COALESCENCE OF BLACK-HOLE BINARIES: FROMTHEORETICAL SOURCE MODELS TO APPLICATIONSIN GRAVITATIONAL-WAVE ASTRONOMY

DISSERTATION
zur erlangung des akademischen grades
doktor der naturwissenschaften(dr. rer. nat.)in der wissenschaftsdisziplin physik
eingereicht an dermathematisch-naturwissenschaftlichen fakultät
der universität potsdam>

vonLUCÍA SANTAMARÍA LARA
geboren am 23. juli 1982 in burgos, spanien
potsdam, april 2010

Lucía Santamaría Lara
Coalescence of Black-Hole Binaries: from Theoretical Source Models to Applications in Gravitational-Wave Astronomy © April 2010
betreuer:Prof. Bernard F. Schutz
supervisoren:
Dr. Badri Krishnan
Dr. Sascha Husa
gutachter:
Dr. Alberto Vecchio
Asst. Prof. Laura Cadonati
Dr. Chris van den Broeck
institut:
Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)
fakultät:
Mathematisch-Naturwissenschaften Fakültat, Universität Potsdam

Sigue, sigue adelante y no regreses
Fiel hasta el fin del camino y tu vida
No eches de menos un destino más fácil
Tus pies sobre la tierra antes no hollada
Tus ojos frente a lo antes nunca visto— Luis Cernuda

ABSTRACT

To date, gravitational waves have not yet been observed directly, but strong
evidence supports their existence. With current ground-based interferometers
operating at their design sensitivity, their upgraded versions approved to becommissioned within the next five years and a space-based antenna in design
stages, the need for optimal data analysis strategies becomes pressing. Signals
coming from binary systems of compact objects, such as neutron stars andblack holes, are expected to stand among the most promising candidates forthe first detection of gravitational waves.
This work focuses on improving existing detection techniques for gravitational-wave signals from binaries of black holes. The current post-Newtonian
approach is valid until the compact objects become so close to each otherthat the weak-field approximation to general relativity breaks down. At thatpoint, one needs to resort to full numerical relativity methods in order to solve
Einstein’s equations. It was only recently that the field succeeded in providing
stable solutions and in extracting the gravitational radiation associated tothe merger of the binary. Ever since, an increasing number of simulations
exploring larger parts of the parameter space have become available.
This dissertation presents a new waveform model to describe the gravitational radiation of the full coalescence of a binary of black holes, fromthe initial inspiral phase through the merger until the final ringdown to astationary black hole. Analytical and numerical approaches to the binary
black-hole problem are brought together to present a joint description of the
coalescence process. The parameter space that our model covers corresponds
to that of comparable-mass, spinning, non-precessing binaries. The inclusionof these waveforms as filters in current searches for gravitational waves withground-based detectors will have an immediate impact in their performance.
On one hand, the extension of the waveforms past the post-Newtonian stage
would allow more massive systems to be surveyed. On the other hand, thespin of the black-hole system affects its detectability, in a way that can now bequantified.
Further related work contained in this dissertation presents the latest searches for compact binaries in the output of the first-generation ground-based
detectors; in addition, we describe the results of a search for numerical relativity signals incorporated into Gaussian data as a way of assessing the performance of current data analysis search strategies; finally, the potential detection
and characterization of black-hole binaries with total mass between hundreds
and tens of thousands of solar masses is suggested, which could be possible
with future-generation interferometers.
The main focus of this dissertation is thus the connection between theoretical solutions of the binary black-hole problem and their direct application in
gravitational-wave data analysis and astrophysics.

v

ZUSAMMENFASSUNG

Bisher wurden Gravitationswellen (GW) noch nicht direkt nachgewiesen,doch gibt es starke Indizien, die auf ihre Existenz hindeuten. Durch die bereits
mit hoher Detektorempfindlichkeit arbeitenden Interferometer, deren bereits
genehmigten verbesserten Ausbaustufen und das geplante LISA-Projekt, steigt
die Notwendigkeit für entsprechende Strategien zur Datenanalyse. Signale vonBinärsystemen kompakter Objekte (Neutronensterne und Schwarze Löcher)sind die aussichtsreichsten Kandidaten für den ersten Nachweis von GW.
Diese Arbeit konzentriert sich auf die Verbesserung existierender Detektortechniken für Signale von GW binärer Schwarzer Löcher. Der gegenwärtige post-Newtonsche Ansatz gilt nur bis sich die kompakten Objekte soweit
annähern, dass die Schwachfeldnäherung der ART zusammenbricht. An
diesem Punkt muss man vollständige numerische Methoden anwenden, um
die Einstein-Gleichungen zu lösen. Erst vor kurzem gelang es hier, stabile
Lösungen anzubieten und die Gravitationsstrahlung aus der Verschmelzungdes Binärsystems zu definieren. Seitdem wurden mehrere Simulationen, dieeinen größeren Teil des Parameterraums untersuchen, veröffentlicht.
Diese Dissertation stellt ein neues Modell zur Beschreibung der GW einer
vollständigen Verschmelzung zweier Schwarzer Löcher vor, von der frühen
Annäherungsphase auf einer Spiralbahn, über die Vereinigung, bis zur Phase
des Abklingens und der Entstehung eine einzelnen stationären Schwarzen
Lochs. Analytische und numerische Ansätze für das Problem binärer Schwarzer
Löcher werden zusammengebracht, um eine gemeinsame Beschreibung desVerschmelzungsprozesses zu präsentieren. Der Parameterraum, den unser
Modell umfasst, korrespondiert mit dem vergleichbar schwerer, rotierender
Binärsysteme ohne Präzession. Die Einbindung dieser Wellenformen als Filter
in die derzeitige Suche nach GW wird einen unmittelbaren Einfluss auf deren
Ergebnis haben. Zum einen würde es die Erweiterung der Wellenform überdie post-Newtonsche Stufe hinaus erlauben, weitaus schwerere Systeme zu
beobachten. Zum anderen beeinflusst der Spin die Nachweisbarkeit in einer
Art und Weise, die nun quantitativ bestimmt werden kann.
Ein weiterer, damit zusammenhängender Teil dieser Dissertation beschäftigt
sich mit der Suche nach Binärsystemen in den Ergebnissen der terrestrischen
Detektoren. Zusätzlich beschreiben wir eine Suche nach numerischen Signalen als eine Methode, die Leistungsfähigkeit heutiger Datenanalyse-Strategienzu bewerten. Zuletzt wird der potentielle Nachweis und die Beschreibung von
Schwarzen Löchern einer Gesamtmasse von hunderten bis zehntausenden
Sonnenmassen erläutert, der mit zukünftigen Interferometern möglich wird.
Der Schwerpunkt dieser Dissertation ist somit die Verbindung von theoretischen Lösungen der Problematik binärer Schwarzer Löcher und deren
direkter Anwendung in der Analyse von GW-Daten und der Astrophysik.

vii

ACKNOWLEDGMENTS

My doctoral experience has been tremendously enriching and I owe gratituteto a large number of people and institutions. I am indebted to the Max Planck
Society and the Albert Einstein Institute for their continued endorsement in
the form of scientific education, facilities and support to pursue my research;
to the LIGO Scientific Collaboration for providing a stimulating environment
for doing gravitational-wave science; to the Relativity Group at the Universityof Jena for giving me the opportunity of starting my research in the field of
numerical relativity; and to the DAAD (German Academic Exchange Service)for their partial funding over the past years.
I am personally grateful to the director of the Albert Einstein Institute,
Professor Bernard F. Schutz, for his strong belief in my capabilities. This work
would not have been possible without the guidance I have received frommy advisors Badri Krishnan and Sascha Husa, who have made sure I neverlacked help and motivation to carry on my doctoral research. I appreciate thesupport received from the leaders of the groups where I have worked, Bernd
Brügmann in Jena; Luciano Rezzolla and Maria Alessandra Papa in Golm;
Steve Fairhurst, Patrick Brady, Bangalore Sathyaprakash and Alan Weinstein
within the LSC. I am thankful to Laura Cadonati, Chris van den Broeck andAlberto Vecchio for accepting to be external reviewers of this dissertation.
I have had the privilege of finding extraordinary people along the way. I
heartily thank Pau Amaro Seoane, Stas Babak, Jolien Creighton, Steve Drasco,
Marc Mars, Alberto Sesana and Ed Porter for their mentoring and advice. It
has been a pleasure to share time and ideas with Diego Fazi, Nick Fotopoulos, Chad Hanna, Michael Jasiulek, Drew Keppel, Christian Reisswig, Emma
Robinson, Jennifer Seiler and John T. Whelan. I especially acknowledge andthank the help of Frank Ohme in the context of our work on phenomenological waveforms. I appreciate the useful comments provided by Ian Hinder,
Mark Hannam and Jocelyn Read, who have read and corrected parts of thisdissertation. Any remaining errors and mistakes are, nonetheless, my full
responsability. For making my doctoral years truly enjoyable I thank all myfellow students and post-doctoral researchers at the Albert Einstein Instituteand the LIGO and Virgo Scientific Collaborations. My friends in Burgos,
Salamanca and Berlin have always encouraged me to pursue my goals; I feel
privileged for having them close to me, despite the distance. Lastly, words
can not express the immense gratitute and love I feel towards my parents andfamily for their affection and understanding. This dissertation is dedicated tothem.
Lucía Santamaría Lara
Potsdam, April 2010ix

CONTENTS
overview3

ithe universe through a new window

1

gravitational waves
1.1

2

7

9

Einstein’s Theory of Gravitation

9

1.1.1

The Einstein Field Equations

9

1.1.2

Effects of Gravitational Waves on Test Particles

1.2

History of Gravitational-Wave Detectors

1.3

Gravitational-Wave Interferometry

1.4

World-Wide International Network of Detectors

1.5

The Future of Gravitational-Wave Astronomy

14

17
19
23

1.5.1

The Advanced Interferometers

1.5.2

Third-Generation Ground-Based Detectors

1.5.3

LISA: The Space Antenna

Stochastic Background

2.2

Periodic Sources

2.3

Bursts2.4

Coalescing Binaries

23
24

25

sources of gravitational waves
2.1

12

27

27

28

29
30

2.4.1

Evidence for Compact Binaries

30

2.4.2

The Coalescence Process

2.4.3

Expected Astrophysical Rates for Compact Binaries

31
34

ii modeling binary black-hole coalescences

37

3

39

analytical modeling of black-hole binaries
3.1

3.2

Post-Newtonian Formalism for Non-Precessing Binaries
3.1.1

Energy and Flux

3.1.2

TaylorT4 Approximant

3.1.3

Stationary Phase Approximation and TaylorF2

3.1.4

Amplitude

39

41
42
43

45

Ringdown and Quasi-Normal Modes

48

3.2.1

Perturbations from Equilibrium

3.2.2

The Ringdown Waveform

48

49

xi

xii

contents

4

black-hole binaries in numerical relativity
4.1

4.2

5

Solving Einstein Equations Numerically

53

53

4.1.1

The 3+1 Decomposition

54

4.1.2

The ADM Equations

4.1.3

The BSSN Formulation

4.1.4

A Recipe for Successful Simulations

56
57
59

Numerical Simulations of Non-Precessing Binaries
4.2.1

NR waveforms and Codes

4.2.2

Going from Ψ4 to h

65

65

67

new waveform model for binary black holes
5.1

Full Models for Binary Black-Hole Coalescence

5.2

Matching Post-Newtonian and Numerical Relativity

5.3

5.4

71

5.2.2

Issues in Matching PN with NR

5.2.3

An Illustration for Non-Spinning Systems

5.2.4

Construction of Hybrids for Aligned-Spin Systems

Phenomenological Model

73
75
82

85

5.3.1

Phase Model

85

5.3.2

Amplitude Model

5.3.3

Mapping the Phenomenological Coefficients86

Summary and Future Work

87

89

6

real searches for gravitational waves

6.3

71

Basic Notions

applications

6.2

69

5.2.1

iii

6.1

69

93
95

The Data Analysis Strategy for Coalescing Binaries
6.1.1

The Optimal Filter

6.1.2

Template Construction

6.1.3

Signal-Based Vetoes

6.1.4

Coincidence Test and Background Estimation

6.1.5

Detection Statistic: False-Alarm Rate

95

95
97
97

The Low-Mass Search in S5 LIGO Data

99

100

101

6.2.1

Description of the Search

102

6.2.2

Search Results

6.2.3

Discussion of the Low-Mass Search

105

The High-Mass Search in S5 LIGO Data

109
109

6.3.1

Motivation and Strategy for a High-Mass Search

110

6.3.2

Preliminary Results of the High-Mass Search

112contents

7

numerical injections in gravitational searches
7.1

7.2

7.3
8

9

The NINJA Project

117

117

7.1.1

Numerical Waveforms

7.1.2

Creation of NINJA Data

118
121

Search with a Phenomenological Template Bank

126

7.2.1

The Non-Spinning Phenomenological Waveform Model

7.2.2

The Search Strategy

7.2.3

Efficiency for Detection

7.2.4

Accuracy for Parameter Estimation

127
130

Looking Ahead: The NINJA 2 Project

133

135

astrophysics of intermediate-mass black holes
8.1

Formation of an Intermediate-Mass Black Hole

8.2

Life of a Massive Binary
Birth

8.2.2

Growing Up (Shrinking Down)

8.2.3

Death

139
140

142

8.3

Horizon Distance and Signal-To-Noise Ratio

8.4

Event Rates

8.5

Summary and Conclusions

bibliography

146

153
155

curriculum vitæpublications

171

137

139

8.2.1

epilogue

169

xiii

150

144

137

126

LIST OF FIGURES

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8

Figure 9
Figure 10
Figure 11

Figure 12

Figure 13
Figure 14
Figure 15

Figure 16

Figure 17

Figure 18
Figure 19

Effect of the two polarizations of a gravitational wave
propagating through a ring of test particles
13
Schematic optical layout of a gravitational-wave interferometer
18
International network of GW interferometers
20
Strain sensitivity curves for the ground-based interferometers in operation during S5/VSR1
23
Expected sources of gravitational waves for future groundbased and space-borne detectors
24
Typical gravitational wave emitted by a coalescing compact binary
33
Pictorial representation of a binary black-hole system
in a quasicircular orbit
40
Different variants of constructing the PN Fourier amplitude in the stationary phase approximation for theequal mass case47
Ringdown signals emitted by sources of total mass
100 M and different spin values
50
Foliation of the space-time in the 3+1 decompositionof the Einstein equations
54
Typical numerical relativity waveform for the late inspiral, merger and ringdown or a binary black-holesystem64
Comparison between the double integration in thetime domain and the division in the frequency domain for the computation of the gravitational-wave
strain68
Contour plot for the fitting error ∆φ0 in the (fL , ∆f)
plane
76
Dependence of the fitting errors in η, φ0 and t0 on
the frequency window (fL , ∆f)
77Best fit value of η as a function of the start frequency
fL of the matching window for the waveform whichcorresponds nominally to a mass ratio 1 : 2
78Distinguishability of different hybrid waveforms constructed from Llama equal-mass waveforms at different resolutions
80
Distinguishability of different hybrid waveforms constructed from NR waveforms produced with either
BAM or Llama
80
Initial LIGO’s ability to distinguish hybrid waveforms
constructed from different PN approximants
81
Illustration of the method for constructing PN-NR
hybrid waveforms in the frequency domain
83

