed exposition of the theory of gravitationalwaves is contained in the recent monograph [37]. Reference [38] is anintroductory review of the theory of gravitational radiation and Ref. [16]is an accessible review of dierent aspects of gravitational-wave research.Some parts of the present chapter closely follow Sections 9.2 and 9.3 ofthe review article [16].

The chapter begins (in Section 1.1) with a discussion of general relativ-ity theory in the limit of weak gravitational elds. In this limit spacetimegeometry is a small perturbation of the at geometry of Minkowski space-time. We restrict our considerations to coordinate systems in which thespacetime metric is the sum of the Minkowski metric and a small per-turbation. We linearize Einstein eld equations with respect to this per-turbation and then we study two classes of coordinate transformationsthat preserve splitting the metric into the sum of Minkowski metric andits small perturbation: global Poincare transformations and gauge trans-formations. Finally we discuss the harmonic gauge, which allows one towrite the linearized Einstein eld equations in the form of inhomogeneouswave equations for the metric perturbation.

In Sections 1.21.4 we introduce gravitational waves as time-dependentvacuum solutions of the linearized Einstein equations. In Section 1.2 westudy the simplest such solution, namely a monochromatic plane grav-itational wave. In Section 1.3 we introduce the TT coordinate systemin which description of gravitational waves is especially simple. We rstdevelop in some detail the TT coordinate system for monochromatic plane

1

2 Overview of the theory of gravitational radiation

gravitational waves and then we remark that the TT coordinates can beintroduced for any kind of gravitational waves. In Section 1.4 we describein the proper reference frame of an observer the eect of gravitationalwaves on freely falling particles.

Section 1.5 is devoted to the problem of dening gravitational wavesin curved backgrounds. In Section 1.6 we present the energymomentumtensor for gravitational waves. Section 1.7 discusses the generation of grav-itational waves within the quadrupole formalism. It also contains approx-imate formulae for the rates of emission of energy and angular momentumcarried away from the source by gravitational waves.

1.1 Linearized general relativity

If the gravitational eld is weak, then it is possible to nd a coordinatesystem (x) such that the components g of the metric tensor in thissystem will be small perturbations h of the at Minkowski metric com-ponents :

g = + h , |h | 1. (1.1)The coordinate system satisfying the condition (1.1) is sometimes calledan almost Lorentzian coordinate system. For the almost Lorentzian coor-dinates (x) we will also use another notation:

x0 c t, x1 x, x2 y, x3 z. (1.2)In the rest of Section 1.1 we assume that the indices of the h will be

raised and lowered by means of the and (and not by g and g).Therefore we have, e.g.,

h = h, h

= h = h . (1.3)

1.1.1 Linearized Einstein equations

In the heart of general relativity there are the Einstein eld equations,

G =8Gc4

T , (1.4)

which relate spacetime geometry expressed in terms of the Einstein ten-sor G with sources of a gravitational eld represented by an energymomentum tensor T . The components of the Einstein tensor Gare constructed from the components of the Riemann curvature tensordescribing spacetime geometry in the following way. We rst need to dene

1.1 Linearized general relativity 3

the Christoel symbols , which depend on the spacetime metric com-ponents and their rst derivatives:

=12g(g + g g

). (1.5)

The Riemann curvature tensor has components

R = g( +

). (1.6)

Next we introduce the symmetric Ricci tensor,

R = gR , (1.7)

and its trace known as the Ricci scalar,

R = gR . (1.8)

Finally, the Einstein tensor is dened as

G := R 12gR. (1.9)If condition (1.1) is satised, one can linearize the Einstein eld equa-

tions (1.4) with respect to the small perturbation h . To do this we startfrom linearizing the Christoel symbols . For the metric (1.1) theytake the form

=12(h + h h

)+O(h2)

=12

(h

+ h

h

)+O(h2). (1.10)

Because Christoel symbols are rst-order quantities, the only contribu-tion to the linearized Riemann tensor will come from the derivatives ofthe Christoel symbols. Making use of Eqs. (1.6) and (1.10) we get

R =12(h + h h h

)+O(h2). (1.11)

Next we linearize the Ricci tensor. Making use of (1.11) we obtain

R =12

(h

+ h

hh

)+O(h2), (1.12)

where h is the trace of the metric perturbation h ,

h := h, (1.13)

and where we have introduced the dAlembertian operator in the atMinkowski spacetime:

:= = 1c2

2t + 2x +

2y +

2z . (1.14)

4 Overview of the theory of gravitational radiation

Finally we linearize the Ricci scalar,

R = R +O(h2)= h h+O

(h2). (1.15)

We are now ready to linearize the Einstein tensor. We get

G =12

(h

+ h

hh +

(h h

))+O(h2). (1.16)

It is possible to simplify one bit the right-hand side of Eq. (1.16) byintroducing the quantity

h := h 12h. (1.17)

From the denition (1.17) follows that

h = h 12 h, (1.18)

where h := h (let us also observe that h = h). Substituting therelation (1.18) into (1.16), one obtains

G =12

(h

+ h

h h

)+O(h2). (1.19)

Making use of Eq. (1.16) or Eq. (1.19) one can write down the linearizedform of the Einstein eld equations.

If spacetime (or its part) admits one almost Lorentzian coordinatesystem, then there exists in spacetime (or in its part) innitely manyalmost Lorentzian coordinates. Below we describe two kinds of coordi-nate transformations leading from one almost Lorentzian coordinate sys-tem to another such system: global Poincare transformations and gaugetransformations.

1.1.2 Global Poincare transformations

Let us consider the global Poincare transformation leading from the oldcoordinates (x) to some new ones (x). The new coordinates are linearinhomogeneous functions of the old coordinates,

x(x) = x + a. (1.20)

Here the constant (i.e. independent of the spacetime coordinates x)numbers are the components of the matrix representing the special-relativistic Lorentz transformation, and the constant quantities a rep-resent some translation in spacetime. The matrix () represents some

1.1 Linearized general relativity 5

Lorentz transformation provided it fullls the condition

= . (1.21)

This condition means that the Lorentz transformation does not changethe Minkowski metric. Let us recall that if the new coordinates are relatedto an observer that moves with respect to another observer related to theold coordinates along its x axis with velocity v, then the matrix built upfrom the coecients is the following

() =

v/c 0 0

v/c 0 00 0 1 00 0 0 1

, := (1 v2c2)1/2

. (1.22)

The transformation inverse to that given in Eq. (1.20) leads from new(x) to old (x) coordinates and is given by the relations

x(x) = (1)(x a), (1.23)

where the numbers (1) form the matrix inverse to that constructedfrom :

(1) = ,

(1) =

. (1.24)

Let us also note that the matrix ((1)) also fullls the requirement(1.21),

(1) (1) = . (1.25)

Let us now assume that the old coordinates (x) are almost Lorentzian,so the decomposition (1.1) of the metric holds in these coordinates. Mak-ing use of the rule relating the components of the metric tensor in twocoordinate systems,

g(x) =

x

xx

xg(x), (1.26)

by virtue of Eqs. (1.1), (1.23), and (1.21) one easily obtains

g = + h, (1.27)

where we have dened

h := (1) (

1) h . (1.28)

This last equation means that the metric perturbations h transformunder the Poincare transformation as the components of a (0, 2) ranktensor. The result of Eqs. (1.27)(1.28) also means that the new coor-dinate system

(x)will be almost Lorentzian, provided the numerical

6 Overview of the theory of gravitational radiation

values of the matrix elements (1) are not too large, because then thecondition |h | 1 also implies that |h | 1.