xv

xvi

List of Figures

Figure 20

Figure 21
Figure 22
Figure 23
Figure 24

Figure 25
Figure 26

Figure 27

Figure 28

Figure 29
Figure 30
Figure 31
Figure 32
Figure 33
Figure 34

Figure 35
Figure 36
Figure 37
Figure 38
Figure 39
Figure 40
Figure 41

Comparison of hybrid waveforms with the numerical data used in their construction in the time domain
84
Fitting procedure for the amplitude87
Map of the phenomenological parameters to the physical parameters of the binary η and χ
88Overlaps between the hybrid waveforms and the proposed phenomenological fit for Advanced LIGO 90
Overlaps between the NR data-sets #4–7ab and the predicted phenomenological waveforms from our model
for advanced LIGO
90
Diagram demonstrating how the χ2 -test discriminates
between true and spurious signals
99
Parameter space of masses surveyed by the low- andhigh-mass searches for coalescing compact binaries in
S5 LIGO data
102
Combined and un-combined inverse false-alarm rates
for triple H1H2L1 and double H1L1 triggers in month
4 of the 2nd year S5 low-mass search
104
Horizon distance in Mpc as a function of the binary’s
total mass for the three interferometers in operation
during month 4 of the 2nd year S5 low-mass search
107
The posterior distribution for the rate of BNS coalescences in the S5 LIGO low-mass search
107
The marginalized 90% rate upper limits as a functionof mass for the LIGO S5 low-mass search
108
Expected horizon distance of the LIGO detectors for
inspiral-only and full signals
111
Found and missed injection during months 1 and 2 ofthe high-mass S5 LIGO search
113
Plot of SNR versus χ2 in L1 during months 1 and 2 ofthe high-mass S5 LIGO search
114
Playground zero-lag events observed compared to background during months 1 and 2 of the high-mass S5
LIGO search
116
Summary of all numerical relativity waveforms submitted to the NINJA project
119
Distribution of power into different spherical harmonics for the waveforms submitted to the NINJA project 120
Design spectra of the first generation LIGO and Virgo
detectors and the NINJA noise curves
124
Total mass and distance of the 126 NINJA injections 125
Two time-domain phenomenological waveforms fromthe template bank used in the NINJA search
128
Found and missed injections in the NINJA search witha phenomenological template bank131
Accuracy in the recovery of chirp mass and end timeof the reduced set of NINJA injections using the phenomenological template bank134

list of figures

Figure 42

Figure 43
Figure 44

Figure 45

Figure 46

Figure 47

Figure 48

Amplitude of equal-mass, non-spinning BBH systems
scaled to various total masses of the IMBH binary
compared with the sensitivity curves of various detectors
138
Triaxiality of the resulting merged cluster for different
mass fractions and the mass fraction 0.5
141
Amplitude of the GW emitted by a system of two
equal-mass IMBHs of total mass 878.4 M placed at
100 Mpc as seen by different GW observatories 143
Sky-averaged horizon distance versus redshifted mass
for three binary configurations obtained with the design sensitivity curves of Advanced LIGO and the Einstein Telescope
144
Signal-to-noise ratio as a function of the total mass of
the BBH for the present and future generations of GW
detectors and LISA 145
Three parameterizations of the stellar formation rate
per unit comoving volume compared in the calculationof the event rates
147
Sky-averaged reach of the Advanced LIGO and the ET
detectors up to z = 5 versus intrinsic total mass of the
IMBH binary
148

xvii

LIST OF TABLES

Table 1
Table 2
Table 3
Table 4

Table 5
Table 6

Table 7

Table 8
Table 9
Table 10
Table 11
Table 12
Table 13
Table 14

Terminology for statements related to astrophysicalrates
34
Compact binary coalescence rates per Mpc3 per Myr
35
Detection rates for compact binary coalescence sources
36
The first QNM frequencies ωM of the spherical, nonspinning Schwarzschild black hole for l = 2, 3, given
in geometrical units
49
Some properties of currently-working NR evolutioncodes
63
NR codes and configurations used for the construction and verification of our hybrid waveforms and phenomenological model
66
Coefficients to map the 9 free parameters Λk of ourphenomenological model to the physical parametersof the BBH binary
89
Detailed results from BNS from the S5 LIGO low-mass
search106
Overview of results from BNS, BBH and BHNS fromthe S5 LIGO low-mass search
106
Initial conditions for the numerical waveforms submitted to the NINJA project
122
Characteristic duration, mass and frequencies of the
waveforms submitted to the NINJA project
123
Results of the search for NINJA signals using the nonspinning phenomenological template bank
130
Overview of the missed injections for the search witha phenomenological template bank132
Event rates for IMBH binaries formed in the singlecluster channel potentially observable by Advanced
LIGO and the ET per year
149

xix

ACRONYMS

ADM Arnowitt-Deser-Misner
AIGO Australian International Gravitational Observatory
BBH Binary Black Hole
BHNS Black Hole–Neutron Star Binary
BNS Binary Neutron Stars
BSSN Baumgarte-Shapiro-Shibata-Nakamura
CBC Compact Binary Coalescence
CLIO Cryogenic Laser Interferometer Observatory
CT Conformal Traceless
DQ Data Quality
EOB Effective One Body
ET Einstein Telescope
FAR False-Alarm RateGRB Gamma-Ray Burst
GW Gravitational Wave
IFAR Inverse False-Alarm RateIFO Interferometer
IMBH Intermediate-Mass Black HoleIMR Inspiral-Merger-Ringdown
ISCO Innermost Stable Circular OrbitΛCDM Λ Cold Dark Matter
LCGT Large-Scale Cryogenic Gravitational-Wave Telescope
LIGO Laser Interferometer Gravitational-Wave Observatory
LHO LIGO Hanford Observatory
LLO LIGO Livingston Observatory
LISA Laser Interferometer Space Antenna
LSC LIGO Scientific Collaboration
NINJA Numerical Injection Analysis

xxi

xxii

acronyms

NR Numerical Relativity
PBH Brimordial Black-hole Binaries
PN Post-Newtonian
QNM Quasi-Normal Mode
SFR Star Formation Rate
SMBH Supermassive Black HoleSMS Supermassive StarSNR Signal-To-Noise Ratio
SPA Stationary-Phase Approximation
SPH Smooth Particle Hydrodynamics
TT Transverse-Traceless
VMS Very Massive Star
WMAP Wilkinson Microwave Anisotropy Probelucía santamaría lara
COALESCENCE OF BLACK-HOLE BINARIES: FROMTHEORETICAL SOURCE MODELS TO APPLICATIONS INGRAVITATIONAL-WAVE ASTRONOMYOVERVIEW

Few other physical laws or mathematical theories have captivated the imagination of generations of scientists the way general relativity has. The geometricaltreatment that Einstein proposed to explain the structure of space-time andits connection with the dynamics of matter possesses a special simplicity andelegance. Particularly fascinating are the far-reaching gedankenexperimentsthat Einstein devised in order to arrive at previously unexplored conclusions
about the consequences of travelling at velocities comparable to that of light.
General relativity is a theory developed from first principles, which applied
novel, abstract mathematical techniques to the understanding of our Universe;
nevertheless, when confronted with experimental phenomena, the predictions
happened to be in excellent agreement with the observations. Relativity goes
one step further than Newton’s theory of gravitation and is able to correctly
calculate the discrepancies in the precession of Mercury’s perihelion. Its accuracy is the basis for the current global positioning systems [44]. And perhaps
more significantly: it predicts the existence of gravitational waves, ripples inspace-time produced by the accelerated movement of massive objects in thefour-dimensional Universe in which we live.

A distorted rubber sheet gives a simple way of visualizing general relativity.
http://www.jrank.org/space/pages/2362/general-relativity.html

The discovery of the Hulse-Taylor binary pulsar in 1974 led to the observation of the decay of its orbit, in precise agreement with the loss of energy
due to gravitational radiation predicted by general relativity [250]. To date,
this constitutes the strongest direct evidence for the existence of gravitational
waves. Direct measurement of such radiation has not yet occurred; nevertheless, active experimental and theoretical efforts are underway to detectgravitational emissions for the first time.
The work presented in this dissertation focuses on the characterization of
the gravitational radiation emitted by binary systems of black holes. The main
focus is the connection of the theoretical predictions for the gravitationalradiation emitted by coalescing black-hole binaries with their applicationin gravitational-wave astronomy. To this end, analytical and numerical approaches to solving the binary black hole problem are brought together withthe goal of accurately modelling the gravitational signature of non-precessing
binary systems. This research is of immediate relevance to current and futuredata-analysis efforts in the context of searches for gravitational waves withthe detectors LIGO and Virgo.

3

The object’s name is
PSR B1913+16, also
known asPSR J1915+1606 andPSR 1913+16

4

overview

This dissertation is structured as follows: chapter 1 introduces the canonicalmathematical formulation of the Einstein equations in the weak-field approximation, which predicts the existence of gravitational waves. It is followed
by a short description of the history of searches for gravitational radiation,
including the development of currently-operating interferometers. Nowadays,
an international network of detectors exists that is actively operating at design
sensitivity and working towards meeting the challenge of the first direct measurement of gravitational waves. Additionally, a brief overview of the futureof gravitational-wave astronomy as it is foreseen for the next decade is given.
Chapter 2 presents various sources of gravitational radiation that can bereasonably expected to exist in our Universe, in particular coalescing binaries,
for they are the main focus of this work. An introduction to the coalescenceprocess is given, describing in detail the three stages in which the full process iscommonly divided — adiabatic inspiral, non-linear merger and perturbative
ringdown to a final Kerr black hole. Finally, the expected astrophysical rates
of occurrences of binaries in our Universe are given for the current and future
generations of gravitational-wave interferometers.
The in-depth description of the theoretical methods that are employed
to solve the binary black hole coalescence can be found in chapters 3 and 4,
which respectively present the analytical and the numerical approaches tothe problem. Chapter 3 first introduces the post-Newtonian treatment of
the inspiral phase to later concentrate on perturbative methods for the final
ringdown. Chapter 4 describes the foundations of numerical relativity andits long-awaited success in performing full, non-linear simulations of the lateinspiral, merger and ringdown parts of the black-hole binary coalescence.
One of the main results of this dissertation is presented in chapter 5. A new
method is proposed to model the full coalescence of non-precessing binary
black hole systems in the frequency domain, that accurately incorporates themost up-to-date contributions from post-Newtonian theory and state-of-theart numerical relativity simulations. The introduction of the novel phenomenological model is accompanied with a general description of the challenges
that need to be taken into account when comparing post-Newtonian andnumerical methods. A discussion of the different possible sources of error in
the construction of hybrid waveforms is provided as well.
In the context of joint work within the LIGO and Virgo Scientific Collaborations, the author has contributed to the analysis of LIGO data corresponding
to the period 2005–2007. The description of two searches for coalescing binaries in different regions of the mass parameter range is given in chapter 6. The
results of one of them provide the latest upper limits on the rate of compact
binaries within the surveyed mass range in our local Universe via direct datataking with gravitational-wave detectors. For the other search, preliminary
results are presented.
Chapter 7 is an introduction to the use of waveforms from numerical
relativity in the context of software injections into simulated detector noise,with the primary goal of assessing the reliability of current data analysis
techniques to signals that include the merger and ringdown of the binary. In
this context, a multi-disciplinary collaboration was established in 2008 withthe task of carrying out this program. I participated in this first challenge in
two direct ways, firstly, as a co-developer of the software required to performthe injections of numerical data into the detector noise and, secondly, as
analyst of such data by means of a family of non-spinning templates modelingoverviewthe full binary black hole coalescence. Chapter 7 summarizes the general
guidelines under which the project was formulated as well as the findings
obtained through our analysis.
Finally, the dissertation concludes with a brief look at astrophysics in chapter 8. The primary goal of gravitational-wave astronomy is to reach a stage
when astrophysically-relevant statements can be made once gravitational radiation is measured and a new window to the Universe is opened. Profoundknowledge of the expected astrophysical sources is certainly key to reach thatstage. Chapter 8 addresses the question of the existence of a particular classof black holes, those with an intermediate mass comprised among hundreds
and tens of thousands of solar masses. Observational evidence for binaries
formed by two of these objects exists but is not conclusive. Hence, there isan unexplored field where gravitational-wave astronomy with second- andthird-generation interferometers might be able to provide further insight.
The consequences that the existence of intermediate-mass black-hole binaries
could have for Advanced LIGO and other future detectors are explored.
Geometric units are used throughout this dissertation, i.e. G = c = 1;
thus, mass, time and distance are measured in the same units. The space-time
metric is assumed to have signature (−, +, +, +). The mass of our sun is1 M = 1.9891 × 1030 kg = 1.4766 × 103 m = 4.92549 × 10−6 s.
Unless otherwise specified, lower case Greek indices (µ, ν, ρ, σ, . . .) range
from 0 to 3, whereas lower case Latin indices (i, j, k, . . .) range from 1 to 3.

5

Part I
THE UNIVERSE THROUGH A NEW WINDOW1GRAVITATIONAL WAVESOver a century ago, physicist Albert Einstein revolutionized the field of gravitation, introducing a shift of scientific paradigm in our understanding of
the structure of space-time. His theory of general relativity is the current
description of gravitation in modern physics. It unifies special relativity and
Newton’s law of universal gravitation. General relativity also predicts the existence of gravitational waves, which have since been measured indirectly;
a direct measurement has not yet occurred, but experiments are currently
underway — certain aspects related to this research field constitute the main
focus of this thesis.
The field of gravitational-wave research has accumulated a rich and somewhat controversial history over its more than five decades of existence [94].
From the primitive Weber cylinders to the low-temperature bars and sphericalresonant detectors, afterwards superseded by sophisticated interferometers,
the search for gravitational waves constitutes not only a fascinating theoreticalendeavour, but also a challenging experimental enterprise.
In this chapter, Einstein’s theory and its prediction of gravitational waves are
introduced, as well as past efforts towards detection of gravitational radiation;in addition, the technology underlying the currently-operating observatories
that are expected to grant positive results within the next few years is described.
1.1 einstein’s theory of gravitation
General relativity asserts that the curvature of space-time causes gravity. The
presence of matter curves space-time and the curvature in turn determines the
behaviour of matter. Einstein arrived at this revolutionary idea by means of aseries of gedankenexperiments or thought experiments based on the assumptions that the speed of light was a constant of our universe and that the motionof free-falling objects was universal. Let us have a look in the next sections at
the elegant mathematical formulation that expresses this phenomenon.
1.1.1

The Einstein Field Equations

In geometric units and covariant notation, the Einstein equations for thegravitational field take the abbreviated form [110]Gµν = 8πTµν ,(1.1)

where Gµν = Rµν − 21 Rgµν is the Einstein tensor, gµν is the metric of
the four-dimensional space-time and Tµν is the stress-energy tensor that
encodes the matter content. In this expression, it is the Einstein tensor Gµνthat is used to express the curvature of the Riemann manifold. The Ricci
scalar R = gµν Rµν , also known as scalar curvature, is the trace of the Ricci
tensor, which in turn is defined as the trace over the first and third indicesof the Riemann curvature tensor, Rµν = Rγµγν . The Riemann tensor is afundamental object in differential geometry that measures the extent to which

9“Space-time grips mass,
telling it how to move,and mass grips
space-time, telling it
how to curve”— John
Archibald Wheeler

10gravitational waves

the metric tensor is not locally isometric to a Euclidean space. The curvaturetensor is given in terms of the Levi-Civita connection ∇ of the space-time as(∇µ ∇ν − ∇ν ∇µ ) υρ = Rρσµν υσ ,Equations 1.4 and 1.5
are known as the firstand second Bianchi
identities(1.2)

for any given vector υµ . The covariant derivatives commute in a flat manifold
but do not if the manifold is curved. The Riemann tensor verifies the following
symmetries and identities [259]Rµνρσ = −Rµνρσ = −Rµνσρ = Rσρµν ,(1.3)

Rµνρσ + Rµρσν + Rµσνρ = 0,

(1.4)

∇λ Rρσµν + ∇ρ Rσλµν + ∇σ Rλρµν = 0.