1.1.3 Gauge transformations

Another family of coordinate transformations, which lead from one almostLorentzian coordinates to another such coordinates, consists of innites-imal coordinate transformations known as gauge transformations. Theyare of the form

x = x + (x), (1.29)

where the functions are small in this sense, that

|| 1. (1.30)Equations (1.29)(1.30) imply that

x

x= +

, (1.31a)

x

x= +O

(()2

). (1.31b)

Let us now assume that the coordinates (x) are almost Lorentzian.Making use of Eqs. (1.1), (1.26), and (1.31), we compute the componentsof the metric in the (x) coordinates:

g = + h +O(h , ()2

), (1.32)

where we have introduced

:= . (1.33)

The metric components g can thus be written in the form

g = + h +O

(h , ()2

), (1.34)

where we have dened

h := h . (1.35)Because condition (1.30) is fullled, the new metric perturbation h issmall, |h | 1, and the coordinates (x) are almost Lorentzian. FromEq. (1.35), making use of the denition (1.17), one obtains the rule forhow the metric perturbation h changes under the gauge transformation,

h = h + . (1.36)

1.2 Plane monochromatic gravitational waves 7

1.1.4 Harmonic coordinates

Among almost Lorentzian coordinates one can choose coordinates forwhich the following additional harmonic gauge conditions are fullled:

h = 0. (1.37)

Let us note that the conditions (1.37) can equivalently be written as

h = 0. (1.38)

If these conditions are satised, the linearized Einstein tensor G fromEq. (1.19) reduces to

G = 12h +O(h2), (1.39)

and the linearized Einstein eld equations take the simple form of thewave equations in the at Minkowski spacetime:

h +O(h2)= 16G

c4T . (1.40)

Harmonic coordinates are not uniquely dened. The gauge conditions(1.37) [or (1.38)] are preserved by the Poincare transformations (1.20),they are also preserved by the innitesimal gauge transformations ofthe form (1.29), provided all the functions satisfy homogeneous waveequations:

= 0. (1.41)

1.2 Plane monochromatic gravitational waves

The simplest way of introducing gravitational waves relies on studyingvacuum solutions of linearized Einstein eld equations in harmonic co-ordinates. In vacuum the energymomentum tensor vanishes, T = 0,and the linearized eld equations in harmonic coordinates, Eqs. (1.40),reduce to homogeneous wave equations for all the components of themetric perturbation h :

h = 0. (1.42)

Time-dependent solutions of these equations can be interpreted as weakgravitational waves propagating through a region of spacetime whereEqs. (1.42) are valid, i.e. where the spacetime metric is almostMinkowskian.

8 Overview of the theory of gravitational radiation

The simplest solution of Eqs. (1.42) is a monochromatic plane wave,which is of the form

h(x) = A cos(kx

()()). (1.43)

Here A and ()() is the constant amplitude and the constant initialphase, respectively, of the component of the wave, and k are anotherfour real constants. We have encircled the indices and of the initialphases by parentheses to indicate that there is no summation over theseindices on the right-hand side of Eq. (1.43). By substituting (1.43) into(1.42) one checks that the functions (1.43) are solutions of Eqs. (1.42) ifand only if

kk = 0, (1.44)

which means that if we dene k := k , then k are the componentsof a null (with respect to the at Minkowski metric) 4-vector.

Let us write the argument of the cosine function in (1.43) in a moreexplicit way. The contraction kx can be written as

kx = k0 x0 +

3i=1

ki xi = k0 x0 +

3i=1

ki xi = c k0 t+ k x. (1.45)

Here we have introduced the two 3-vectors: k with components (k1, k2, k3)and x with components (x1, x2, x3). If we additionally introduce the quan-tity := c k0, then the plane-wave solution (1.43) becomes

h(t,x) = A cos(t k x+ ()()

). (1.46)

We can assume, without loss of generality, that 0. Then is angularfrequency of the wave; it is measured in radians per second. We will alsouse frequency f of the wave, measured in hertz (i.e. cycles per second).These two quantities are related to each other by the equation

= 2f. (1.47)

The 3-vector k is known as a wave vector, it points to the direction inwhich the wave is propagating and its Euclidean length is related to thewavelength ,

|k| = 2. (1.48)Equation (1.44) written in terms of and k takes the form

= c|k|. (1.49)This is the dispersion relation for gravitational waves. It implies that boththe phase and the group velocity of the waves are equal to c.

1.2 Plane monochromatic gravitational waves 9

Summing up: the solution (1.46) represents a plane gravitational wavewith frequency f = /(2) and wavelength = 2/|k|, which propagatesthrough the 3-space in the direction of the 3-vector k with the speed oflight.

Einstein eld equations take the simple form (1.42) only if the har-monic gauge conditions (1.37) are satised. Therefore we must addition-ally require that the functions h(x) from Eq. (1.46) fulll (1.37). Letus rewrite these functions in the form

h(t,x) = C cos( t k x) + S sin( t k x), (1.50)where we have introduced new quantities

C := A cos()(), S := A sin()(). (1.51)Then the requirements (1.37) imply that

Ck = 0, Sk = 0. (1.52)

Equations (1.52) provide constraints on the wave amplitudes C and S :they must be orthogonal to the 4-vector k. As a consequence the wholeplane-wave solution h , Eq. (1.50), is orthogonal to k:

hk = 0. (1.53)

The Poincare transformations preserve the form of the plane-wave solu-tion (1.43). Making use of the transformation rule (1.28) one can showthat the coordinate transformation (1.20) transforms the metric pertur-bations (1.43) into the new perturbations h ,

h(x) = A cos

(kx

()()), (1.54)

where the new constants k are related to the old ones k by

k = (1) k, (1.55)

so they transform as components of a (0, 1) rank tensor, and the newamplitudes and initial phases are dened through the relations

A cos()() = (

1)(1)A cos(()() + ka), (1.56)

A sin()() = (

1)(1)A sin(()() + ka). (1.57)

Because the Poincare transformations preserve the harmonicity condi-tions (1.37), as a consequence the orthogonality relations (1.53) are alsopreserved.

10 Overview of the theory of gravitational radiation

1.3 Description in the TT coordinate system

Equations (1.53) restrict the number of independent components of thegravitational plane wave from 10 to 6. These equations are a consequenceof the harmonic gauge conditions (1.37), which are preserved by gaugetransformations (1.29), provided each function is a solution of homo-geneous wave equation [see Eq. (1.41)]. Because we have at our disposalfour functions (for = 0, 1, 2, 3), we can use them to further restrictthe number of independent components of the plane wave from 6 to 2.The choice of the functions is equivalent to the choice of a coordi-nate system. We describe now the choice of leading to the so calledtransverse and traceless (TT in short) coordinate system.

At each event in the spacetime region covered by some almostLorentzian and harmonic coordinates (x) let us choose the timelike unitvector U, gUU = 1. Let us consider a gauge transformation gen-erated by the functions of the form

(t,x) = B cos(t k x+ ()

), (1.58)

with = c|k| and k the same as in the plane-wave solution (1.46). Itis possible to choose the quantities B and () in such a way, that inthe new almost Lorentzian and harmonic coordinates x = x + thefollowing conditions are fullled

hU = 0, (1.59a)

h = 0. (1.59b)

Furthermore, the gauge transformation based on the functions (1.58) pre-serves the condition (1.53):

hk = 0. (1.59c)

Let us also note that, as a consequence of Eq. (1.59b),

h = h . (1.60)

Equations (1.59) dene the TT coordinate system related to the 4-vectoreld, U . These equations comprise eight independent constraints on thecomponents of the plane-wave solution h . This means that any planemonochromatic gravitational wave possesses two independent degrees offreedom, often also called the waves polarizations.

The simplest way to describe the two gravitational-wave polarizationsis by making more coordinate changes. We have used all the freedomrelated to the gauge transformations, but we are still able to performglobal Lorentz transformations, which preserve equations (1.59) deningthe TT gauge. One can rst move to coordinates in which the vector U

1.3 Description in the TT coordinate system 11

(from now we omit the primes in the coordinate names) has componentsU = (1, 0, 0, 0). Then Eqs. (1.59a) imply

h0 = 0. (1.61)

Further, one can orient spatial coordinate axes such that the wave propa-gates in, say, the +z direction. Then k = (0, 0, /c), k = (/c, 0, 0, /c),and Eqs. (1.59c) together with (1.61) give

h3 = 0. (1.62)

The last constraint on the plane-wave components h providesEq. (1.59b) supplemented by (1.61) and (1.62). It reads

h11 + h22 = 0. (1.63)

It is common to use the following notation:

h+ := h11 = h22, h := h12 = h21. (1.64)The functions h+ and h are called plus and cross polarization of thewave, respectively.

We will label all quantities computed in the TT coordinate system bysuper- or subscript TT. Equations (1.61)(1.63) allow us to rewrite theplane-wave solution (1.46) in TT coordinates in the following matrix form:

hTT (t,x) =

0 0 0 00 h+(t,x) h(t,x) 00 h(t,x) h+(t,x) 00 0 0 0

, (1.65)where the plus h+ and the cross h polarizations of the plane wave withangular frequency traveling in the +z direction are given by

h+(t,x) = A+ cos((t z

c

)+ +

),

h(t,x) = A cos((t z

c

)+

). (1.66)

Any gravitational wave can be represented as a superposition of planemonochromatic waves. Because the equations describing TT gauge,

h = 0, hU = 0, h = 0, (1.67)

are all linear in h , it is possible to nd the TT gauge for any grav-itational wave. If we restrict ourselves to plane (but not necessarilymonochromatic) waves and orient the spatial axes of a coordinate systemsuch that the wave propagates in the +z direction, then Eqs. (1.61)(1.63)are still valid. Moreover, because all the monochromatic components of

12 Overview of the theory of gravitational radiation

the wave depend on spacetime coordinates only through the combina-tion t z/c, the same dependence will be valid also for the general wave.Therefore any weak plane gravitational wave propagating in the +z direc-tion is described in the TT gauge by the metric perturbation of the fol-lowing matrix form:

hTT (t,x) =

0 0 0 00 h+(t z/c) h(t z/c) 00 h(t z/c) h+(t z/c) 00 0 0 0

. (1.68)If we introduce the polarization tensors e+ and e by means of equations

e+xx = e+yy = 1, exy = eyx = 1, all other components zero, (1.69)then one can reconstruct the full gravitational-wave eld from its plusand cross polarizations as

hTT (t,x) = h+(t,x) e+ + h(t,x) e

. (1.70)

It turns out that hTT is a scalar under boosts and behaves like a spin-two eld under rotations. This means that in two dierent reference framesrelated by a boost in some arbitrary direction (with the spatial axes of theframes unrotated relative to each other) the gravitational-wave elds hTTare the same; whereas if one rotates the x and y axes in the transverseplane of the wave through an angle , the gravity-wave polarizations arechanged to (

hnew+hnew

)=(

cos 2 sin 2 sin 2 cos 2

)(hold+hold

). (1.71)

For further details see Section 2.3.2 of [38].It is useful to write explicitly all the non-zero components of the

gravitational-wave Riemann tensor in the TT coordinates. Componentsof the Riemann tensor linearized with respect to metric perturbations hare given by Eq. (1.11). Making use of Eqs. (1.60) and (1.68) one thencomputes

RTTx0x0 = RTTy0y0 = RTTx0xz = RTTy0yz = RTTxzxz = RTTyzyz = 1220h+,

(1.72a)

RTTx0y0 = RTTx0yz = RTTy0xz = RTTxzyz = 1220h. (1.72b)

All other non-zero components can be obtained by means of symmetriesof the Riemann tensor.

1.4 Description in the observers proper reference frame 13

1.4 Description in the observers proper reference frame

Let us consider two nearby observers freely falling in the eld of a weakand plane gravitational wave. The wave produces tiny variations in theproper distance between the observers. We describe now these variationsfrom the point of view of one of the observers, which we call the basicobserver. We endow this observer with his proper reference frame, whichconsists of a small Cartesian latticework of measuring rods and syn-chronized clocks. The time coordinate t of that frame measures propertime along the world line of the observer whereas the spatial coordi-nate xi (i = 1, 2, 3)1 measures proper distance along his ith axis. Space-time coordinates (x0 := c t, x1, x2, x3) are locally Lorentzian along thewhole geodesic of the basic observer, i.e. the line element of the metric inthese coordinates has the form

ds2 = c2 dt2 + ij dxi dxj +O((xi)2

)dx dx , (1.73)

so this line element deviates from the line element of the at Minkowskispacetime by terms that are at least quadratic in the values of the spatialcoordinates xi.

Let the basic observer A be located at the origin of his proper ref-erence frame, so his coordinates are xA(t) = (c t, 0, 0, 0). A neighboringobserver B moves along the nearby geodesic and it possesses the coordi-nates xB(t) = (c t, x

1(t), x2(t), x3(t)). We dene the deviation vector

describing the instantaneous relative position of the observer B withrespect to the observer A:

(t) := xB(t) xA(t) =(0, x1(t), x2(t), x3(t)

). (1.74)

The relative acceleration D2 /dt2 of the observers is related to the space-time curvature through the equation of geodesic deviation,

D2

dt2= c2 Ru u , (1.75)

where u := dx/(cdt) is the 4-velocity of the basic observer. Let us notethat all quantities in Eq. (1.75) are evaluated on the basic geodesic, so theyare functions of the time coordinate t only. Equation (1.73) implies thatChristoel symbols vanish along the basic geodesic. Therefore theirtime derivative, d/dt, also vanishes there, and D

2 /dt2 = d2 /dt2

along that geodesic. Also taking into account the fact that u = (1, 0, 0, 0)

1 We will also use another notation for the spatial coordinates: x1 x, x2 y, x3 z.

14 Overview of the theory of gravitational radiation

and making use of Eq. (1.74), from (1.75) one nally obtains

d2xi

dt2= c2 Ri0j0xj = c2 Ri0j0xj +O

((xi)3

). (1.76)

If one chooses the TT coordinates in such a way that the 4-velocityeld needed to dene TT coordinates coincides with the 4-velocity of ourbasic observer, then the TT coordinates (t, xi) and the proper-reference-frame coordinates (t, xi) dier from each other in the vicinity of the basicobservers geodesic by quantities linear in h. It means that, up to theterms quadratic in h, the components of the Riemann tensor in bothcoordinate systems coincide,

Ri0j0 = RTTi0j0 +O(h2). (1.77)

Now we make use of the fact that in the TT coordinates hTT0 = 0 [see Eq.(1.61)], then from the general formula (1.11) for the linearized Riemanntensor we get [remembering that in the TT coordinate system hTT = hTT ,Eq. (1.60)]

RTTi0j0 = 1

2 c22hTTijt2

+O(h2) = 12 c2

2hTTij

t2+O(h2). (1.78)

In the derivation of the above relation we have not needed to assume thatthe wave is propagating in the +z direction of the TT coordinate sys-tem, thus this relation is valid for the wave propagating in any direction.Collecting Eqs. (1.76)(1.78) together, after neglecting terms O(h2), oneobtains

d2xi

dt2=

122hTTij

t2xj , (1.79)

where the second time derivative 2hTTij /t2 is to be evaluated along the

basic geodesic x = y = z = 0.Let us now imagine that for times t 0 there were no waves (hTTij = 0)

in the vicinity of the two observers and that the observers were at restwith respect to each other before the wave has come, so

xi(t) = xi0 = const.,dxi

dt(t) = 0, for t 0. (1.80)

At t = 0 some wave arrives. We expect that xi(t) = xi0 +O(h) for t > 0,therefore, because we neglect terms O(h2), Eq. (1.79) can be replaced by

d2xi

dt2=

122hTTij

t2xi0. (1.81)

1.4 Description in the observers proper reference frame 15

Making use of the initial conditions (1.80) we integrate (1.81). The resultis

xi(t)=(ij +

12hTTij

(t))

xi0, t > 0. (1.82)

Let us orient the spatial axes of the proper reference frame suchthat the wave is propagating in the +z direction. Then we can employEq. (1.68) (where we put z = 0 and replace t by t) to write Eqs. (1.82) inthe more explicit form

x(t)= x0 +

12

(h+(t)x0 + h

(t)y0

), (1.83a)

y(t)= y0 +

12

(h(t)x0 h+

(t)y0

), (1.83b)

z(t)= z0. (1.83c)

Equations (1.83) show that the gravitational wave is transverse: it pro-duces relative displacements of the test particles only in the plane per-pendicular to the direction of the wave propagation.