(1.5)

The Bianchi identities imply ∇µ Tµν = 0, which is the equation of local
conservation of energy and momentum. In absence of sources or in a region
far away from them, equation 1.1 further simplifies to Gµν = 0.
In this geometric language it is hard to notice that 1.1 represents a set ofdifferential equations. But as a matter of fact, the curvature tensor depends
on the metric and its first and second derivativesRρσµν = ∂µ Γ ρνσ − ∂ν Γ ρµσ + Γ ρµα Γ ανσ − Γ ρνα Γ αµσ1Γ ρµν = gρα (∂µ gνα + ∂ν gµα − ∂α gµν ) .
2

(1.6)
(1.7)

Furthermore, we are interested in the study of gravitational radiation, treatedas a small perturbation that propagates through an otherwise flat space-time.
In this weak-field situation there exist coordinate systems where the components of the metric can be decomposed asgµν = ηµν + hµν ,|hµν |  1 throughout space-time,(1.8)

where ηµν is the Minkowski metric that describes a space-time with nocurvature. Such coordinates are called nearly Lorentz coordinates and areparticularly suitable to solve equation 1.1, which predicts gravitational waves.
Einstein showed that the perturbative hµν field can be calculated in a manner
analogous to that of the retarded electrodynamic potentials.
Applying ansatz 1.8 to equation 1.1 and solving for the perturbative radiation
field to first order in hµν yields the linearized Einstein equations−∂α ∂α hµν − ∂α ∂β hαβ + ∂α ∂µ hνα + ∂α ∂µ hµα = 16πTµν (1.9)
on the field hµν ≡ hµν − 12 ηµν hµν , which is introduced for simplicity. Notethat hµν = hµν , and in the case when h(= −h) = 0, then hµν = hµν .
The three last terms on the left-hand side of equation 1.9 serve to keep the
expression gauge-invariant. In general relativity there is a gauge freedom
corresponding to the group of diffeomorphisms. In the linear approximation
0 represent the same physicalthis implies that two perturbations hµν and hµν
phenomenon if they are related by a transformation of the form
0hµν
→ hµν + ∂µ ξν + ∂ν ξµ ,(1.10)

where ξα is a vector field. Without loss of generality, a field ξα can be foundsuch that the following gauge condition is verified∂α hµα = 0(Lorentz gauge),(1.11)

1.1 einstein’s theory of gravitation11

in clear analogy with the Lorentz gauge condition for the electromagnetic
tensor ∂α Aα = 0. However, it is interesting to note that whereas the Maxwell
equations in the Lorentz gauge are valid in every source-free region of thespace, in the case of the linearized Einstein equations we need to impose theadditional condition of being far away from the sources, so that the weak-field
condition 1.8 is satisfied. This taken into account, equation 1.9 in the Lorentz
gauge simplifies to∂α ∂α hµν = 0(in vacuum).

(1.12)Aside of the choice of Lorentz gauge, there remains the freedom to makefurther gauge transformations of the form given in equation 1.10 provided
that ∂α ∂α ξβ = 0, for they leave equation 1.11 unchanged. This means that
the Lorentz condition does not uniquely fix the degrees of freedom in thefield variables. It can be shown [235] that the gauge can be changed whileremaining within the Lorentz class of gauges using any vector solving
∂2(1.13)− 2 + ∇2 ξα = 0.
∂t

The Lorentz gauge is infact a class of gauges

Thus, we can always arrive at the gauge
h=0(1.14)

h0i = 0(i = 1, 2, 3)

h00 = 0

if no sources are present anywhere,

in a source-free region(1.15)(1.16)

which is referred to as Coulomb gauge, also known as radiation gauge. In thistransverse-traceless (TT) gauge hµν = hµν , as already noted. The Einsteinfield equations in vacuum far away from the source of the field take the form∂22
− 2 + ∇ hµν = 0,(1.17)∂ta wave equation for the gravitational radiation that admits the plane-wave
solutionαhµν = aµν eikα x ,(1.18)

where aµν is a four-dimensional symmetric tensor containing the amplitudeof the different components of the wave and kα is the wave vector. Substituting 1.18 in 1.17 yields the condition kα kα = 0, i.e. kα = (ω, ~k) is a null
vector tangent to the world line of a photon, which shows that gravitational
waves propagate at the speed of light. The nullity of kα implies ω2 = |~k|2 .
The choice of Coulomb gauge imposes several constraints on the components of the tensor aµν . The traceless condition 1.14 translates to a = 0; 1.15
and 1.16 imply that only the spatial components of aµν are non-zero. If wefurther orient the direction of propagation of the wave along the z-axis sothat kα = (ω, 0, 0, ω) then aαz = 0 for all α. These conditions reduce thenumber of independent components of aµν from ten to only two


0 0
0
0


0 axx axy 0,aµν = (1.19)

0 axy −axx 00 000

It is natural thatgravitational waves
propagate with speed c,
as c is the only relevant
speed in the theory.
Astrophysicist Sir
Arthur Eddington,
however, liked toexpress his skepticism
by saying that“gravitational waves
propagate at the speedof thought”

12

gravitational waves

hence the final form of the solution to the source-free, linearized Einsteinequations for the perturbative field is


0 0
0
0


0 h+ h× 0 iω(z−t)
ehµν = ,(1.20)
0 h× −h+ 00 000with h+ and h× representing the two polarization states of the wave. A general
gravitational wave can be written as a linear combination of the plus and crosscomponents h = h+ ê+ + h× ê× in the orthonormal basis of vectors
!
!
1 0
0 1
ê+ =
ê× =
.
(1.21)0 −1
1 0
Rotations of the x- and y-axes in the transverse plan by an angle ψ change
the polarization components in the following way
0h+= h+ cos 2ψ + h× sin 2ψ
0h×= −h+ sin 2ψ + h× cos 2ψ,(1.22)

which indicates that general relativistic gravitational waves have spin two,
because the source of gravity is the stress-energy tensor, which is a secondrank tensor. This is of importance for issues related to quantization of gravity,
which fall beyond the scope of this thesis.
1.1.2 Effects of Gravitational Waves on Test Particles

These threegravitational-wave
detection mechanisms
are, respectively, thefunctional principles ofinterferometers, pulsar
timing arrays andresonant bars

The formalism developed in section 1.1.1 shows how gravitational radiationfrom a far source propagates throughout the universe in the form of ripples
in the space-time, deforming its geometry. Thus, in principle, gravitational
waves could be detected by accurately tracking the separation of a pair offreely-suspended masses. Using the same principle, one could try to measure
the perturbation on the paths of photons coming from a far object —such as amillisecond-period pulsar— when they are affected by a passing gravitational
wave. Alternatively, masses connected by a solid piece of material would react
to the gravitational tidal forces that stress the material.
Light constitutes an excellent way of measuring proper distances between
free-falling objects. Pulses of electromagnetic radiation propagating in a region affected by a gravitational field react to it in a measurable way that can
provide us with information about the metric of the space-time. Consider
two free-falling particles A and B in a background Lorentz frame and choose
the TT gauge introduced in section 1.1.1 associated to this frame. We will workin the TT gauge, for it is the most appropriate when dealing with radiation
space-times. Besides, test particles in a given spatial location remain at thosesame TT coordinates, which is particularly convenient.
We shall denote ξα the connecting vector between A and B. Free particles
obey the geodesic equation for their 4-velocity uα [259]uα ∇α uβ = 0.

(1.23)

1.1 einstein’s theory of gravitation

y

x
(a)

(b)

(c)

Figure 1: Effect of the two polarizations of a gravitational wave propagating througha ring of test particles. (a) A ring of free particles before a wave travellingin the z-direction reaches them. (b) Distortion produced in the ring by
the plus polarization h+ , which modifies the proper distance between theparticles in the ring, altering its geometry as the phase of the gravitational
wave changes through a complete oscillation cycle. (c) Same as (b) but forthe cross polarization h× . The effects are not drawn to scale but have been
greatly magnified.

In a curved space-time, the second derivative of the vector ξα is non-zero,
which means that there is an acceleration between particles A and B given by
the equation of geodesic deviation
aα =

d2 ξα
β µ ν
α
α
= −Rα
βµν ξ u u = −Rβ00 ξ ,dt2(1.24)

since the 4-velocity uα has coordinates (1, 0, 0, 0) in this coordinate basis.
In the TT gauge, the relevant components of the Riemann tensor, defined inequation 1.6, can be easily calculated
Rν00µ =

1 d2 hµν,
2 dt2(1.25)

which, according to equation 1.20, are only non-zero for µν = ij with i, j =1, 2 representing the x- and y-directions. Thusd2 ξi
1 d2 hij j
=
ξ.
dt2
2 dt2

(1.26)

Figure 1 illustrates the meaning of equation 1.26. A ring of particles placed at
rest on the xy-plane in an initially wave-free region of space-time encountersa gravitational wave travelling along the z-direction. The arrival of the wave
modifies the proper distance between the particles. The plus polarization of
the wave stretches and squeezes the ring along the x- and y-axes, oscillating
between the shapes displayed on panel (b) of figure 1. The √cross polarization
distorts the ring along the directions given by (êx ± êy )/ 2. The effects ofthe two polarizations are rotated 45◦ relative to one another, in contrast to
the electromagnetic field, where the rotation angle is 90◦ .
Once we know the effect of a passing gravitational wave on a pair of test
particles, we can proceed to devise mechanisms for detecting that effect, for

13

14

gravitational waves

instance, by means of a laser and an interferometer of arm’s length L. An
assumption that is usually made for ground-based detectors is that the wavelength of the gravitational wave is much larger than the size the interferometer,
λGW  L. This simplifies the integration of equation 1.26
δx =

hxx
x,
2

δy =

hyy
y.
2

(1.27)

Thus, the variation of the proper lengths of the interferometer’s arms is proportional to the original distance between them. When a gravitational wave
passes along the z-direction with polarization hxx = −hyy = h+ (t), thechange in proper distance and the phase difference between the two beams at
the origin are
δL(t)= h+ (t),L∆φ = 2πL
h+ (t).
λ(1.28)The interferometer response, i.e. the difference in phase between the beams
recombining at the beam splitter is proportional toh(t) ∝ δ(∆φ) ≡ F+ h+ (t) + F× h× (t),(1.29)

where F+ and F× are the antenna patterns of the detector, which encode theprojections between the wave’s polarizations in the radiation’s and detector’sreference frames. The quantity h is the gravitational wave strain.
In this derivation it is implicitly assumed that the spatial variation of thegravitational wave in the interferometer’s arms is negligible. The temporalvariation of h(t) in during the short time ≈ 2L that it takes for the light totravel back and forth to the mirror is not taken into account. The result of
the derivation is right, however it is in principle not correct to use the spatial
distance. As a matter of fact, the photons travel along null geodesics, notspatial. One can calculate how the frequency of the photon ω changes whentravelling through a space-time affected by gravitational radiation:δω1= − x̂i x̂i ḣij L.
ω2

For concrete examples
of correct derivations of
the response functions
for LIGO and LISA werefer the reader to thearticles [226] and [117]respectively(1.30)The passing gravitational wave affects the photon by altering its path; tracking
the Doppler shift of a sinusoidal electromagnetic signal allows us to measure
the effect of the gravitational radiation. For a complete, general derivationof the Doppler shift of a signal transmitted and transponded from a distantspacecraft, including the results obtained when dropping the assumption oflong wavelength, the reader is referred to [113]. This paper is fundamental
for understanding the physics that underlie the operation of the space-borne
antenna LISA, and also generalizes commonly-used formulations of the effectof gravitational waves on ground-based detectors like LIGO and Virgo.
1.2 history of gravitational-wave detectors
Weber studied gravitational radiation with Wheeler and pioneered the field ofgravitational-wave detectors at a time —late 1950s and early 1960s— when themere existence of gravitational waves was not widely accepted. He designed
and built the first gravitational-wave detector in the form of an aluminum
cylinder carefully suspended and connected to instrumentation to observe1.2 history of gravitational-wave detectors

15

oscillations of the bar’s fundamental mode [261]. A passing gravitational wave
would set the cylinder vibrating at its resonant frequency —about 1660 Hz—
and piezoelectric crystals firmly attached around the cylinder’s waist would
convert that ringing into an electrical signal. Due to the extreme weakness ofthe gravitational radiation, the cylinder had to be massive and the piezoelectric
sensors very sensitive and capable of detecting a change in the cylinder’s lengthby about 10−16 meters.
This is indeed what Weber reported a couple of years after the constructionof his first detector [262, 263]. By that time his group in Maryland had built
several cylindric bars and moved one of them to Argonne National Laboratory,
near Chicago, about 1000 km away. Reported were observations of abovebackground coincidences between both detectors with very low false alarmprobability, which were interpreted as “good evidence” for gravitational waves.
In his publication from 1970 [263], Weber even claimed to have measured
largely anisotropic radiation that peaked in the direction of the galactic center.
In the years subsequent to Weber’s detection claim, a number of independent groups built their own bars and tried to reproduce his results. None of the
instruments —located in Argonne, Glasgow, Moscow, Reading, Rutherford,
Tokyo, Munich, Frascati, Stanford, Rochester and the Bell Laboratories in
New Jersey— were able to measure anything other than random noise, thus
discrediting Weber’s work. By the late 1970s the consensus in the scientificcommunity was that Weber’s results were spurious. His recognition as fatherof gravitational-wave detection efforts is nevertheless well deserved, for hedrew many others into the field.
Soon it became evident that improved detectors were needed in order toapproach the sensitivity region that would allow astrophysicists’ predictions
to be tested. Resonant bars cooled to liquid helium temperatures of ∼3K andbelow at millikelvin —cryogenic bars— promised to reach a sensitivity fiveorders of magnitude better than that of Weber’s bars. Ideally, that would makethem sensitive to the strongest potential sources of gravitational waves in our
Galaxy and in the local group. Thus, a number of ultra-low temperature bars
were designed and constructed during the 1980s and 1990s: the cryogenic
resonant gravitational radiation detector ALLEGRO [180] at Louisiana State
University, the Italian-built bar EXPLORER [46] located at CERN in Geneva,
the coldest and more sensitive AURIGA [215] in Legnaro near Padua, theniobium-made bar NIOBE [63] at the University of Western Australia, themillidegree bar NAUTILUS [47] at Frascati.
The experimental challenge faced by the groups operating resonant detectors is better understood with a simple calculation of the tiny effects inducedin the bars by gravitational radiation. Gravitational waves are generated by theacceleration of masses with quadrupolar distributions. A very simple upper
limit for the amplitude of the gravitational waves can be given by



1 GM
GM
h 4
,
(1.31)cr
R

AURIGA: Antenna
Ultracriogenica
Risonante perl’Indagine
Gravitazionale
Astronomica

where M is the mass, r is the distance and R is the radius of the astrophysicalobject. Hence, a neutron star with a typical mass of 1M and R = 10 km
located at a distance of 10 Mpc would produce a strain h  10−21 . Theenergy flux of a gravitational wave is given by

NAUTILUS: Nuova
Antenna a Ultrabassa
Temperatura per
Investigare nel Lontano
Universo le Supernovæ

F=c3|ḣ|2 .
16πG

(1.32)

Physicist Joseph Weber(1919–2000) studied
electronics before
joining the engineering
faculty of the Universityof Maryland, College
Park, where hecompleted his PhD witha thesis on microwave
spectrometry. He gave
the earliest public
lecture on the principles
behind the laser and themaser

The creativity of bar
researchers is shown inthe made-up acronymsof their detectors.
ALLEGRO: A
Louisiana Large
Experimental
Gravitational Radiation
Observatory

16

See [119] for a
description of the
posible orthogonal sets
proposed by Forward
MiniGRAIL:
Gravitational Radiation
Antenna In Leiden

gravitational waves

For a wave of h = 10−21 and frequency 1 kHz the energy flux is about
0.3W/m2 . This flux is comparable to that of the Moon light on the Earth,
however it is very difficult to detect with a resonant bar detector, given thevery small absortion cross section of matter.
One way to improve the detectability of weak signals is to increase thesize of the effect they induce in the detector, which depends on the lenght ofthe bar, the amplitude of the wave and the quality factor Q of the material.
The value of Q is a characteristic of each oscillator, defined as the ratio of
the stored energy to the energy dissipated per one radian of the oscillation.
Unfortunately, the size of a bar detector can not be increased arbitrarily, for an
object of dimensions larger than a few meters would be almost impossible toisolate from external noise sources. The amplitude of the gravitational waves
is likewise determined by the position, distance and characteristics of thesources. Hence, the only alternative to achieve a larger and easier-to-measureeffect in the detector is to build it of a high Q-value material, such as sapphire.
A high Q means that the bar stays for a longer time in an excited state, which
results in an increased signal-to-noise ratio. Bars made of sapphire were
proposed and constructed, but no sustantially superior results were reported.
In fact, a major drawback of cylindric detectors comes intrinsically fromtheir geometric design: when a bar is hit by gravitational wave, it rings; more
specifically it expands and contracts along its length. These detectors are
therefore relatively insensitive to gravitational waves travelling along theiraxes, since gravitational waves are transverse waves. A more optimal design
choice is that of spherical detectors, which are responsive to radiation arrivingfrom any direction. The transducers that measure the small disturbances inthe detector can be placed everywhere on the sphere. As a matter of fact, abasis of five tensors suffices to determine all directions, polarizations, andmagnitudes of incident gravitational waves.
Two projects based on spherical detectors are currently operating or preparing commissioning runs. The pioneer is MiniGRAIL [106], located at the
Leiden University in the Netherlands. The MiniGRAIL detector is a cryogenic
68 cm diameter spherical gravitational wave antenna made of CuAl with amass of 1400 kg, a resonance frequency of 2.9 kHz and a bandwidth around
230 Hz. It aims to operate at 20 mK. The “Mario Schenberg” detector [21] at
the University of São Paulo in Brazil has a diameter of 65 cm and weights
1150 kg. The two detectors will ideally operate in coincidence, searchingfor high frequency events in the 3.0–3.4 kHz frequency bandwidth, such as
rotating neutron star instabilities or small black hole mergers. The quantumlimited strain sensitivity dL/L of these antennas ranges from 4 × 10−21 to10−22 .
The theory of how to detect below the quantum limit and the challenge of
how to manipulate the Heisenberg uncertainty principle in a macroscopic
object has not yet been met in practice and that is one aspect that makes
detection of gravitational waves with resonant detectors so fascinating and yet
so arduous. Another serious problem bars encounter is the narrow frequency
band —around their resonance frequency, typically above 600 Hz— that they
are capable of surveying. Most strong sources of gravitational waves, suchas binaries of moderate to massive black holes, emit at lower frequencies
than that and on a wide range of frequencies. Moreover, the relatively small
dimensions of resonant detectors seriously limit the maximum size of thetidal stretching induced in them by a passing gravitational wave and make

1.3 gravitational-wave interferometry

them more prone to be affected by quantum, thermal and vibration noise
sources. All these inherent difficulties for measuring gravitational ratiation
with resonant detectors have led the field to a gradual decline after the year
2000, while gravitational-wave interferometry has progresively taken theleading role in the detection effort.
1.3 gravitational-wave interferometry
Laser interferometers use light to measure the distance between two freefalling mirrors. Nowadays they are the most sensitive operating gravitationalwave detectors and constitute the most promising technology for performing
the first measurements within the next years. Their superiority over resonant
detectors comes from their larger size, which translates in an inherently better sensitivity, and from their ability to detect signals in a broad range of
frequencies.
A typical astrophysical source of interest for gravitational-wave astronomy
has frequency components at fGW ∼ 100 Hz, corresponding to a wavelengthof λGW ∼ 3000 km. Maximum sensitivity of the interferometer is achieved
when the wave spends half of its period in the arms, this isL=

λGW∼ 1500 km,
2(1.33)an unrealistic length for a ground-based detector. In practice, a simple Michelson interferometer is enhanced by means of two additional near mirrors
placed near the beam splitter. The Fabry–Pérot cavities store the beams andincrease the effective path length.
A simple diagram showing the basics of a gravitational-wave interferometer is given in figure 2. The design consists of a power-recycled Michelsoninterferometer with Fabry–Pérot arms. Two mirrors are suspended at eachend of the L–shaped detector arms. The light originates in a pre-stabilized
laser that passes through an optical mode cleaner and is divided in two paths
at the beam splitter, travelling back and forth within the Fabry-Pérot cavities.
The two arms form cavities that trap much of the light that enters, due to thealmost perfect reflectance of the near mirrors. The associated power gain increases the sensitivity. All mirrors, including the beam splitter, are suspended
in order to remove noise associated to mechanical vibration. The light thatexits the cavity after being reflected in the far mirrors eventually returns to thebeam splitter and the two separate beams recombine. The returning beams
are kept out of phase so that when the arms are both in resonance —when
there is no gravitational wave passing through— their light waves subtract,and no light should arrive at the photodiode.
When a gravitational wave passes through the interferometer, the distancesalong the arms of the interferometer are shortened and lengthened as shownin section 1.1.2, causing the beams to become slightly less out of phase, so somelight arrives at the photodiode, indicating a signal. Light that does not containa signal is not wasted but it is returned to the interferometer using a power
recycling mirror, thus increasing the power of the light in the arms. During
normal operation of the interferometer, noise sources can cause movementin the optics, producing similar effects to real gravitational waves — as amatter of fact, a major complexity factor in the effort of detecting gravitational

17

18

gravitational waves

Laser

Photo
detectorPower
recyclingmirrorNearmirrorBeamsplitter

Arm 2
Arm 1

Far
mirrorFigure 2: Schematic optical layout of a gravitational-wave interferometer, showing its
main components. The diagram is not to scale and does not display otherrefinements of the optical design.

radiation is finding ways to extract the real signals without confusing themwith these spurious motions of the mirrors.
In designing an interferometer for gravitational waves, the same principles
operating on bar detectors apply: an increase in the instrument’s size entails alarger signal and thus a reduction in the noise. Contrary to resonant detectors,
interferometers can expand in size and obtain a larger signal by means of alonger arm. Nevertheless, the signal will always be masked by a variety of
instrumental sources of noise. The significance of the main noise sources will
be briefly reviewed in the next subsections.
Seismic Noise
Undesired ground vibrations, due to seismic and human activities, limit thesensitivity of the interferometers at low frequencies. To filter out these disturbances, the optical components are suspended to a series of several pendulums,
each hanging from the above. Above their resonance frequency, a simple pendulum is actually a second-order low-pass filter, with a response function
attenuated as 1/f2 for the mirror motion. The interferometer Virgo [257] uses
seven-stage pendulums —called superattenuators— which effectively filter
out the seismic noise, enabling detection down to ∼ 4 Hz. In the case of the
LIGO [173] and GEO600 [127] detectors, double and triple pendulums areused. The lower cut-off frequencies below which the seismic noise dominates
are slightly larger than that of Virgo, around 40 and 50 Hz respectively.
Thermal Noise
This source of noise is associated to thermal vibrations of the mirrors and theirsuspensions. The steel wire that suspends the mirror is at room temperature,in thermal equilibrium with the environment. Its thermal fluctuations induce
a motion in the mirror that changes the length of the interferometer’s arm.
In order to keep thermal noise as low as possible, the use of ultra-high-Q

1.4 world-wide international network of detectors

19materials in the construction of mirrors and suspension wires is advisable.
Suspension thermal noise limits the sensitivity of the detectors mainly in the
50–200 Hz band.
Shot Noise
Photon shot noise is the major limiting source of disturbance at frequencies
above 200 Hz and results from the finite number of photons arriving at thephoto-detector. Phase and amplitude of the light field are conjugate variables
which can not be determined simultaneously with arbitrary precision. Thus,
the power of the light beam is expected to fluctuate randomly, affecting themeasurements in the photodiode. Shot noise can be reduced by increasing thelaser power. In order to be able to detect gravitational waves with frequency∼ 100 Hz, the intensity of the laser would need to be of the order of ∼ 100 W,
a value beyond the capability of any existing continuous laser. To circumvent
this limitation, power reciclying techniques are used to increase the power in
the interferometer’s arm and Fabry–Pérot cavities are installed which amplify
the phase shift accummulated. Using such a design, the typically-used 10 W
lasers suffice to achieve the desired sensitivity at high frequencies.
Other Quantum Sources of Noise
Although photon shot noise can be reduced by increasing the laser power, thisin turn will increase the size of the fluctuations in the laser intensity and in
the laser pressure on the mirrors. This quantum limit does eventually become
the limiting noise factor at high powers. Future-generation interferometers
will need to deal with this source of noise by means of signal recycling andthe use of squeezed light. Such investigations are already taking place incurrently-operating interferometers, like the GEO600 detector.
1.4 the world-wide international network of detectors
As noted in section 1.3, the extreme sensitivity required for measuring the tiny
effects that gravitational radiation induce in matter, together with the unavoidable presence of spurious noise sources in the detector imply that random
internal disturbances might masquerade as real signals. In practice, any kindof detection claim needs to be corroborated by coincident observations from
more than one interferometer in order for the scientists to have confidence.
Besides, it is convenient to have more than one detection instrument, since
the millisecond-scale time delays between observations at different sites canprovide information on the location of the source on the sky, a crucial aspect
for gravitational-wave astronomy. Each new interferometer improves thesensitivity of all existing ones.
In this spirit, several independent projects that now form an international
network of ground-based gravitational-wave interferometers have been designed, funded, constructed and brought into operation in the course of thelast 20 years. In figure 3 the locations of both currently-operating and planned
detectors all over the world are marked. They correspond to the sites of the
LIGO [173], Virgo [257], GEO600 [127], TAMA300 [249] and AIGO [48]detectors.
LIGO operates two gravitational wave observatories taking data simultaneously: the LIGO Livingston Observatory (30◦ 33’46.42” N 90◦ 46’27.27” W)

LIGO: Laser
Interferometer
Gravitational-wave
Observatory

20

gravitational waves

LHO

GEO
Virgo

TAMA

LLO

AIGO

Figure 3: International network of gravitational-wave interferometers. Shown in the
map are the locations of the three American LIGO detectors at two sites,
LHO in Hanford, WA and LLO in Livingston, LA, the Dutch-French-ItalianVirgo detector in Cascina near Pisa, the British-German GEO600 locatednear Hannover, the Japanese detector TAMA300 in Tokyo and the Australian AIGO project near Perth.

in Livingston, Louisiana, USA and the LIGO Hanford Observatory, on the
Hanford Nuclear Reservation (coordinates of central complex: 46◦ 27’18.52” N
119◦ 24’27.56” W), located near Richland, Washington, USA. LHO hosts two
detectors of arm lengths 2 km and 4 km; LLO hosts one 4 km-arm detector.
The two LIGO sites are separated by 3002 km. Since gravitational waves travelat the speed of light, this distance corresponds to a difference in gravitationalwave arrival times of up to ten milliseconds. LIGO is the largest and mostambitious project ever funded by the American National Science Foundation.
The construction of the facilities was completed in 1999, a series of sciencedata-taking runs started in 2002 and the LIGO design sensitivity was reachedin November 2005, initiating a two-year period that culminated after one fullyear of coincident data among the three LIGO detectors was taken, the S5 run.
As of spring 2010 the analysis of these data, split in searches for stochastic,
continuous, burst and inspiral signals is close to completion. Chapter 6 of thisthesis focuses on the work done by the author in the context of two searches
for coalescent binaries in S5 data.
From its beginnings, LIGO was envisioned not as a one-shot experiment,
but as an ongoing scientific quest with increasing reach. To that end, the two
4 km-arm LHO and LLO detectors have undergone equipment upgrades
and system improvements that have led to LIGO’s sixth science run (S6), in
progress since July 2009. The enhanced interferometers have been boosted
by new optics, increased laser power, advanced seismic isolation tables, andimproved signal sensing. But this is by no means the end of LIGO’s potential.
In addition to enlarging the present scope of gravitational-wave searches,
Enhanced LIGO and S6 are already providing an important testing groundfor yet a new set of improvements, which is expected to be fully implemented
in 2015. This advanced-generation upgrade program, known as Advanced
LIGO, will increase the detectors’ sensitivities by a factor of 10 and probe avolume of space a thousand times greater than initial LIGO. If our present
knowledge of general relativity and expected rates of astrophysical sources in
the local universe are correct, the advanced interferometers should be able todetect gravitational waves within the next five years.

1.4 world-wide international network of detectors

Virgo, named after the Virgo Cluster, is located within the site of EGO (European Gravitational Observatory) at Cascina, Italy (43◦ 37’53” N 10◦ 30’16” E).
With its 3 km-long arms, it is the largest interferometer built outside the USA.
Virgo is funded by EGO, a consortium created by CNRS for France and INFN
for Italy with the aim of fostering gravitational-wave research in Europe. The
construction of Virgo finalized in 2003 and the detector started its first sciencerun, VSR1, in May 2007, joining LIGO’s S5 run in a three-site data-taking effort
that extended until October 2007. In contrast to LIGO, that is built aroundhaving two facilities and using the combined power to make observations,
Virgo is a single powerful detector optimized to extend the bandwidth tolower frequencies with a very sophisticated seismic isolation system.
In a manner similar to that of LIGO, Virgo has planned a two-step upgrade
towards a second-generation detector. Virgo+, the enhanced version of Virgo,
includes a system for thermal compensation and a laser with increased power.
In this configuration, a VSR2 data-taking period started coincidentally with
LIGO’s S6 in July 2009. A one- to two-year joint S6/VSR2 science run withthese enhanced interferometers is foreseen, that will extend until the beginning of 2011. New payloads —dielectric reference mass, fused silica fibers, newmirrors allowing to achieve a higher cavity finess— will be installed in 2010.
The interferometer optical configuration does not change at this stage. On
the contrary, the second-generation detector, Advanced Virgo, will require amajor upgrade, with the goal of increasing the sensitivity by about one orderof magnitude with respect to Virgo in the whole detection band. Its starting
operation date, most likely around 2015, shall be selected in agreement with
the Advanced LIGO installation in order to optimize the efficiency of theworld-wide network of gravitational-wave detectors.
The British-German gravitational-wave detector GEO600 is situated near
Sarstedt in the proximity of Hannover, Germany (52◦ 14’49” N 9◦ 48’30” E).
Its construction started in September 1995 with funds from the Max Planck
Society, the Federal republic of Germany and the Particle Physics and Astronomy Research Council of the UK. With its 600 m-long arms, GEO600 is thefirst large-scale instrument already now using second-generation technology,
such as electro-static actuators and signal recycling. Between 2002 and 2006
GEO600 participated in several data runs in coincidence with the LIGO detectors, the last of which had GEO600 joining S5 from May to October 2006.
In November 2007, as soon as the LIGO and Virgo detectors went off-line,
GEO600 entered continuous operations —the “Astrowatch mode”— supplemented when possible by the 2-km LHO detector. Astrowatch terminatedwhen the enhanced detectors began operating in July 2009.
Due to limiting infrastructure and the topology of the site, which prevents
any increase in the arm length, GEO600 can not undergo a major upgrade in
the fashion of the advanced LIGO and Virgo detectors. However, significant
improvements through small sequential changes can be made to reduce noise
sources at high frequencies in the kHz region where, for example, normal
modes in neutron stars or quasi-normal modes in black holes provide interesting gravitational wave sources. The goal of this GEO-HF program [265]is to improve the sensitivity of the detector in the high-frequency mode by
reducing the effect of two limiting noise sources, namely shot noise andcoating thermal noise. In such a configuration, GEO will be the most sensitive
detector at high frequencies and will be used to search for sources whenever
the detection strategy of the worldwide detector network allows it.