We will study in more detail the eect of the gravitational wave onthe cloud of freely falling particles. To do this let us imagine that thebasic observer is checking the presence of the wave by observing someneighboring particles that form, before the wave arrives, a perfect ring inthe (x, y) plane. Let the radius of the ring be r0 and the center of the ringcoincides with the origin of the observers proper reference frame. Thenthe coordinates of any particle in the ring can be parametrized by someangle [0, 2] such that they are equal

x0 = r0 cos, y0 = r0 sin, z0 = 0. (1.84)

Equations (1.83) and (1.84) imply that in the eld of a gravitational wavethe z coordinates of all the rings particles remain equal to zero: z

(t)= 0,

so only x and y coordinates of the particles should be analyzed.

1.4.1 Plus polarization

The gravitational wave is in the plus mode when h = 0. Then, makinguse of Eqs. (1.83) and (1.84), one gets

x(t)= r0 cos

(1 +

12h+(t))

, y(t)= r0 sin

(1 1

2h+(t))

. (1.85)

Initially, before the wave arrives, the ring of the particles is perfectlycircular. Does this shape change after the wave has arrived? One can

16 Overview of the theory of gravitational radiation

Fig. 1.1. The eect of a plane monochromatic gravitational wave with + polar-ization on a circle of test particles placed in a plane perpendicular to the directionof the wave propagation. The plots show deformation of the circle measured inthe proper reference frame of the central particle at the instants of time equalto nT/4 (n = 0, 1, 2, . . .), where T is the period of the gravitational wave.

treat Eqs. (1.85) as parametric equations of a certain curve, with beingthe parameter. It is easy to combine the two equations (1.85) such that is eliminated. The resulting equation reads

x2(a+(t)

)2 + y2(b+(t)

)2 = 1, (1.86)where

a+(t) := r0(1 +

12h+(t))

, b+(t) := r0(1 1

2h+(t))

. (1.87)

Equations (1.86)(1.87) describe an ellipse with its center at the origin ofthe coordinate system. The ellipse has semi-axes of the lengths a+(t) andb+(t), which are parallel to the x or y axis, respectively. If h+

(t)is the

oscillatory function, which changes its sign in time, then the deformationof the initial circle into the ellipse is the following: in time intervals whenh+(t)> 0, the circle is stretched in the x direction and squeezed in the y

direction, whereas when h+(t)< 0, the stretching is along the y axis and

the squeezing is along the x axis. This is illustrated in Fig. 1.1.Let us now x a single particle in the ring. The motion of this parti-

cle with respect to the origin of the proper reference frame is given byEqs. (1.85), for a xed value of . What is the shape of the particlestrajectory? It is again easy to combine Eqs. (1.85) in such a way that thefunction h+

(t)is eliminated. The result is

x

r0 cos+

y

r0 sin 2 = 0. (1.88)

Equation (1.88) means that any single particle in the ring is movingaround its initial position along some straight line.

1.5 Gravitational waves in the curved background 17

1.4.2 Cross polarization

The gravitational wave is in the crosss mode when h+ = 0. Then fromEqs. (1.83) and (1.84) one gets

x(t)= r0

(cos+

12sinh

(t))

, y(t)= r0

(sin+

12cosh

(t))

.

(1.89)

Let us now introduce in the (x, y) plane some new coordinates (x, y).The new coordinates one gets from the old ones by rotation around the zaxis by the angle of = 45. Both coordinate systems are related to eachother by the rotation matrix,(

xy

)=(

cos sin sin cos

)(xy

)=22

(1 11 1

)(xy

). (1.90)

It is easy to rewrite Eqs. (1.89) in terms of the coordinates (x, y). Theresult is

x(t)=22

r0(sin+ cos)(1 + h

(t))

, (1.91a)

y(t)=22

r0(sin cos)(1 h

(t))

. (1.91b)

After eliminating from Eqs. (1.91) the parameter , one gets

x2(a(t)

)2 + y2(b(t)

)2 = 1, (1.92)where

a(t) := r0(1 +

12h(t))

, b(t) := r0(1 1

2h(t))

. (1.93)

Equations (1.92)(1.93) have exactly the form of Eqs. (1.86)(1.87). Thismeans that the initial circle of particles is deformed into an ellipse withits center at the origin of the coordinate system. The ellipse has semi-axesof the lengths a(t) and b(t), which are inclined by an angle of 45 tothe x or y axis, respectively. This is shown in Fig. 1.2.

1.5 Gravitational waves in the curved background

So far we have considered gravitational waves in such regions of spacetime,where the waves are the only non-negligible source of spacetime curvature.If there exist other sources it is not possible in a fully precise manner,because of the non-linear nature of relativistic gravity, to separate the

18 Overview of the theory of gravitational radiation

Fig. 1.2. The eect of a plane monochromatic gravitational wave with polar-ization on a circle of test particles placed in a plane perpendicular to the directionof the wave propagation. The plots show deformation of the circle measured inthe proper reference frame of the central particle at the instants of time equalto nT/4 (n = 0, 1, 2, . . .), where T is the period of the gravitational wave.

contribution of a gravitational wave to the spacetime curvature from thecontributions of the other sources. Such separation can only be madeapproximately. We describe now a method of dening a gravitational wavethat is a special case of a standard technique in mathematical physicscalled (among other names) shortwave approximation.

Let us consider a gravitational wave with a wavelength . This wavecreates spacetime curvature that varies on the scale of the order of thereduced wavelength of the wave, where

:=

2. (1.94)

In many realistic astrophysical situations the lengthscale is very shortcompared to lengthscales L on which all other non-gravitational-wavecurvatures vary:

L. (1.95)This inequality allows one to split the full Riemann curvature tensorRinto a background curvature Rb and a gravitational-wave producedpart Rgw . The background curvature R

b is the average of the full

Riemann tensor R over several gravitational-wave wavelengths

Rb := R, (1.96)whereas the gravitational-wave curvature Rgw is the rapidly varyingdierence:

Rgw := R Rb. (1.97)It is possible to introduce a TT coordinate system for the gravitational

wave propagating in the curved background. In this coordinate systemthe spacetime metric is nearly Minkowskian and can be written in theform

g = + hb + hTT , (1.98)

1.6 Energymomentum tensor for gravitational waves 19

where hb (|hb | 1) is the background metric perturbation that varieson a long lengthscale L, and hTT (|hTT | 1) is the gravity-wave metricperturbation that varies on a short lengthscale . The timetime hTT00 andthe spacetime hTT0i = h

TTi0 components of the gravity-wave perturbation

vanish in the TT coordinate system, and if the wave propagates in the +zdirection, then the metric perturbation hTT may be written in the formgiven in Eq. (1.68).

The extent of the TT coordinates in both time and space must be farsmaller than the radius of background curvature. In typical astrophysicalsituations one can stretch TT coordinates over any small region comparedto the distance at which curvature of the cosmological background of ouruniverse becomes important (the Hubble distance), cutting out holesin the vicinities of black holes and neutron stars.

1.6 Energymomentum tensor for gravitational waves

A fully satisfactory mathematical description of the energy carried by agravitational wave was devised by Isaacson [39, 40], who introduced anenergymomentum tensor for gravitational waves. This tensor is obtainedby averaging the squared gradient of the wave eld over several wave-lengths. In the TT gauge it has components

T gw =c4

32G

h

TT h

TT

. (1.99)

The gravitational-wave energymomentum tensor T gw , like the back-ground curvature, is smooth on the lengthscale . If one additionallyassumes that hTT0 = 0, then Eq. (1.99) reduces to

T gw =c4

32G

3i=1

3j=1

h

TTij h

TTij

. (1.100)

For the plane gravitational wave propagating in the +z direction, thetensor T gw takes the standard form for a bundle of zero-rest-mass particlesmoving at the speed of light in the +z direction, which can be immediatelydemonstrated by means of Eqs. (1.68) and (1.100):

T gw00 = T gw0z = T gwz0 = T gwzz =c2

16G

(h+t

)2+(ht

)2(1.101)

(all other components are equal to zero).