21

CNRS: Centre nationalde la recherche
scientifique
INFN: Istituto
Nazionale di Fisica
Nucleare22

Tama is the area ofTokyo where theobservatory is situated,and “tamatama” isJapanese for “casually,
unexpectedly, by
chance”

CLIO: Cryogenic Laser
Interferometer
Observatory

AIGO: AustralianInternational
Gravitational
Observatory“Sweet spot” is thecommon name to referto the most sensitive
part of the detector

gravitational waves

TAMA300 is a 300-m baseline gravitational-wave detector located at the Mitaka campus of the National Astronomical Observatory of Japan (35◦ 40’31” N
139◦ 32’4” E). It is a project of the gravitational-wave studies group at theInstitute for Cosmic Ray Research (ICRR) of the University of Tokyo. Itsconstruction started in 1995; joint operation with LIGO and GEO600 took
place until 2004. Due to its modest size, TAMA300 is seen as a step towards alarger-scale interferometer in the sense of technology and construction budget.
That future project is the Large Scale Cryogenic Gravitational Wave Telescope(LCGT), which pursues the goal of detecting at least one gravitational wave
event per year at a sensitivity comparable to that of Advanced LIGO and Virgoand two orders of magnitude better than TAMA300. LCGT is currently being
constructed and will consist of two sets of interferometers with 3 km-longarms located underground in a tunnel of the Kamioka mine in Japan. Whereas
TAMA300 plays an important role as test bed for interferometer operations
and advanced vibration-isolation systems, CLIO, a 100 m-baseline underground cryogenic interferometer, shall prove the feasibility of a cryogenic
installation in the Kamioka mine. The LCGT mirrors, made of sapphire andcooled down to temperatures of 20 K, will reduce thermal vibrations of thematerial.
Given the fact that all the detectors described so far are located in the Northern hemisphere, it becomes apparent that the addition of one in the Southernhemisphere would greatly improve the world-wide network of interferometers [236]. The AIGO site at Gingin, north of Perth in Western Australia(31◦ 21’27.6” S 115◦ 42’50” E) was carefully chosen for its convenience to hostsuch a facility. Currently, the AIGO research facility consists of a 80 m interferometer, but a proposal is underway to build a 5 km-long detector which
could join Advanced LIGO, Advanced Virgo, GEO-HF and LCGT in what
would constitute the most sensitive gravitational-wave data-taking effort todate.
The sensitivity of currently-operating gravitational-wave interferometers
has steadily improved over the last decade. With every new experimental
challenge, an improved noise curve has been obtained, which means that thedetectors’ reach to astrophysical sources of gravitational radiation increases.
Figure 4 displays the strain sensitivity curves corresponding to the groundbased interferometers in operation during S5/VSR1. The curves show thestatus of the three LIGO detectors in summer 2007 in red, blue and green, theVirgo detector in May 2008 in purple and the GEO600 detector in June 2006
in black.
Figure 4 illustrates in practice the diverse noice sources affecting the detectors that were described in section 1.3. At low frequencies below 50 Hz
for GEO600 and 40 Hz for LIGO, the seismic wall appears. The special suspension technology of Virgo makes it specially competitive down to 10 Hz,
which should enable the observation of many more cycles for the merger ofcompact objects and a broader frequency band to search for signals fromspinning pulsars. In the intermediate frequency band between 50 and 200 Hz,
the suspension thermal noise becomes dominant. The LIGO detectors are
the most sensitive in this region, with their sweet spot of maximum sensitivity
located at around 150 Hz. At high frequencies, the sensitivity of all detectors
worsens again due to the presence of photon shot noise. The future GEO-HF
will be optimized for these high frequencies and is expected to show improved
sensitivity in the kHz region. For comparison, the design sensitivity curves

1.5 the future of gravitational-wave astronomy

LHO 4km (S5)
LHO 2 km (S5)
LLO 4 km (S5)
Virgo (VSR1)
GEO600 (S5)
adv LIGO (design)
adv Virgo (design)

10−14
10−15

p
√
Sn ( f ) (1/ Hz)

10−16
10−17
10−18
10−19
10−20
10−21
10−22
10−23
10−24

101

102

Frequency (Hz)103

Figure 4: Strain sensitivity curves for the ground-based interferometers in operation
during S5/VSR1, the last run before the LIGO and Virgo detectors underwent
their “enhancement” upgrades. The solid curves correspond to the three
LIGO detectors as of summer 2007, the Virgo detector as of May 2008 and
the GEO600 detector as of June 2006. For comparison, the dashed curves
show the design reference noise budget of Advanced LIGO and Virgo.

of the second-generation detectors Advanced LIGO and Virgo described in
section 1.5.1 are shown in figure 4 as well.
1.5

the future of gravitational-wave astronomy

1.5.1 The Advanced Interferometers
The following years will see significant sensitivity improvement of the detectors
and more extensive upgrades in what will constitute a second generation ofgravitational-wave interferometers. Advanced LIGO and Virgo will replace
their existing hardware with new technology, with the goal of gaining a factorof 10 in improved sensitivity with respect to the first-generation detectors.
One of the most significant consequences of the upgrades in the suspension
systems of LIGO will be the reduction of the seismic cut-off frequency fromthe existing 40 Hz value in initial LIGO to 10 Hz for the advanced detector.
To improve the sensitivity limited by the quantum noise, the laser power will
be increased from the 10 W of initial LIGO to ∼ 200 W. A signal recycling
mirror will give the advanced detectors the ability to tune the interferometerfrequency response, so that the sensitivity can be optimized for detection ofdifferent kinds of astrophysical sources.
The second generation of ground-based interferometers will most likely
inaugurate an era of routine gravitational-wave observations, as its physicalreach during their first several hours of operation will exceed the integrated
observations of the first year LIGO science run. If the current instruments do

23

24

gravitational waves

inspiralof massive
black holes

10−16
10−17

p
√
Sn ( f ) (1/ Hz)

10−18
10−19

BNS and BBH
coalescence

10−20
10−21

unresolved
galacticbinaries

10−22

resolvedgalacticbinaries

Advanced LIGOAdvanced Virgo
Einstein Telescope
ET (Xylophone config.)
LISA

10−23
10−24
10−25

extrememass-ratio
inspirals

SN corecollapse

10−4

10−3

10−2

10−1

100

Frequency (Hz)

101

102

103

104

Figure 5: Expected sources of gravitational waves for future ground-based and spaceborne detectors. Advanced and third-generation ground-based detectors
are expected to detect compact binaries formed by neutron stars, stellarmass black holes and, eventually, intermediate-mass black holes, as wellas signals from supernovæ core collapse. The space antenna LISA focuseson low-frequency signals coming from coalescence of supermassive black
holes, extreme mass-ratio inspirals and galactic binaries.

not make the first detection of GWs, the second-generation interferometers
should succeed.
1.5.2 Third-Generation Ground-Based Detectors

http://www.et-gw.eu/

The fundamental low-frequency limitations of the second-generation detectors are given by thermal, gravity gradient and seismic noise, as explained
in section 1.3. To circumvent these problems, yet a third generation of interferometers to be operated underground is currently being proposed. The
Einstein Telescope (ET) will be a 10 km laser-interferometer with a sensitivity
100 times larger than that of the current detectors. Moreover it will cover
the frequency range between 1 Hz and 104 Hz, increasing the ability to detect massive BBHs which merge at frequencies lower than the cut-off valuesof LIGO and Virgo. Once the design study and the technical preparation
phase are completed, construction could begin after the second-generation
observatories have started operation, probably before the end of the next
decade.
The frequency range that the ET will be able to probe and its expected
sensitivity could make this third-generation, ground-based interferometera complementary companion for the space antenna LISA described in section 1.5.3, a very advantageous fact, since these two detectors might well be
operating simultaneously in the future. Whereas the geometry of the current ground-based detectors requires a multi-site network to measure thepolarization of the GW signal, the ET design will be able to do so by itself,

1.5 the future of gravitational-wave astronomy

25

benefiting from two coaligned, coplanar detectors at a single site. The currently favored design contains three independent detectors arranged in an
equilateral-triangle geometry. The expected sensitivity curve for this “baseline”
design of the ET is shown in Figure 5. An alternative “xylophone” configuration of the ET has been proposed [151], which trades off improved sensitivity
near 10 Hz for decreased sensitivity at higher frequencies. The ability to operate either in broad- or narrow-band mode —within the frequency range
where the noise budget is limited by photon-shot noise— in order to optimize
the sensitivity to targeted astrophysical sources is a common characteristic of
the proposed ET and the Advanced LIGO and Virgo detectors.
1.5.3 LISA: The Space Antenna
LISA completes the family of gravitational-wave detectors and represents aqualitative step forward in the exploration of the gravitational-wave spectrum.
LISA will operate in orbit around the Sun, and will consist of three widely
separated spacecrafts arranged in a triangular configuration. The entire arrangement has been designed with a size ten times larger than the orbit of the
Moon.
LISA will perform low-frequency gravitational-wave astronomy in the bandfrom 0.03 mHz to above 0.1 Hz; its sources are massive black holes merging
at the center of galaxies, binaries of compact stars and stellar remnants in
our Galaxy, extreme mass-ratio inspirals in which a star-sized compact object
falls into a massive black hole at the center of a distant galaxy and, eventually,other sources of cosmological origin, including the relic radiation from thevery early phase of the Big Bang, and speculative astrophysical objects suchas cosmic strings. The antenna is expected to be launched within the next ten
years. A schematic depiction of the expected sources for the advanced, thirdgeneration and space-borne gravitational-wave detectors is shown in figure 5,
including galactic binaries, extreme mass-ratio inspirals, supernovæ corecollapse and compact binary coalescences.
Chapter 2 describes the main sources of gravitational radiation reasonably
expected to exist in our local universe. According to our present understandingof general relativity and the current progress in the detectors’ sensitivity figures,
signals arriving from these sources ought to be detected with the enhanced oradvanced detectors over the course of the next years.

LISA: Laser
Interferometer Space
Antenna

2

SOURCES OF GRAVITATIONAL WAVES

Astrophysicists and gravitational-wave physicists work towards understanding
the kind of potentially-detectable sources that exist in our Universe. Throughthe study of electromagnetic radiation by means of traditional astronomy,
enormous progress and discoveries of exciting phenomena have taken place
during the last century. Most regions of the electromagnetic spectrum havebeen studied at some level of sensitivity.
The gravitational-wave spectrum constitutes a completely new scenario on
its own. In the case of gravity, the conservation of energy, linear momentum
and angular momentum prevent radiation due to the acceleration of massmonopoles, mass dipoles and current dipoles respectively. This implies thatthe leading order radiation comes from the acceleration of mass quadrupoles.
Our current knowledge of the composition of the Universe states that only
4% of the mass-energy of the Universe exists in the form of charged particles
capable of emitting electromagnetic radiation. Nevertheless, the remaining
96% does couple with gravity and could perhaps emit gravitational waves.
Nearly all interesting astrophysical phenomena indeed do so, and usually incopious amounts. The information encoded in gravitational waves is pristine;
it comes to us unaffected by the matter it encounters, from the heart of themost fabulous astrophysical events, such as supernovæ explosions or mergersof black-hole binaries.
This chapter succinctly describes the principal sources of gravitational
waves that are being searched for with the currently-operating interferometers.
These are stochastic signals that fluctuate randomly over a long time compared
to an observing run; periodic waves modeled as superposition of sinusoids
that are roughly constant over a long time compared to an observing run;bursts that last only a few cycles or for short times compared to an observing
run; signals from coalescing binaries corresponding to systems that inspiraland/or merge within the observation time. Special emphasis shall be placedon coalescing black-hole binaries, as they constitute the main focus of thisthesis.
2.1

stochastic background

The very early Universe must have been the source of a random sea of gravitational radiation that even today permeates space and acts as an omnidirectional background. Once produced, gravitational waves emitted shortly after
the Big Bang would forever carry unaltered information about the physicalprocesses that generated them. The waves are expected to be originated ina large number of unresolved sources, nowadays superimposed forming astochastic gravitational-wave background. Examples of this type of radiation
are relic gravitational waves from inflation, from Galactic binary white-dwarf
systems and from slow-spinning Galactic pulsars. Estimations of the stochastic background are usually made by considering the relation between the27

Quasars, pulsars,
gamma-ray bursts andthe cosmic microwave
background radiation
were not discovered
until after 1950

28

sources of gravitational waves

energy density of the gravitational waves ΩGW and the one needed to close
the UniverseΩGW (f) =f dρGW
.
ρc df

(2.1)

The quantity dρGW is the energy density of gravitational radiation containedin the frequency range f to f + df and ρc is the critical density of the Universe.
The expansion of the Universe has cooled down the stochastic radiation— an uncertain aspect in our knowledge of it is its current intensity. Theoretical predictions for the value of ΩGW are affected by large uncertainties
depending on the applied model. The optimum detection and characterization strategy for this kind of signal is based on a cross-correlated analysis. A
stochastic gravitational-wave background signal would cause random fluctuations in the phase of the output laser, which are indistinguishable from various
instrumental noise sources. If a signal is present, however, the randomness
is correlated among detectors. Such an analysis has been performed in the
LIGO S5 data [12], resulting in a constraint for ΩGW of ∼ 6.9 × 10−6 at 95%
confidence in the frequency band around 100 Hz, a result that already rulesout certain cosmic (super)string models. It is hoped that the advanced detectors will be able to push the upper limit on ΩGW down to 10−10 , a result thatwould lead to further meaningful constraints on other theories of the early
Universe. A measurement of the stochastic background of gravitational waves
is possibly the most important observation that gravitational-wave detectors
can make from the point of view of fundamental cosmology.
2.2 periodic sources

The word “pulsar” is acontraction of“pulsating star”. The first
pulsar was observed in1967 by astrophysicist
Jocelyn Bell Burnell

Neutron stars are very compact objects originated from the gravitational
collapse of massive stars. Pulsars are highly magnetized, rotating neutronstars that emit a beam of electromagnetic radiation. X-ray observations indicate that most of the rapidly accreting, weakly magnetic neutron stars in the
Galaxy have a rotation frequency contained within a narrow band. This apparently remarkable fact can be explained by assuming that the loss of angularmomentum is due to emission of gravitation waves [61] through some given
mechanism. Currently there are over 200 known pulsars with frequencies
larger than 20 Hz, which would fall in the detection band of the ground-based
interferometers.
These objects radiate gravitational waves due to a variety of physical processes. Regardless of their origin, they emit a quasiperiodic continuous signalof slowly-changing frequency over the observation time, due to loss of energy
in the form of gravitational radiation and, possibly, other mechanisms. Thus,
the period of the pulsar steadily lengthens as the pulsar loses energy, a phenomenon known as spindown. By measuring the pulsar’s spindown rate via
electromagnetic observations, an upper limit on its gravitational emission canbe set. One of the most relevant results from the LIGO detectors to date has
been the determination of a new upper limit on the gravitational wave emission from the Crab pulsar that beats indirect limits inferred from spindown
and energy conservation arguments [9].
The mechanisms responsible for an accreting neutron star developing aquadrupolar asymmetry are numerous. Among those that have been proposed are internal r-mode oscillations, magnetic deformations and crustal