20 Overview of the theory of gravitational radiation

The energymomentum tensor for gravitational waves dened inEq. (1.99) has the same properties and plays the same role as the energymomentum tensor for any other eld in the background spacetime. It gen-erates background curvature through the Einstein eld equations (aver-aged over several wavelengths of the waves); it has vanishing divergencein regions where there is no wave generation, absorption, and scattering.

Let us compute the components of the energymomentum tensorfor the monochromatic plane wave with angular frequency . Thegravitational-wave polarization functions h+ and h for such a wave aregiven in Eqs. (1.66). Making use of them, from Eqs. (1.101) one gets

T gw00 = T gw0z = T gwz0 = T gwzz =c22

16G

(A2+

sin2

((t z

c

)+ +

)+A2

sin2

((t z

c

)+

)).

(1.102)

Averaging the sine squared terms over one wavelength or one wave periodgives 1/2. After substituting this into (1.102), and replacing by thefrequency f = /(2) measured in hertz, one obtains

T gw00 = T gw0z = T gwz0 = T gwzz =c2f2

8G

(A2+ +A

2). (1.103)

Typical gravitational waves that we might expect to observe at Earthhave frequencies between 104 and 104 Hz, and amplitudes of the orderof A+ A 1022. The energy ux in the +z direction for such wavescan thus be estimated as

T gwtz = c T gw0z = 1.6 106(

f

1Hz

)2A2+ +A2(1022)2

ergcm2 s

.

1.7 Generation of gravitational waves and radiation reaction

Quadrupole formalism. The simplest technique for computing thegravitational-wave eld hTT is delivered by the famous quadrupole formal-ism. This formalism is especially important because it is highly accuratefor many astrophysical sources of gravitational waves. It does not require,for high accuracy, any constraint on the strength of the sources internalgravity, but requires that internal motions inside the source are slow. Thisrequirement implies that the sources size L must be small compared tothe reduced wavelength of the gravitational waves it emits.

Let us introduce a coordinate system (t, xis) centered on a gravitatio-nal-wave source and let an observer at rest with respect to the source

1.7 Generation of gravitational waves and radiation reaction 21

measure the gravitational-wave eld hTT generated by that source. Letus further assume that the observer is situated within the local wave zoneof the source, where the background curvature both of the source andof the external universe can be neglected. It implies (among other things)that the distance from the observer to the source is very large comparedto the sources size. Then the quadrupole formalism allows one to writethe gravitational-wave eld in the following form:

hTT0 (t, xis) = 0, h

TTij (t, x

is) =

2Gc4

1R

d2J TTijdt2

(t R

c

), (1.104)

where R :=

ijxisxjs is the distance from the point (xis) where the

gravitational-wave eld is observed to the sources center, t is proper timemeasured by the observer, and tR/c is retarded time. The quantity Jijis the sources reduced mass quadrupole moment (which we dene below),and the superscript TT at Jij means algebraically project out and keeponly the part that is transverse to the direction in which wave propagatesand is traceless. Quite obviously Eqs. (1.104) describe a spherical gravi-tational wave generated by the source located at the origin of the spatialcoordinates.

Let ni := xis/R be the unit vector in the direction of wave propagationand let us dene the projection operator P ij that projects 3-vectors to a2-plane orthogonal to ni,

P ij := ij ninj. (1.105)Then the TT part of the reduced mass quadrupole moment can be com-puted as (see Box 35.1 of [34])

J TTij = Pki P lj Jkl 12Pij(PklJkl

). (1.106)

For the wave propagating in the +z direction the unit vector ni hascomponents

nx = ny = 0, nz = 1. (1.107)

Making use of Eqs. (1.105)(1.106) one can then easily compute the TTprojection of the reduced mass quadrupole moment. The result is

J TTxx = J TTyy =12(Jxx Jyy), (1.108a)

J TTxy = J TTyx = Jxy, (1.108b)J TTzi = J TTiz = 0 for i = x, y, z. (1.108c)

22 Overview of the theory of gravitational radiation

Making use of these equations and the notation introduced in Eq. (1.64)one can write the following formulae for the plus and the cross polar-izations of the wave progagating in the +z direction of the coordinatesystem:

h+(t, xis) =G

c4 R

(d2Jxxdt2

(t R

c

) d

2Jyydt2

(t R

c

)), (1.109a)

h(t, xis) =2Gc4 R

d2Jxydt2

(t R

c

). (1.109b)

Polarization waveforms in the SSB reference frame. Let us now intro-duce another coordinate system (t, xi) about which we assume that somesolar-system-related observer measures the gravitational-wave eld hTTin the neighborhood of its spatial origin. In the following chapters ofthe book, where we discuss the detection of gravitational waves by solar-system-based detectors, the origin of these coordinates will be located atthe solar system barycenter (SSB). Let us denote by x the constant 3-vector joining the origin of our new coordinates (xi) (i.e. the SSB) withthe origin of the (xis) coordinates (i.e. the center of the source). Wealso assume that the spatial axes in both coordinate systems are parallelto each other, i.e. the coordinates xi and xis just dier by constant shiftsdetermined by the components of the 3-vector x,

xi = xi + xis. (1.110)

It means that in both coordinate systems the components hTT of thegravitational-wave eld are numerically the same.

Equations (1.108) and (1.109) are valid only when the z axis of the TTcoordinate system is parallel to the 3-vector x x joining the observerlocated at x (x is the 3-vector joining the SSB with the observers location)and the gravitational-wave source at the position x. If one changes thelocation x of the observer, one has to rotate the spatial axes to ensure thatEqs. (1.108) and (1.109) are still valid. Let us now x, in the whole regionof interest, the direction of the +z axis of both the coordinate systemsconsidered here, by choosing it to be antiparallel to the 3-vector x (so forthe observer located at the SSB the gravitational wave propagates alongthe +z direction). To a very good accuracy one can assume that the sizeof the region where the observer can be located is very small comparedto the distance from the SSB to the gravitational-wave source, whichis equal to r := |x|. Our assumption thus means that r r, wherer := |x|. Then x = (0, 0,r) and the Taylor expansion of R = |x x|

1.7 Generation of gravitational waves and radiation reaction 23

and n := (x x)/R around x reads

|x x| = r(1 +

z

r+

x2 + y2

2(r)2+O((xl/r)3)), (1.111a)

n =( x

r+

xz

2(r)2, y

r+

yz

2(r)2, 1 x

2 + y2

2(r)2

)+O((xl/r)3).

(1.111b)

Making use of Eqs. (1.105)(1.106) and (1.111b) one can compute the TTprojection of the reduced mass quadrupole moment at the point x in thedirection of the unit vector n (which, in general, is not parallel to the +zaxis). One gets

J TTxx = J TTyy =12(Jxx Jyy) +O

(xl/r

), (1.112a)

J TTxy = J TTyx = Jxy +O(xl/r

), (1.112b)

J TTzi = J TTiz = O(xl/r

)for i = x, y, z. (1.112c)

To obtain the gravitational-wave eld hTT in the coordinate system (t, xi)one should plug Eqs. (1.112) into the formula (1.104). It is clear that ifone neglects in Eqs. (1.112) the terms of the order of xl/r, then in thewhole region of interest covered by the single TT coordinate system, thegravitational-wave eld hTT can be written in the form

hTT (t,x) = h+(t,x) e+ + h(t,x) e

+O

(xl/r

), (1.113)

where the polarization tensors e+ and e are dened in Eq. (1.69), andthe functions h+ and h are of the form given in Eqs. (1.109).

Dependence of the 1/R factors in the amplitudes of the wave polariza-tions (1.109) on the observers position x (with respect to the SSB) isusually negligible, so 1/R in the amplitudes can be replaced by 1/r [thisis consistent with the neglection of the O(xl/r) terms we have just madein Eqs. (1.112)]. This is not the case for the second time derivative of J TTijin (1.109), which determines the time evolution of the wave polarizationphases and which is evaluated at the retarded time tR/c. Here it isusually enough to take into account the rst correction to r given byEq. (1.111a). After taking all this into account the wave polarizations

24 Overview of the theory of gravitational radiation

(1.109) take the form

h+(t, xi) =G

c4 r

(d2Jxxdt2

(t z + r

c

) d

2Jyydt2

(t z + r

c

)), (1.114a)

h(t, xi) =2Gc4 r

d2Jxydt2

(t z + r

c

). (1.114b)

The wave polarization functions (1.114) depend on the spacetime coor-dinates (t, xi) only through the combination t z/c, so they represent aplane gravitational wave propagating in the +z direction.

Mass quadrupole moment of the source. We shift now to the denition ofthe sources mass quadrupole moment Jij . We restrict only to situationswhen the source has weak internal gravity and small internal stresses, soNewtonian gravity is a good approximation to general relativity insideand near the source. Then Jij is the symmetric and trace-free (STF)part of the second moment of the sources mass density computed in aCartesian coordinate system centered on the source:

Jij(t) :=(

(xk, t)xixjd3x)STF

=

(xk, t)(xixj 1

3r2ij

)d3x.

(1.115)

Equivalently, Jij is the coecient of the 1/r3 term in the multipolarexpansion of the sources Newtonian gravitational potential ,

(t, xk) = GMr 3G

2Jij(t)xixj

r5 5G

2Jijk(t)xixjxk

r7+ . (1.116)

Gravitational-wave luminosities. From the quadrupole formula (1.104)and Isaacsons formula (1.100) for the energymomentum tensor of thegravitational waves, one can compute the uxes of energy and angularmomentum carried by the waves. After integrating these uxes over asphere surrounding the source in the local wave zone one obtains the ratesLgwE and LgwJi of emission respectively of energy and angular momentum:

LgwE =G

5c5

3i=1

3j=1

(d3Jijdt3

)2, (1.117a)

LgwJi =2G5c5

3j=1

3k=1

3=1

ijk

d2Jjdt2

d3Jkdt3

. (1.117b)

Formula (1.117a) was rst derived by Einstein [41, 42], whereas formula(1.117b) was discovered by Peters [43].

1.7 Generation of gravitational waves and radiation reaction 25

The laws of conservation of energy and angular momentum implythat radiation reaction should decrease the sources energy and angu-lar momentum at rates just equal to minus rates given by Eqs. (1.117),

dEsource

dt= LgwE , (1.118a)

dJ sourceidt

= LgwJi . (1.118b)Justication of the validity of the balance equations [examples of whichare Eqs. (1.118)] in general relativity is thorougly discussed e.g. inSection 6.15 of Ref. [44] (see also references therein).

2Astrophysical sources of

gravitational waves

It is convenient to split the expected astrophysical sources of gravitationalwaves into three main categories, according to the temporal behavior ofthe waveforms they produce: burst, periodic, and stochastic sources. InSections 2.12.3 of the present chapter we enumerate some of the mosttypical examples of gravitational-wave sources from these categories (moredetailed reviews can be found in [45], Section 9.4 of [16], and [46, 47]).Many sources of potentially detectable gravitational waves are relatedto compact astrophysical objects: white dwarfs, neutron stars, and blackholes. The physics of compact objects is thoroughly studied in the mono-graph [48].

In the rest of the chapter we will perform more detailed studies of gravi-tational waves emitted by several important astrophysical sources. In Sec-tion 2.4 we derive gravitational-wave polarization functions h+ and h fordierent types of waves emitted by binary systems. As an example of peri-odic waves we consider, in Section 2.5, gravitational waves coming froma triaxial ellipsoid rotating along a principal axis; we derive the functionsh+ and h for these waves. In Section 2.6 we relate the amplitude of gravi-tational waves emitted during a supernova explosion with the total energyreleased in gravitational waves and with the time duration and the fre-quency bandwidth of the gravitational-wave pulse. Finally in Section 2.7we express the frequency dependence of the energy density of stationary,isotropic, and unpolarized stochastic background of gravitational wavesin terms of their spectral density function.

26

2.1 Burst sources 27

2.1 Burst sources

2.1.1 Coalescing compact binaries

Binary systems consisting of any objects radiate gravitational waves andas a result of radiation reaction the distance between the components ofthe binary decreases. This results in a sinusoidal signal whose amplitudeand frequency increases with time and which is called a chirp.

For the binary of circular orbits the characteristic dimensionless ampli-tude h0 of the two gravitational-wave polarizations [see Eq. (2.40) and itsderivation in Section 2.4 later in this chapter] is equal:

h0 = 2.6 1023(MM

)5/3 ( fgw100 Hz

)2/3 ( R100 Mpc

)1, (2.1)

whereM is the chirp mass of the system [see Eq. (2.34) for the denitionof M in terms of the individual masses of the binary components], fgw isthe frequency of gravitational waves (which is twice the orbital frequency),and R is the distance to the binary. The characteristic time gw := fgw/fgw(where fgw := dfgw/dt) for the increase of gravitational-wave frequencyfgw is given by [here Eq. (2.38) was employed]

gw = 8.0 s(MM

)5/3 ( fgw100 Hz

)8/3. (2.2)

Among all coalescing binaries the most interesting are those made ofcompact objects, neutron stars (NS), or black holes (BH), in the last fewminutes of their inspiral. There are three kinds of such compact binaries:NS/NS, NS/BH, and BH/BH binaries. The nal merger of the two NSin a NS/NS binary is a promising candidate for the trigger of some typesof gamma-ray bursts. At the endpoint of a NS/BH inspiral, a neutronstar can be tidally disrupted by its BH companion, and this disruption isanother candidate for triggering gamma-ray bursts. For heavier BH/BHbinaries, most of the detectable gravitational waves can come from themerger phase of the evolution as well as from the vibrational ringdown ofthe nal BH.

The number densities per unit time of dierent compact binary coales-cences are rather uncertain. Their estimates crucially depend on the eventrate RGal of binary mergers in our Galaxy. Recent studies (see Section 2.3of [47] and references therein) give 106 yr1 RGal 5 104 yr1 forNS/NS mergers, and 107 yr1 RGal 104 yr1 for NS/BH inspirals.For BH/BH binaries two distinct estimates can be made: one for thebinaries not contained in globular and other types of dense star clusters(in eld binaries), and the other for binaries from these clusters. The

28 Astrophysical sources of gravitational waves

BH/BH event rate estimates are: 107 yr1 RGal 105 yr1 for ineld binaries, and 106 yr1 RGal 105 yr1 for binaries in clusters.

2.1.2 Supernovae

Neutron stars and black holes (of stellar masses) form in the gravitationalcollapse of a massive star, which leads to a supernova type II explosion(core-collapse supernova). Because of our incomplete knowledge of theprocess of collapse (we do not know how non-spherical the collapse mightbe in a typical supernova) and the diversity of emission mechanisms, wecannot predict the gravitational waveform from this event accurately. Agravitational-wave burst might be rather broad-band with frequency cen-tered on 1 kHz, or it might contain a few cycles of radiation at a frequencyanywhere between 100Hz and 10 kHz, chirping up or down.

The dimensionless amplitude h0 of the gravitational-wave pulse fromsupernova explosion can be estimated by [see Eq. (2.77) in Section 2.6]

h0 1.4 1021(

Egw102 Mc2

)1/2 ( 1ms

)1/2 (fgw1 kHz

)1( R15Mpc

)1,

(2.3)

where Egw is the total energy carried away by gravitational waves duringthe explosion, is the duration of the pulse and fgw is its frequencybandwidth, R is the distance to the source. The value of Egw is veryuncertain, it can dier from the above quoted number (around 102 Mc2)by orders of magnitude. We expect around ten such sources per year inthe Virgo cluster of galaxies (15Mpc is the distance to the center of theVirgo cluster).

For a recent review of the theory of core-collapse supernovae see [49],and recent studies of gravitational waves emitted during the core-collapsesupernova can be found in [50, 51], see also the review article [52].