2.3 bursts

29

mountains. A rapidly rotating neutron star with small deformations from its
axially-symmetric shape emits quasiperiodic gravitational waves. This deviation can be expressed by the ellipticity ε ≡ (Ixx − Iyy )/Izz , where Ixx , Iyy
are the components of the star’s quadrupole moment along the principal axes
in its equatorial plane and Izz is the moment of inertia along its rotation
axis. Typical neutron stars with masses M ∼ 1.4 M have 3 × 1044 g cm2 .
Izz . 3 × 1045 g cm2 , depending on the assumed equation of state. The
uncertainties in the typical values of ε are much larger. The gravitational wave
emitted by a neutron star located at a distance r has a frequency twice its
rotation frequency f and can be expressed as [253]
4π2 G Izz f2
ε ∼ 2 × 10−26
h∼
c4
r



10 kpc
r



ε 
10−6



f
1 kHz

2
. (2.2)The intrinsic amplitude of the signal associated to rotating neutron stars isvery small, of the order of h ∼ 10−25 for a pulsar of f = 60 Hz and ε = 10−5
at r = 1 kpc. Detection can therefore only be achieved by means of long,
day-to-weeks integration times that allow to extract the signal from the noise
it is buried in. The Doppler effect due to the Earth changing its position by anon-negligible fraction during this long observation times needs to be taken
into account. In practice, this kind of searches might become computationallyprohibitive, and thus, distributed methods of computation, such as the project“Einstein@Home” are being developed. If gravitational waves from rotating
neutron stars are detected, they will provide important information about thestar’s structure.
2.3

burstsOur Universe’s violent nature manifests itself sporadically in the form of highlyenergetic, short-duration bursts of radiation. A variety of astrophysical sources
can give rise to such gravitational-wave bursts, being supernovæ explosions
and other generic gamma-ray bursts the canonical examples. A supernova
explosion happens when the compact core of a giant star collapses after having
exhausted its supply of energy from nuclear reactions. The burst of radiation
released in such an explosion is so energetic that it might outshine an entire
galaxy. The exact mechanisms that commence in the star’s core and trigger the
explosion are unfortunately not yet fully understood. The complexity of the
problem requires that any realistic simulation incorporates a large number ofphysical phenomena: three-dimensional hydrodynamics, neutrino transport,
realistic nuclear physics, magnetic fields, rotation. At present the computationof a detailed waveform is not feasible, thus making optimal searches using
matched filtering unattainable.
In absence of complete models for the gravitational emission, typical dataanalysis strategies to search for gravitational-wave bursts involve looking forexcess power in certain frequency bands at specific times, which might or
might not be correlated with electromagnetic or neutrino observations —these are known as externally-triggered searches. Additionally, burst searches
rely on finding evidence of transient signals in coincidence among several
gravitational-wave detectors. An estimate of the amplitude of the radiation

http://www.
einsteinathome.
org/

30

sources of gravitational waves

from a typical supernova at 10 kpc that emits in 1 ms the energy equivalentof 10−7 M at 1 kHz can be given by [230]h ∼ 6×10

−21

E
−7
10 M

1/2  
1 ms 1/2 1 kHz
10 kpc. (2.3)Tfr

A typical search for burst signals in the detectors’ output consists on the
combination of a signal-processing algorithm together with post-processing
and diagnostics tools. The output of a search is a list of “triggers” that are
subject to consistency and coincident tests. Upon passing those tests, they
are considered gravitational-wave candidates. By definition, and due to theirunmodeled nature, burst searches make use of non-optimal and therefore
less sensitive detection methods than match-filtered searches. However, they
have the advantage that, by not making any assumption on the functional
form of the wave to be detected, they might allow detection of sources that weare so far unaware of. Certainly, if another supernova explosion like SN1987aoccurred in our galaxy within the next decade, burst detection algorithms
would turn out to be excellent tools for its characterization.
2.4 coalescing binaries
Gravitationally-bound binaries formed by compact objects orbiting aroundeach other —such as neutron stars or black holes— undergo a coalescenceprocess in the course of which they emit part of their energy as gravitationalradiation. These coalescing binaries are of particular interest to gravitationalwave astronomy because they are among the most promising candidates tobe detected in the first place. The gravitational radiation emitted by pairs ofneutron stars and/or black holes can be accurately modeled by a variety ofanalytical and numerical theoretical methods. This implies that searches forthese systems can be designed that employ optimal algorithms based on the
expected waveforms to be detected. Additionally, coalescing binaries can beused as standard candles for measuring astrophysical distances, since theiramplitude is uniquely determined by their phase evolution and the luminosity
distance to the source. Finally, coalescing binaries can serve as test probesof General Relativity, for their gravitational-wave signature originates fromobjects which massively curve space-time and travel at speeds approachingthat of light.
2.4.1 Evidence for Compact Binaries
Both observational and theoretical reasons strongly support the existence ofneutron stars and black holes in our Universe, as well as their occurrence ingravitationally-bound binary systems, formed either by two neutron stars,
two black holes or a neutron star and a black hole. The Chandrasekhar limitof ∼ 1.4 M gives a theoretical upper bound for the mass of an electrondegenerate star. This limit is of importance in multiple processes of stellar
explosion, such as TypeIa/b/c and II supernovæ, in particular processes thatend up in a white dwarf (Type Ia) or when the core of a massive star explodes(Type 1b/c II).
Compact objects with an orbiting companion in a tightly gravitationallybound orbit have been observed and measured. By tracking down the elliptical2.4 coalescing binaries

movement of the binary components, an estimation of the masses of bothobjects is possible. If the masses of the compact objects are larger than the
Chandrasekhar limit, then the binary must be formed by either neutron stars
or black holes. Associated phenomena such as thermonuclear explosions
observed as matter falls onto the compact object help determine whether weare dealing with neutron stars. In absence of matter-associated processes, thebinary is formed by black holes, whose spins, among other values, can be
determined.
Further experimental evidence, for instance the observation of the orbital
period of the Hulse-Taylor pulsar [155, 250], supports the existence of inspiralling compact objects. Determination of the orbital energy loss due toemission of gravitational waves allows for the masses of the binary to be estimated as roughly 1.44 M and 1.39 M . The two objects orbit each otherwith a period of 8 hours, which corresponds to a separation of ∼ 1R . At
this separation and given their orbital frequency, it is ruled out that the objects could be ordinary stars, for a much weaker force would result if they
were. On the other hand, the absence of observations of electromagnetic
counterparts excludes the possibility that these objects were white dwarfs.
Recently, the observation of the highly relativistic double pulsar system PSR
J0737-3039A/B [171] has provided the most stringent test to date of strong-field
gravity. Its mean orbital velocities and accelerations are much higher thanthose of other known binary pulsars. This system is unique in the sense thatboth neutron stars are detectable as radio pulsars. Precise timing observations
taken over the first few years since its discovery have turned this object into
an excellent candidate for testing general relativity.
Confident in the existence of gravitationally-bound binary systems formedby neutron stars and/or black holes in our Universe, the next sections proceed
to describe the coalescence process as well as the expected event rates forcoalescing binaries.
2.4.2 The Coalescence Process
For a number of practical and theoretical reasons, the process of coalescenceof a compact binary is usually divided upon three distinct phases. During thefirst stage, the two objects are still far from each other and move with nonrelativistic velocities in orbits that slowly spiral inwards. This adiabatic phasecan be treated perturbatively under the assumptions of a weak gravitational
field inside the source and of slow internal motions. The typical velocity ofthe system is the orbital velocity v/c ∼ , which can be thought of as aperturbative parameter indicating the deviation from the Newtonian regime.
When the Einstein equations introduced in section 1.1.1 are expanded
in terms of , an approximation which is valid if the deviation from the
Minkowski flat metric is small, the result is the post-Newtonian formalism.
The lowest term in this expansion, corresponding to  → 0, gives the wellknown quadrupole formula for the gravitational field in the TT coordinate
system [64]hTijT =

2G
d2
P
(
n̂)
Qmn (t − D/c),
ijmn
c4 D
dt2

(2.4)

where D is the distance from the observer to the source, n̂ = ~x/D is the unit
vector in the direction connecting the source with the observer and the symbol31

32
Pijmn = Pim Pjn −
1
δ P P
with
2 ij ij mn
Pij = δij − ni nj the
projector operator

sources of gravitational waves

Pijmn represents the TT projection operator onto the plane orthogonal to n̂.
The object Qij is the quadrupole moment of the source, which can be written
in terms of the mass density ρ as follows


Z
1
2
Qij = ρ(~x, t) xi xj − δij x d3 x .
(2.5)
3During the long inspiral phase, the two compact objects forming the binary
follow an adiabatic sequence of quasi-circular orbits. The movement of theircenter of mass can be modelled by a series expansion on two variables, theparameter  and the symmetric mass ratio of the system, defined asη=

At approximatelyv/c ∼ 0.3 one canconsider that the system
is in the strong-fieldregimem1 m 2,M

M = m1 + m2 .

(2.6)

The parameter η ranges from 1/4 in the equal-mass case to 0 in the test-masslimit. The validity of the post-Newtonian expansion is limited to situations
where  → 0 and η → 1/4. This implies that the series does not converge as
the two bodies draw closer and enter the strong-field regime — likewise, theapproximation is more exact the more similar the masses of the two compact
objects are. The more detailed description of the post-Newtonian formulation
applied to the binary black-hole problem is given in section 3.1 of chapter 3.
The emission of gravitational radiation progressively leads to the shrinkingof the orbit of the binary system, as the two bodies follow the quasi-circular
trajectory and enter the strong-curvature phase, when their motion becomes
more and more relativistic. Eventually, the system transitions from the adiabatic inspiral to the plunge phase, being the frequency associated to the Innermost Stable Circular Orbit (ISCO) of a test particle in a Schwarzschild spacetime a traditionally-used transition point. The ISCO frequency is reachedwhen the two bodies are separated by a distance of R = 6M resulting in
c3.
fISCO = √6 6πGM(2.7)

This mass-dependent frequency can be computed for several systems of interest for ground-based interferometers. For instance, fISCO |M=2.8 M ≈1570 Hz whereas fISCO |M=30 M ≈ 150 Hz. The merger of binaries of neutron stars occurs at frequencies at which the detector’s response is dominatedby shot noise — the post-Newtonian description is then adequate to describesuch binaries while they are within the most sensitive band of the detectors.
On the contrary, binaries formed by solar-mass black holes reach the plunge
and ringdown stages of their coalescence while in the LIGO/Virgo/GEO600
band, hence the corresponding post-Newtonian waveforms terminate too
early. Intermediate-mass and supermassive black-hole binaries merge at much
lower frequencies and are therefore the target of advanced and/or space-borne
detectors, as discussed in chapter 8.
Beyond the ISCO frequency, the real evolution of the system starts diverging significantly from the predictions given by the post-Newtonian formalism.
The orbital frequency of the inspiralling system speeds up and the power series
breaks up. Non-perturbative methods need to be employed and a full-GR
approach is required, i.e. the exact, non-truncated equations for the gravitational field need to be solved. An analytical solution of the Einstein equations
is unfortunately not available for the two-body problem and thus, numericalmethods become very relevant in this regime. The particular 4-dimensional

2.4 coalescing binaries

D hGW

ringdown
wave f orm
mergerwave f orm
inspiralwave f orm

time

Figure 6: Typical gravitational wave emitted by a coalescing compact binary. The
system undergoes a long adiabatic inspiral phase (only the last piece of
which is plotted here) while the two bodies approach each other due to lossof energy in the form of gravitational radiation. The inspiral terminates
when the two bodies merge, after which they ring down to a final Kerr black
hole. The wave depicted in this plot is a hybrid waveform corresponding toan equal-mass, non-spinning, binary black-hole system.

nature of the Einstein equations imply that a 3+1 decomposition in their spatial
and time components is required before any attempt of discretizing them ismade. This fact has given rise to a whole field of research on well-posedness ofthe diverse formulations of the decomposed Einstein equations, ever since thefirst numerical simulations of black-hole binaries were performed by Hahn
and Lindquist [139] in the 1960s.
A large body of research exists that studies the plunge of coalescing binaries, both those formed by neutrons stars, black holes and mixed binaries.
The presence of a non-zero stress-energy tensor on the right-hand side ofequation 1.1 makes numerical simulations with matter considerably more
complex than those of black-hole binaries. Nevertheless, and for a numberof reasons, fully self-consistent numerical relativity simulations of binary
black-hole space-times were inviable in practice until a few years ago — aftera series of breakthroughs occurred in 2005 [212, 88, 52], complete numerical simulations of the last stages of the binary black-hole coalescence finallybecame feasible. The field has quickly blossomed ever since, turning into a
powerful tool to test predictions of general relativity in the strong-curvature
regime, and becoming an important element in the obtaining of accurate andcomplete waveform templates for binary black-hole coalescence to be usedin gravitational-wave searches. The key results achieved by the numerical
relativity community over the past years, which constitute one of the pillarsof the work developed in this dissertation, are presented in chapter 4.

33

34

sources of gravitational waves

The final fate of the binary after its plunge and merger is a single spinningblack hole that relaxes towards stationarity while it radiates all their perturbations away, a process usually called ringdown. The end-state is the only
possible axisymmetric and stationary solution in vacuum, the Kerr blackhole [164], that according to the no-hair theorem only carries information
about its mass and spin. The gravitational wave emitted by the system canbe analytically calculated making use of perturbation theory, as explained in
section 3.2 of chapter 3.
Figure 6 summarizes in a graphical way what has been exposed so far about
the gravitational signature of compact binaries. Plotted is the typical gravitational wave emitted by a coalescing compact binary formed by a pair ofblack holes. The figure shows the last part of the long adiabatic inspiral phasethat eventually transitions into the plunge regime after which the two objects
merge and ring down to the final Kerr black hole. The increasing frequency
and amplitude of the wave as the two coalescing objects approach each otheris clearly visible. The signal displayed in figure 6 is a hybrid waveform corresponding to a non-spinning, equal-mass black-hole binary system obtainedwith the method described in chapter 5.
2.4.3 Expected Astrophysical Rates for Compact Binaries
The volume of the Universe that current gravitational-wave detectors are capable of surveying and the increased observable volume that the advanced
detectors would be able to explore make a natural question arise: what are
the expected rates of coalescing binaries thought to exist in our neighbouring
Universe? Theoretical predictions of astrophysical event rates are crucial tothe detection process. A clear understanding of these numbers can provide
essential input for questions relevant to which one of the various possible
configurations of the advanced detectors is optimal. Additionally, as the sensitivity of the detectors improve, gravitational-wave observations will startyielding astrophysically-interesting upper limits. Some models that predicthigh detection rates could be ruled out, and constraints could be placed on
quantities such of the strength of massive-star winds and the kick velocitiesof merged compact objects.
Based on the ample work done in the field of predictions of astrophysicalrates for compact binaries, a compilation of the state-of-the-art expectations
is presented here. The results follow the discussion presented in [1]. Significant
uncertainties affect the predictions of astrophysical rates for compact binary
coalescences, that can have an error bar of a couple of orders of magnitude.

Table 1: Terminology for statements related to astrophysical rates. The symbols R
refer to rates per galaxy; the symbols N represent detection rates
Abbreviation

Rate statement

Physical significanceRmax , Ṅmax

Upper limit

Rates should be no higher than...

Rhigh , Ṅhigh

Plausible optimistic estimate

Rates could reasonably be as high as...

Rre , ṄreRealistic estimate

Rates are likely to be...

Rlow , Ṅlow

Plausible pessimistic estimate

Rates could reasonably be as low as...