2.2 Periodic sources

The primary example of sources of periodic gravitational waves are spin-ning neutron stars. Because a rotating body, perfectly symmetric aroundits rotation axis, does not emit gravitational waves, the neutron star willemit waves only if it has some kind of asymmetry. Several mechanismsleading to such an asymmetry have been discovered and studied. Thesemechanisms include elastic deformations of the stars solid crust (or core)developed during the crystallization period of the crust and supported byanisotropic stresses in it. The strong magnetic eld present in the star may

2.3 Stochastic sources 29

not be aligned with the rotation axis, consequently the magnetic pressuredistorts the entire star. Some mechanisms result in a triaxial neutron starrotating about a principal axis. Detailed computations of gravitationalwaves emitted by a triaxial ellipsoid rotating about a principal axis arepresented in Section 2.5 later in the current chapter.

The dimensionless amplitude of the gravitational waves emitted by arotating neutron star can be estimated by [this is Eq. (2.67) in which thephysical constants are replaced by their numerical values]

h0 = 4.2 1025 105Izzs

1045 g cm2

(f0

100Hz

)2 ( R10 kpc

)1, (2.4)

where is the stars ellipticity [dened in Eq. (2.63)], Izzs is its momentof inertia around the rotation axis, f0 is the rotational frequency of thestar, and R is the distance to the star.

The LIGO Scientic Collaboration has recently imposed, using datafrom LIGO detectors, a non-trivial upper limit on h0 for the Crab pul-sar (PSR B0531+21 or PSR J0534+2200). The upper limit is h95%0 =3.4 1025 [53], where h95%0 is the joint 95% upper limit on the gravitatio-nal-wave amplitude using uniform priors on all the parameters. This limitis substantially less than the spin-down upper limit hsd0 = 1.4 1024 thatcan be inferred assuming that all the energy radiated by the Crab pulsaris due to gravitational-wave emission. This result assumes that the Crabsspin frequency f0 = 29.78Hz, spin-down rate f0 = 3.7 1010 Hz s1,principal moment of inertia Izzs = 1045 g cm2, and distance R = 2 kpc.Moreover the analysis assumes that the gravitational-wave emission fol-lows the observed radio timing.

2.3 Stochastic sources

A stochastic background of gravitational radiation arises from an ex-tremely large number of weak, independent, and unresolved gravitatio-nal-wave sources. Such backgrounds may arise in the early universe fromination, phase transitions, or cosmic strings. It may also arise from pop-ulations of astrophysical sources (e.g., radiation from many unresolvedbinary star systems). See Ref. [54] for a comprehensive review of stochas-tic gravitational-wave sources.

There is a broadband observational constraint on the stochastic back-ground of gravitational waves that comes from a standard model of big-bang nucleosynthesis. This model provides remarkably accurate ts tothe observed abundances of the light elements in the universe, tightlyconstraining a number of key cosmological parameters. One of the

30 Astrophysical sources of gravitational waves

parameters constrained in this way is the expansion rate of the universe atthe time of nucleosynthesis. This places a constraint on the energy densityof the universe at that time, which in turn constrains the energy densityin a cosmological background of gravitational radiation. This leads to thefollowing bound [54], which is valid independently of the frequency f (andindependently of the actual value of the Hubble expansion rate)

h2100 gw(f) 5 106, (2.5)where gw is the dimensionless ratio of the gravitational-wave energydensity per logarithmic frequency interval to the closure density of theuniverse [see Eq. (2.88)] and h100 is related to the Hubble constant H0 by

H0 = h100 100 km s1

Mpc. (2.6)

From observations it follows that h100 almost certainly lies in the range0.6 h100 0.8 (see e.g. Ref. [55]). In terms of the nucleosynthesis bound(2.5) we have the following numerical expression for the characteristicdimensionless amplitude hc of the gravitational-wave stochastic back-ground [see Eq. (2.92)]:

hc(f) 2.8 1023(h2100 gw(f)5 106

)1/2 (f

100Hz

)1. (2.7)

There are other, tighter constraints on h2100 gw(f) that come fromobserved isotropy of the cosmic microwave background and timing of themillisecond pulsars, but they are valid for very small frequencies, wellbelow the bands of the existing and planned gravitational-wave detectors.

2.4 Case study: binary systems

The relativistic two-body problem, i.e. the problem of describing thedynamics and gravitational radiation of two extended bodies interactinggravitationally according to general relativity theory, is very dicult (seethe review articles [44, 56]). Among all binary systems the most impor-tant sources of gravitational waves are binaries made of compact objects:white dwarfs, neutron stars, or black holes. Only compact objects canreach separations small enough and relative velocities large enough toenter the essentially relativistic regime in which using Einsteins equa-tions is unavoidable. There are two approaches to studying the two-bodyproblem in its relativistic regime. The rst approach is to solve Einsteinsequations numerically. Another is to employ an analytic approximationscheme. Among the many approximation schemes that were developed,

2.4 Case study: binary systems 31

the most eective approach turned out to be the post-Newtonian (PN)one, in which the gravitational eld is assumed to be weak and relativevelocities of the bodies generating the eld are assumed to be small (soit is a weak-eld and slow-motion approximation).

We focus here on binary systems made of black holes. The time evo-lution of a black-hole binary driven by gravitational-wave emission canbe split into three stages [57]: adiabatic inspiral, merger or plunge, andringdown. In the inspiral phase, the orbital period is much shorter thanthe time scale over which orbital parameters change. This stage can beaccurately modeled by PN approximations. During the merger stage gra-vitational-wave emission is so strong that the evolution of the orbit isno longer adiabatic, and the black holes plunge together to form a sin-gle black hole. Full numerical simulations are needed to understand thisphase. Finally, in the ringdown phase the gravitational waves emitted canbe well modeled by quasi-normal modes of the nal Kerr black hole.

Only recently there was a remarkable breakthrough in numerical sim-ulations of binary black-hole mergers (see the review article [57]). Herewe are more interested in explicit approximate analytical results con-cerning motion and gravitational radiation of compact binary systems.Such results were obtained by perturbative solving Einstein eld equa-tions within the PN approximation of general relativity.

Post-Newtonian approximate results. Post-Newtonian calculations pro-vide equations of motion of binary systems and rates of emission of energyand angular momentum carried by gravitational waves emitted by thebinary (gravitational-wave luminosities) in the form of the PN series, i.e.the power series of the ratio v/c, where v is the typical orbital veloc-ity of the binary members. Let us mention that the most higher-orderPN explicit results were obtained under the assumption that the binarymembers can be modeled as point particles. Dierent PN formalisms arepresented in Refs. [58, 59, 60, 61, 62, 63, 64, 65, 66, 67].

The PN dynamics of binary systems can be split into a conservativepart and a dissipative part connected with the radiation-reaction eects.The conservative dynamics can be derived from an autonomous Hamil-tonian. Equations of motion of compact binary systems made of non-rotating bodies (which can be modeled as spinless point particles) wereexplicitly derived up to the 3.5PN order, i.e. up to the terms of the order(v/c)7 beyond Newtonian equations of motion. The details of derivationsof the most complicated 3PN and 3.5PN contributions to the equationsof motion can be found in Refs. [68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,79, 80, 81]. The rate of emission of gravitational-wave energy by binarysystems was computed up to the terms of the order (v/c)7 beyond theleading-order quadrupole formula for binaries along quasi-circular orbits

32 Astrophysical sources of gravitational waves

[82, 83, 84, 85, 86, 87, 88] and up to the terms of the order (v/c)6 beyondthe leading-order contribution for general quasi-elliptical orbits [89].

Analytical results concerning compact binaries made of rotating bodieswere also obtained in the form of a PN expansion. There are variousspin-dependent contributions to the PN-expanded equations of motionand gravitational-wave luminosities of a binary system: terms exist thatare linear, quadratic, cubic, etc., in the individual spins S1 and S2 of thebodies. Gravitational spin-orbit eects (i.e. eects that are linear in thespins) were analytically obtained to the next-to-leading order terms (i.e.one PN order beyond the leading-order contribution) both in the orbitaland spin precessional equations of motion [90, 91, 92] and in the rate ofemission of gravitational-wave energy [93, 94]. Higher-order in the spinseects (i.e. the eects of the order of S1 S2, S21, S22, etc.) were recentlydiscussed in Refs. [95, 96, 97, 98, 99].