2.4 coalescing binaries

35

Table 2: Compact binary coalescence rates per Mpc3 per Myr. Quoted are the expected rates for neutron-star, mixed and black-hole binaries, where the latest
are taken to be stellar-mass black holes of 10 M . For a discussion on the
expected rates of intermediate-mass black-hole binaries, see chapter 8.
Rlow

Source

Rre

Rhigh

Rmax

NS-NS (Mpc−3 Myr−1 )0.01 [162]

1 [162]

10 [162]

50 [166]

NS-BH (Mpc−3 Myr−1 )6 × 10−4 [196]0.03 [196]

1 [196]—

1 × 100.005 [160]0.3 [160]—−3

BH-BH (Mpc

−1

Myr

)

−4

[160]

Nevertheless, a range or rates can be quoted that expresses the plausible
optimistic, likely and plausible pessimistic estimates. Table 1 introduces theterminology used to refer to these concepts. Rre refers to the mean of theposterior probability density function for the rates, Rlow and Rhigh are the95% pessimistic and optimistic confidence intervals, respectively, and Rmax isthe upper limit.
Rates are usually expressed in events per Myr per Milky Way Equivalent
Galaxy, events per Myr per Mpc3 or events per Myr per L10 , where L10 is1010 times the blue solar luminosity LB, . Here we present the most up-todate compact binary merger rates per unit volume. The references in table 2
provide further information on the assumptions that these rates are based on.
In order to convert these merger rates into detector rates, a number ofassumptions need to be made regarding the characteristic of the detectors, thenetwork of interferometers and the data analysis strategies that are followed in
the searches. The figures quoted in table 3 have been computed considering a
volume of the observable Universe such that the LIGO/Virgo network would
detect a compact binary with a signal-to-noise ratio of 8. This is a conservative
choice if the detector noise is Gaussian and stationary and if there are two
or more detectors operating in coincidence. Additionally, a post-Newtonian
stationary-phase approximation is assumed for the functional form of thegravitational wave originated by the binary coalescence. This neglects the factthat the most massive binaries merge and ring down within the detector bandand thus, the signal-to-noise ratio is underestimated. A further assumption
is made by considering that all neutron stars have a mass of 1.4 M and all
black holes a mass of 10 M — our knowledge of the mass distribution is not
sufficient at present to warrant more detailed models, and the uncertainties in
the coalescence rates dominate errors from the simplified assumptions about
component masses.
Given the constraints mentioned above, the detection rates expected for the
initial and advanced ground-based interferometers are quoted in table 3 forneutron-star, mixed and black-hole binaries of stellar mass. It is immediate tonote that even the most optimist estimates predict relatively low for the initialinterferometers; nonetheless, a lucky detection with the enhanced detectors
should not be ruled out. The chances that compact binaries are detectedwith the advanced detectors look however much more promising, for tens
to hundreds of compact binaries are realistically expected to exist within thereach of the interferometers. In this section we have concentrated of neutronstars and stellar-mass black-hole binaries. For a more detailed discussion on

LB, =2.16 × 1033 erg/s36sources of gravitational waves

Table 3: Detection rates for compact binary coalescence sources. Event the optimist
estimates predict low rates in the initial interferometers — however, detectionis plausible with the advanced instruments.
Detector

InitialAdvanced

SourceṄlow (yr−1 )NS-NS

2 × 10−4

NS-BH

7 × 10−5

BH-BH

2 × 10−4Ṅre (yr−1 )Ṅhigh (yr−1 )Ṅmax (yr−1 )

0.02

0.2

0.6

0.004

0.1

—

0.007

0.5

—

NS-NS

0.4

40

400

1000

NS-BH

0.2

10

300

—

BH-BH

0.4

20

1000

—

rates and possible detection of intermediate-mass black holes with advanced
detectors we refer the reader to chapter 8.
This chapter has provided an overview of the possible sources of gravitational waves that are being searched for in the output of the current interferometers. The focus has been put on the description of coalescing binaries, forthey are the main topic of this thesis. The next chapters describe in more detailthe analytical and numerical tools relevant for computing the gravitationalradiation associated to black-hole binaries with aligned spins and introduce
the phenomenological model developed for these systems.

Part II
MODELING BINARY BLACK-HOLE COALESCENCES

3

ANALYTICAL MODELING OF BLACK-HOLE BINARIESThe complex, non-linear structure of the Einstein equations poses a non-trivial
challenge when one tries to find an exact analytical solution. The final object
to search for is a suitable metric describing the structure of the space-time,
including the inertial motion of objects in it. In order to do so, the stressenergy tensor on the right-hand side needs to be calculated, which in turndepends on the unknown metric. If only interested in the weak-field limit of
the theory, the dynamics of matter can be computed using special relativity
methods and/or Newtonian laws of gravity and then placing the resulting
stress-energy tensor into the Einstein field equations. But if the exact solution
is required or a solution describing strong fields, the evolution of the metricand the stress-energy tensor must be solved for together.
The particular case of the two-body problem —the dynamics of two structureless point-particles, characterized solely by their masses and, possibly,
their spins, moving under their mutual, purely gravitational interaction— has
no known complete solution. Even writing down the equations of motion forthis system is difficult. Unlike in Newton’s theory, it is impossible to express
the acceleration by means of the positions and velocities, in a way which
would be valid within the exact theory. Therefore we are obliged to resort
to approximation methods. The nature of the adiabatic inspiral phase of thebinary black-hole coalescence makes the post-Newtonian approximation particularly suitable to describe the evolution of the system until the two compact
objects approach the merger and the strong-field regime dominates. Likewise,
once the two compact objects merge, the final state is reduced to a single
black hole, for which known, exact solutions to the Einstein equations do
exist. Perturbations of these solutions around the equilibrium give rise toemission of gravitational radiation, and analytical methods exist to describe
this phenomenon.
In this chapter these two analytical approaches to the modeling of the inspiral and ringdown stages of the binary black-hole coalescence are reviewed.
These formalisms are of importance for the construction of the phenomenological model presented in chapter 5.
3.1 post-newtonian formalism for non-precessing binaries
Coalescing compact binaries such as the binary black-hole system can be accurately modeled by the post-Newtonian approximation to general relativity atleast during the major part of the long inspiral phase, under the assumptionsof a weak gravitational field. The full general relativity equations are approximated by a series on the expansion parameter  ∼ (v/c)2 ∼ (Gm/rc2 ),where r is the separation between the two point masses and v the magnitudeof their relative velocity. In order to compute a post-Newtonian waveform forthe gravitational radiation, it is necessary to solve the equations of motion
and the generation of gravitational waves.
The two-body equations of motion can be reduced to relative equationsof motion in the center-of-mass frame, and expressions for them up to dif-

39

This approach is basedon a formal expansionof the Einsteinequations when thevelocity of light c tends
to infinity

40

analytical modeling of black-hole binaries

~L
a
~2
θ2
a
~1
θ1
m2
m1
Figure 7: Pictorial representation of a binary black-hole system in a quasicircularorbit. In the most general case the black holes are rotating with their spins
given by a
~ 1 and a~ 2 . The particular subset of physical configurations that
we are interested in modeling here are those for which the orbit is nonprecessing, i.e. θ1 = θ2 = 0.

PN orders are named
according to the powerin the (v/c)n
expansion. Thus,
3.5PN represents termsof order (v/c)7 beyond
quadrupole

See e.g. [40]ferent PN orders can be found in the literature. For a complete overview
and description of the achievements in the field, the reader is referred to thereview article [64]. In the most general configuration of the binary black-hole
problem, the spins of the black holes are randomly oriented and, thus, theorbit precesses. A pictorial representation can be seen in figure 7.
The dimensionality of the parameter space of a spinning binary is considerably larger than the equivalent non-spinning system. Assuming the black holes
to be well described by a mass and a spin vector (ignoring higher multipoles),and their trajectory to be well described by adiabatically evolving Keplerian
orbits, we need 17 parameters to describe the binary system. Besides the twomasses and spin vectors, we also need a fiducial time t0 and orbital phase φ0
at t0 , the distance to the source and its sky-location, two parameters for theunit vector normal to the orbital plane, and finally if we consider eccentricity
then we additionally need the eccentricity and the direction of the semi-majoraxis. Sufficiently close to or during the merger, this description in terms of
Keplerian orbit will break down, and higher order black hole multipoles mightplay a role as well.
As a first step towards the understanding of the full parameter step, a subset
of configurations can be studied. This dissertation deals with the study ofbinaries where both black holes have aligned or antialigned spins with respect
to the direction of the total angular momentum. Previous works on postNewtonian evolutions show evidence that, given generic initial conditions,the system evolves towards stable configurations with spin vectors arounda fixed orientation [233], although this restriction seems to apply only tocertain sections of the parameter space. Similar hints pointing to the existenceof a rich structure for some regions of initial inspiral conditions have beenpresented in [176, 150].
Hence, as a natural expansion of the parameter space encompassed by
our waveform models, the physical configurations that we will study in thisdissertation are those corresponding to non-precessing, spinning binaries.

3.1 post-newtonian formalism for non-precessing binaries

41

Additionally we will work in the frequency rather than in the time domain.
The justification for this choice will become clear in view of the matching andfitting techniques developed in chapter 5.
In this chapter we focus on the post-Newtonian treatment for this kind ofsystems, which we know is accurate at least during the major part of the longinspiral part of the coalescence, under the assumptions of a weak gravitational
field [64]. In order to obtain an analytical description of the early inspiral in
the Fourier domain we construct the TaylorF2 phase [100, 101, 102, 43] andthe 3PN amplitude [67, 42] for compact binaries with comparable masses
and spins aligned with the orbital angular momentum. Each black hole ischaracterized by its mass mi and the magnitude of its spin vector
Si = |χi |m2i ,i = 1, 2.

(3.1)

The spin vectors are aligned or antialigned with the orbital angular momentum~L, where the sign of ~L · ~Si is given by the sign of χi . Useful quantities in the
post-Newtonian formalism are the symmetric mass ratio 2.6, the total massand the chirp massM = η3/5 M.

(3.2)

The PN expansion is written in the dimensionless variable x, which is related
to the orbital frequency ω of the binary via
x = ω2/3 .
3.1.1

(3.3)

Energy and Flux

Two key quantities in the post-Newtonian formalism are the binding energy of the center-of-mass E and the gravitational-wave flux L. The postNewtonian expansion for the energy of inspiralling compact binary systems
in the adiabatic approximation is published in the the literature, see for instance [64, 104, 66, 81] and references therein. For our purposes we include
leading order and next-to-leading order spin-orbit effects [165, 40, 65] as well
as spin-spin effects that appear at relative 2PN order [165, 97, 205]. The explicit
expression for the energy for non-precessing spinning black-hole binaries canbe written asxη X
ek xk/2 ,
2
6

E=−

(3.4)

k=0

with
e0 = 1,e1 = 0,3ηe2 = − − ,4 12

8(S1 + S2 ) + 2η(χ1 + χ2 ),
3



27
19
η2
2
2
2
2
e 4 = − − χ 1 m1 + χ 2 m2 + η− 2χ1 χ2 −,8824
72 − 31ηη(9 − 10η)
(S1 + S2 ) +e5 =(χ1 + χ2 ),
9
3

675
34445 205π2
155η2 35η3
e6 = −
+η
−
−
−
.
64
576
96
96
5184

e3 =

Note that the square
terms in the individual
spins are only valid forblack holes, as discussed
in [97, 205, 81]

42analytical modeling of black-hole binaries

Since the numerical relativity data that we will use in our phenomenologicalmodel presents constant spin magnitudes, we use here the corresponding
post-Newtonian spin definition [116, 65].
The other necessary component in the description of an inspiralling blackhole binary as a sequence of quasi-circular orbits is the flux L, which we takeat 3.5PN order including the same spin effects as for the energy. Additionally,
we take into account the 2.5PN correction of the flux due to the energy flow
into the black holes, as calculated in [32]. The final expression reads
X32
L = η2 x5`k xk/2 ,
5
7

(3.5)

k=0

with the coefficients `k given by`0 = 1,`1 = 0,`2 = −

1247 35η
−
,
336
12

5η
11
(S1 + S2 ) −(χ1 + χ2 ),
4
4

44711
`4 = −+ 2 χ21 m21 + χ22 m22
9072
9271 3165η2+η+ χ1 χ2 +
,
504
8
18




135 1189
8191 583+
η + (S1 + S2 )+
η
`5 = −π
672
24
112
126


51
15
3
+ (χ1 + χ2 )
η − η2 − (χ31 m31 + χ32 m32 ),
16
7
4
2
16π
6643739519 1712γE 856
`6 =
+
−
−
ln (16x)
3  69854400
105
105

41π2 134543
94403η2 775η3
+η
−
−
−
,
48
7776
3024
324


16285 214745η 193385η2`7 = π −
+
+
.
504
1728
3024
`3 = 4π −(3.6)

The symbol γE ≈ 0.5772 represents the Euler constant.
3.1.2 TaylorT4 Approximant
The post-Newtonian adiabatic model assumes that the motion of the binary
components proceeds along an adiabatic sequence of quasi-circular orbits.
In particular, the assumption is that ṙ = 0 and the following energy-balance
equation holdsdE(t)= −L(t),dt(3.7)

which, when using 3.3, translates into an evolution equation for the orbital
frequency, or equivalently
dx
L(x)
=−
.
dt
dE(x)/dx(3.8)

Taking equation 3.8 as starting point, diverse manipulations can be made thatlead to different waveform models, as summarized in [69, 86]. For instance,3.1 post-newtonian formalism for non-precessing binaries

numerically solving 3.8 leads to the TaylorT1 post-Newtonian approximant.
Alternatively, we may explicitly construct the TaylorT4 approximant, which
is obtained by expanding the right-hand side of 3.8 to 3.5PN order.
X
dx
64
= ηx5
ak xk/2 ,
dt
5
7

(3.9)

k=0

with the following coefficientsa0 = 1,a1 = 0,a2 = −

743 11η
−
,
336
4

25η
113
(S1 + S2 ) −(χ1 + χ2 ),
12
4

34103
+ 5 χ21 m21 + χ22 m22
a4 =18144

13661 7959η2
+η
+ χ1 χ2 +
,
2016
8
18




26387 6427
4159 189+
η − (S1 + S2 )−
η
a5 = −π
672
8
1008
168



1245 2
3 3 3
473
η−
η −
χ1 m1 + χ32 m32 ,− (χ1 + χ2 )
84
56
4
2
16π
856
16447322263 1712−
γE +−ln (16x)a6 =139708800
105
3
105


451π2 56198689
541 2 5605 3
+η
−
+
η −
η ,
48
217728896
2592

4415 358675
91495 2a7 = −π−η−η .
(3.10)
4032
6048
1512
a3 = 4π −Note that the formal re-expansion of the denominator and the multiplication with the numerator in 3.8 yields contributions to higher orders thanthose in 3.9. However, since 4PN and higher terms in flux and energy are notcompletely determined so far, the expressions that one can compute for akwith k > 7 are incomplete. The same applies for contributions of the spins at
relative PN orders higher than 2.5PN. When we later use the TaylorT4 expression 3.9, we only expand it to 3.5PN order but keep all the spin expressions,i.e. (incomplete) contributions in a6 and a7 are not neglected.
3.1.3

Stationary Phase Approximation and TaylorF2

Consider a gravitational-wave complex signal in the time domain of the formh(t) = A(t)eiφ(t) .