PN eects modify not only the orbital phase evolution of the binary,but also the amplitudes of the two independent wave polarizations h+and h. In the case of binaries along quasi-circular orbits the PN ampli-tude corrections were explicitly computed up to the 3PN order beyondthe leading-order formula given in Eqs. (2.26) (see [100] and referencestherein). For quasi-elliptical orbits the 2PN-accurate corrections to wavepolarizations were computed in Ref. [101].

It is not an easy task to obtain the wave polarizations h+ and h asexplicit functions of time, taking into account all known higher-order PNcorrections. To do this in the case of quasi-elliptical orbits one has to dealwith three dierent time scales: orbital period, periastron precession, andradiation-reaction time scale. Explicit results concerning the phasing ofthe inspiraling binaries along quasi-elliptical orbits were obtained e.g. inRefs. [102, 103] (see also references therein).

Eective-one-body approach. We should also mention here the eective-one-body (EOB) formalism, which provides an accurate quasi-analyticaldescription of the motion and radiation of a coalescing black-hole binary atall stages of its evolution, from adiabatic inspiral to ringdown. The quasi-analytical means here that to compute a waveform within the EOBapproach one needs to solve numerically some explicitly given ordinarydierential equations (ODEs). Numerical solving ODEs is extremely fast,contrary to computationally very expensive (3 + 1)-dimensional numeri-cal relativity simulations of merging black holes.

The core of the EOB formalism is mapping, through the use of invari-ant (i.e. gauge-independent) functions, the real two-body problem (twospinning masses orbiting around each other) onto an eective one-body problem: one spinless mass moving in some eective backgroundmetric, which is a deformation of the Kerr metric. The EOB approach

2.4 Case study: binary systems 33

was introduced at the 2PN level for non-rotating bodies in [104, 105].The method was then extended to the 3PN level in [106], and spin eectswere included in [107, 108]. See Ref. [109] for a comprehensive introduc-tion to the EOB formalism.

2.4.1 Newtonian binary dynamics

In the rest of this section we consider a binary system made of two bodieswith masses m1 and m2 (we will always assume m1 m2). We introduce

M := m1 +m2, :=m1m2M

, (2.8)

so M is the total mass of the system and is its reduced mass. It is alsouseful to introduce the dimensionless symmetric mass ratio

:=m1m2M2

=

M. (2.9)

The quantity satises 0 1/4, the case = 0 corresponds to thetest-mass limit and = 1/4 describes equal-mass binary. We start fromderiving the wave polarization functions h+ and h for waves emitted by abinary system in the case when the dynamics of the binary can reasonablybe described within the Newtonian theory of gravitation.

Let r1 and r2 denote the position vectors of the bodies, i.e. the 3-vectors connecting the origin of some reference frame with the bodies. Weintroduce the relative position vector,

r12 := r1 r2. (2.10)The center-of-mass reference frame is dened by the requirement that

m1 r1 +m2 r2 = 0. (2.11)

Solving Eqs. (2.10) and (2.11) with respect to r1 and r2 one gets

r1 =m2M

r12, r2 = m1M

r12. (2.12)

In the center-of-mass reference frame we introduce the spatial coor-dinates (xc, yc, zc) such that the total orbital angular momentum vectorJ of the binary is directed along the +zc axis. Then the trajectories ofboth bodies lie in the (xc, yc) plane, so the position vector ra of the athbody (a = 1, 2) has components ra = (xca, yca, 0), and the relative positionvector components are r12 = (xc12, yc12, 0), where xc12 := xc1 xc2 andyc12 := yc1 yc2. It is convenient to introduce in the coordinate (xc12, yc12)plane the usual polar coordinates (r, ):

xc12 = r cos, yc12 = r sin. (2.13)

34 Astrophysical sources of gravitational waves

Within Newtonian gravity the orbit of the relative motion is an ellipse(here we consider only gravitationally bound binaries). We place the focusof the ellipse at the origin of the (xc12, yc12) coordinates. In polar coordi-nates the ellipse is described by the equation

r() =a(1 e2)

1 + e cos( 0) , (2.14)

where a is the semi-major axis, e is the eccentricity, and 0 is theazimuthal angle of the orbital periapsis. The time dependence of the rel-ative motion is determined by Keplers equation,

=2P

(1 e2)3/2(1 + e cos( 0))2, (2.15)where P is the orbital period of the binary,

P = 2

a3

GM. (2.16)

The binarys binding energy E and the modulus J of its total angularorbital momentum are related to the parameters of the relative orbitthrough the equations

E = GM2a

, J =

GMa(1 e2). (2.17)

Let us now introduce the TT wave coordinates (xw , yw , zw) in whichthe gravitational wave is traveling in the +zw direction and with the originat the SSB. The line along which the plane tangential to the celestialsphere at the location of the binarys center-of-mass [this plane is parallelto the (xw , yw) plane] intersects the orbital (xc, yc) plane is called theline of nodes. Let us adjust the center-of-mass and the wave coordinatesin such a way that the x axes of both coordinate systems are parallelto each other and to the line of nodes. Then the relation between thesecoordinates is determined by the rotation matrix S,xwyw

zw

=xwyw

zw

+ Sxcyc

zc

, S :=1 0 00 cos sin 0 sin cos

, (2.18)where (xw , yw , zw) are the components of the vector x joining the SSBand the binarys center-of-mass, and ( 0 ) is the angle betweenthe orbital angular momentum vector J of the binary and the line of sight(i.e. the +zw axis). We assume that the center-of-mass of the binary is atrest with respect to the SSB.

2.4 Case study: binary systems 35

In the center-of-mass reference frame the binarys moment of inertiatensor has changing in time components that are equal

Iijc (t) = m1 xic1(t)xjc1(t) +m2 xic2(t)xjc2(t). (2.19)Making use of Eqs. (2.12) one easily computes that the matrix Ic builtfrom the Iijc components of the inertia tensor equals

Ic(t) = (xc12(t))2 xc12(t) yc12(t) 0xc12(t) yc12(t) (yc12(t))2 0

0 0 0

. (2.20)The inertia tensor components in wave coordinates are related to thosein center-of-mass coordinates through the relation

Iijw (t) =3

k=1

3=1

xiwxkc

xjwxc

Ikc (t), (2.21)

which in matrix notation reads

Iw(t) = S Ic(t) ST. (2.22)To obtain the wave polarization functions h+ and h we plug the com-

ponents of the binarys inertia tensor [computed by means of Eqs. (2.18),(2.20), and (2.22)] into general equations (1.114) [note that in Eqs. (1.114)the components Jij of the reduced quadrupole moment can be replaced bythe components Iij of the inertia tensor, compare the denitions (1.115)and (2.56)]. We get [here R = |x| is the distance to the binarys center-of-mass and tr = t (zw +R)/c is the retarded time]

h+(t,x) =G

c4R

(sin2

(r(tr)2 + r(tr)r(tr)

)+ (1 + cos2 )

(r(tr)2 + r(tr)r(tr) 2r(tr)2(tr)2

)cos 2(tr)

(1 + cos2 )(4r(tr)r(tr)(tr) + r(tr)2(tr)) sin 2(tr)),(2.23a)

h(t,x) =2Gc4R

cos ((

4r(tr)r(tr)(tr) + r(tr)2(tr))cos 2(tr)

+(r(tr)2 + r(tr)r(tr) 2r(tr)2(tr)2

)sin 2(tr)

). (2.23b)

Polarization waveforms without radiation-reaction eects. Gravitationalwaves emitted by the binary diminish the binarys binding energy andtotal orbital angular momentum. This makes the orbital parameters

36 Astrophysical sources of gravitational waves

changing in time. Let us rst assume that these changes are so slow thatthey can be neglected during the time interval in which the observationsare performed. Making use of Eqs. (2.15) and (2.14) one can then elimi-nate from the formulae (2.23) the rst and the second time derivatives ofr and , treating the parameters of the orbit, a and e, as con