(3.11)

If the amplitude of the signal varies slowly in a time scale compatible with thefrequency of the oscillation, i.e.
Ȧ
 φ̇
A

∀t,

(3.12)

then the analytical Fourier transform of the wave can be easily calculated. This
stationary phase approximation holds well in the case of the long adiabatic

43

44

analytical modeling of black-hole binaries

inspiral. As a result, the TaylorF2 expression for the phase can be obtained [100,101, 102, 43]. Firstly, the expansion of the the inverse of equation 3.8,
dt
dE/dx
=−
,
dx
L

(3.13)

allows us to analytically integrate t(x). The orbital phase φ can be integrated
viadφ= ω = x3/2
dt

⇒

dφ
dE(x)/dx= −x3/2dx
L(x)(3.14)

to obtain φ(x). This is the TaylorT2 approximant. The decomposition of
the gravitational wave into modes using spin-weighted spherical harmonics−2 Y
lm allows us to express each component in the time domain as [67]hlm (t) = Alm (t) e−imφ(t) ,(3.15)The transformation to the frequency domain is made in the framework of thestationary phase approximation,
s
Z∞
lm
2π
lm
2πift
lm
lm
eiψ (f) , (3.16)h̃ (f) =h (t) edt ≈ A (tf )mφ̈(tf )−∞where tf is defined as the moment of time at which the instantaneous frequency coincides with the Fourier variable, i.e., mω(tf ) = 2πf. The phasein the frequency domain is given byψlm (f) = 2πf tf − mφ(tf ) −π.
4(3.17)

Given t(x) and φ(x) the change to the Fourier variable is done through thechangex(tf ) = [ω(tf )]

2/3

=

2πf
m

2/3
.

(3.18)

Applying this machinery to the energy 3.4 and flux 3.5 leads to an expression
for the phase of the dominant mode l = 2, m = 2 of the gravitational-wave
signalXπ3
ψ (f) = 2πft0 − φ0 − +(πf)−5/3
αk (πf)k/3 , (3.19)
4 128η
7

22

k=0

with
α0 = 1,α1 = 0,α2 =3715 55η
+
,
756
9

113
(S1 + S2 ) + 25η(χ1 + χ2 ) ,3
15293365
α4 =− 50 χ21 m21 + χ22 m22
508032

27145 395
3085η2+η−χ1 χ2 +,504
472

α3 = −16π +3.1 post-newtonian formalism for non-precessing binaries

45

 



38645 65
683665 3790
α5 = ln (πf) π+ η − (S1 + S2 )
+
η
756
9
2268
63


3400 237265
η+η
− (χ1 + χ2 )
252
63


10 3 33
3−
χ 1 m1 + χ 2 m2 ,36848
11583231236531 640π2 6848−−
γE −ln (64πf)
α6 =4694215680
3
21
63


76055 2 127825 32255π2 15737765635
−
+
η −
η ,
+η
12
3048192
1728
1296


77096675 378515
74045 2
α7 = π+η−η .
(3.20)
2540161512
756
From 3.17 it can be noticed that, in fact, the expression for the phase 3.19 isvalid not only for the l = 2, m = 2 mode, but also for all spherical harmonics
with m = 2. The quantities t0 and φ0 are arbitrary and arise as constantsof integration when calculating t(x) and φ(x). Note that the calculation of
α5 also leads to contributions not proportional to ln(πf). For given mass
ratio and spins, however, these terms are just an additional constant due thepre-factor of the sum in 3.19 and we absorb them in the definition of φ0 . In our
implementation of this Fourier-domain phase we also take into account the
spin terms that appear after re-expanding at 3PN and 3.5PN order, although
they are not complete and therefore not given here.
3.1.4

Amplitude

The time-domain amplitude of the gravitational wave has been recently calculated at 3PN order by Blanchet et al. [67]. We use the expression given there
for the l = 2, m = 2 mode in combination with the spin corrections provided
in [58, 42]. In our notation, the time-domain amplitude is expressed asr68η x π X
22
A (x) =
(3.21)Ak xk/2 ,DL
5
k=0

where DL is the luminosity distance between the source and the observer andthe coefficients are given by
107 55
+ η,
42
42
2
2
A3 = 2π + (m2 − m1 )(χ1 − χ2 ) − (1 − η)(χ1 + χ2 ),
3

 3
2173
1069
2047 2
A4 = −−η− 2χ1 χ2 +
η ,
1512
216
1512


34π
107π
A5 = −+η− 24i ,
21
21

27027409 856γE 428iπ 2π2
41π2 278185
A6 =
−
+
+
+η
−
646800
105
105
3
96
33264
2
3
20261η
114635η428−+−
ln(16x).
(3.22)
2772
99792
105
In order to construct the Fourier-domain amplitude as given by 3.16, an explicit
expression for φ̈ = d2 φ/dt2 = ω̇ is needed in the denominator. In [42]A0 = 1,A1 = 0,A2 = −

A similar discussion
can be found in [43]

46

analytical modeling of black-hole binariespthe approach consists of a re-expansion of 1/ω̇ using the same ingredientsas those underlying the TaylorTn approximants. One could, however, take a√different route, since ω̇ = 3 xẋ/2 allows to choose different prescriptions
for ẋ without re-expanding the quotient.
One can for instance use the Newtonian ẋ, i.e. the leading term given by
ẋ =

64η 5
x .
5M

(3.23)

Incorporating 3.23 in 3.16 leads to the following expression for the 3PN amplitude in the frequency domain assuming a Newtonian ẋr
M2 π η
22Ã (f) '(πMf)−7/6 A22 (tf ).
(3.24)
R
6
Alternatively, one can re-expand the denominator as
s
s
2π
5π
'
M(πMf)−11/6 T[(πMf)1/3 , 5],96η
φ̈

(3.25)

where T[u, n] represents a Taylor series in the variable u up to order un .
Finally, we might as well replace ẋ by its TaylorT1 or TaylorT4 description andthe re-expansion of the form [42]sr5π5π −11/4 X'
x
Sk xk/2 ,(3.26)
ω̇
96η
k=0

with
743 11+ η,
S1 = 0,
S2 =
672
8


113 19113δ
S3 = −2π +− η χs +
χa ,
24
6
24

1379 2 79
7266251 18913+
η+η − η χ2s − χ2a ,S4 =812851216128 1152
16
4757 57
S5 = π −+ η .
1344 16
S0 = 1,(3.27)In the expression above, we have introduced the three auxiliary variablesδ = (m1 − m2 )/M, χs = (χ1 + χ2 )/2, χa = (χ1 − χ2 )/2. The transfer
to the the Fourier domain is completed by using 3.21 in 3.16 in combination
with 3.18. Let us recall that these expressions are only valid for systems withaligned or antialigned spins.
This finally allows to write down the expression for the Fourier-domain
amplitude of the dominant mode for the TaylorT4 approximant asr
52η πM2 −7/6 X
22Ã (f) =ΩÃk Ωk/3 ,(3.28)
3 DL
k=0

where Ω ≡ πMf and

323 451Ã0 = 1,Ã1 = 0,Ã2 = −+
η ,224 168


27δ
27 11
Ã3 =
χa +
− η χs ,886

3.1 post-newtonian formalism for non-precessing binaries

1000

Num. simulation
TaylorT4
TaylorT1
re-expanded 22Èh È DL

100

10

1
0.001

0.01

0.1
Mf

Figure 8: Different variants of constructing the PN Fourier amplitude in the stationary
phase approximation for the equal mass case. The labels refer to the different
ways of treating (π/φ̈)1/2 in equation 3.16. The thick curve shows data obtained by a numerical simulation in full general relativity.p
The straight grayline illustrates the restricted PN amplitude, |h̃22 | DL = π 2η/3 (πf)−7/6.

105271 2 79
27312085 1975055−
η+η + η χ2a − χ2s ,8128512338688
24192
16
85π
13577
Ã5 = −
(1 − 4η) − 24iη −(χs + δχa )64
1008
4367δχa + 13985χs
583χs 2+η−
η .
(3.29)
1008
252
Ã4 = −Taking the norm in 3.28 is necessary due to a small imaginary contribution at
the highest PN order.
The comparison between the different approaches proposed to deal withthe presence of φ̈ in the denominator of 3.16 is shown in figure 8. We plot theFourier-domain amplitude of the dominant mode in the case of re-expanded,
TaylorT1 and TaylorT4 φ̈. Additionally, the restricted post-Newtonian amplitude that scales like f−7/6 is shown, together with the Fourier-transform ofdata from a numerical relativity simulation. It is interesting to observe that thecorrections to the restricted amplitude are already noticeable at frequencies
below Mf = 0.005, therefore affecting the reliability of gravitational-wave
searches based on templates without amplitude corrections for parameter
estimation of stellar-mass black-hole binaries.
From figure 8 it can be concluded that all variants of the 3PN Fourier
amplitude agree reasonably well with the numerical relativity data roughly
up to the frequency of the last stable circular orbit in Schwarzschild given
by equation 2.7, fISCO ≈ 0.022. Due to the comparable behavior even beyond this point we choose to construct the Fourier-domain amplitude of ourpost-Newtonian model by using the TaylorT4 for ẋ given in 3.9. This is the
final choice that will be employed in the construction of our inspiral-mergerringdown waveform model described in detail in chapter 5.

47

48

analytical modeling of black-hole binaries

3.2 ringdown and quasi-normal modes

The name “ringdown”
reminds of a bell that,when struck, rings at agiven frequency before
returning toequilibrium

The final fate of two compact objects undergoing an inspiral process during
which they emit a fraction of their energy in the form of gravitational waves
is the eventual merge into a single, distorted black hole. When the two objects
that form the binary merge into the final black hole, this becomes perturbed
and undergoes a ringdown process in the course of which gravitational radiation is emitted. During the ringdown, all perturbations are radiated away until
a state of stationary, spinning black hole is reached, after an infinite amount oftime. By means of the no-hair theorem, the final black hole can be completely
characterized by only three externally observable classical parameters: mass,electric charge, and angular momentum. All other information —for which“hair” is a metaphor— about the matter which formed a black hole or is falling
into it, disappears behind the black-hole event horizon.
The following sections introduce the concept of quasi-normal modes andthe associated gravitational radiation. The interested reader will find a comprehensive review of this topic, including both black holes and neutron stars,
in [168].
3.2.1 Perturbations from Equilibrium
The first studies on stability of Schwarzschild black holes date back from
1957, when Regge and Wheeler [216] discovered that a disturbance from theblack hole’s sphericity would not grow with time, but would oscillate about theequilibrium configuration. Zerilli [267] extended this study for the even-parity
case. The existence of the associated quasi-normal modes was pointed out by
Vishveshwara [258] in calculations of the scattering of gravitational waves by
a Schwarzschild black hole. Solving the Einstein equations for perturbations
around a spherically-symmetric solution leads to wave equations, assuming
a decomposition of the full solution χ(t, r, θ, φ) into spherical harmonics
Ylm (θ, φ)χ(t, r, θ, φ) =X χlm (t, r)l,mr

Ylm (θ, φ).

(3.30)

Essentially, the equation to solve is of the forms2 χ − χ 00 + Vχ = 0,

(3.31)

where prime denotes differentiation with respect to a radial variable and V isan effective potential depending on the nature of the perturbations under study.
The solutions for the quasi-normal ringing can be approximated by a damped
exponential. Thus, for each mode there exists a characteristic complex angularfrequency ωlm ; the real part is the angular frequency of the oscillation andthe imaginary part is the decay rate, or the inverse of the damping time τ. It
is perhaps more common to express these physical characteristics in terms of
the oscillation frequency flm and the quality factor Qlm
ωlm = 2πflm − i τ−1 = 2πflm − i π
The solutions present a
2l + 1 degeneracy on m

flm
Qlm

(3.32)

Leaver [174] determined the fundamental l = 2, 3 frequencies of the oscillations of a black hole around the Schwarzschild solution, as well as the3.2 ringdown and quasi-normal modes

Table 4: The first QNM frequencies ωM of the spherical, non-spinning
Schwarzschild black hole for l = 2, 3, given in geometrical units.

n

l=2

l=3

0

0.3737 − i 0.0890

0.5994 − i 0.0927

1

0.3467 − i 0.2739

0.5826 − i 0.2813

20.3011 − i 0.4783

0.5517 − i 0.4791

30.2515 − i 0.7051

0.5120 − i 0.6903

overtones, classified by an index n. The values of the first quasi-normal frequencies are given in table 4. The fundamental l = 2, n = 0 mode is the mostslowly-damped mode, i.e. the one with the lowest value of the imaginary partof the frequency. When n increases, it can be shown that also the imaginary
part of the frequency grows very quickly. This indicates that higher-ordermodes do not contribute significantly to the emitted gravitational radiation;
the statement is true as well for the higher l modes (octupole, etc). In contrast,
the real part of the frequency asymptotes to a constant value.
So far, we have dealt with Schwarzschild black holes only; however, theaxisymmetric, black hole solution to the source free Einstein equations, i.e
the Kerr solution, is a more accurate description of a spinning black hole,which is the case that we intend to model in chapter 5. The problem of theperturbations from a spinning black hole was addressed by Teukolsky [251].
The solution for the separation in angular and radial variables is of the form
χ(r, θ, φ) = R(r, ω)S(θ, ω)ei mφ ,(3.33)

which, unlike the Schwarzschild case, entails a dependency in ω. The solutions
decouple into spin-weighted spheroidal, rather than spherical, harmonics
2s Slm with s = −2. For real ω , the spheroidal harmonics can be expanded
into spherical harmonics of fixed m.
The calculation of the quasi-normal frequencies of the Kerr black holeis considerably more involved than the Schwarzschild case. The reason isthe complexity of the perturbation equations and, in particular, their nonseparability. For this reason, only partial results for these frequencies areavailable in the literature. In [174], Leaver presented the l = 2 modes fordifferent m and spin, showing that the spin removes the 2l + 1 degeneracy
in m. In particular, for the l = 2, m = 2 mode, the value of the real part ofthe orbital frequency ranges from MωR = 0.37 for a non-spinning blackhole to MωR = 0.9 for a maximally-spinning black hole. This fact will be ofimportance as theoretical input for the phenomenological model developedin chapter 5.
3.2.2

The Ringdown Waveform

Since the gravitational waves emitted by a ringing black hole will be dominated,
after an initial transient period, by a superposition of quasi-normal modes,
it seems reasonable to try to model the ringdown radiation in such a way.
If the characteristic parameters of the radiation —ω and τ— introduced in

49

50

analytical modeling of black-hole binaries

a=0
a = 0.75
a = 0.9

0

20

40

60

80

100

tM
Figure 9: Ringdown signals emitted by sources of total mass 100 M and spin values
a = 0, 0.75 and 0.9, corresponding to frequencies 119, 189 and 221 Hz andQ–factors 2, 3.73 and 5.67 respectively. The amplitude of the waveforms
has been arbitrarily rescaled.

section 3.2.1, could be expressed as a function of the black hole’s M and a,then it would be certainly possible to infer its physical parameters in terms of
the emitted waves. In fact, this approach constitutes the basis of the modeled
ringdown searches made by the LIGO and Virgo collaborations [14].
Motivated by the fact that a coalescing binary has a rotating shape corresponding to the spheroidal harmonic l = 2, m = 2, Echeverria [109]performed an analytical fit to Leaver’s data for the fundamental quasi-normal
frequency in terms of the black hole’s physical parametersi
1 c3 h1 − 0.63(1 − a)3/10
2π GM
Q220 (a) = 2(1 − a)−9/20 ,f220 (a, M) =(3.34)(3.35)

with aM2 being the spin of the final black hole after the binary has merged.
These relations can be inverted to yield the mass and spin of the black hole interms of the characteristic parameters of the radiation, which motivates thealready-mentioned ringdown searches with LIGO and Virgo.
Assuming that the gravitational radiation of a ringing black hole far fromthe source can be approximated by the most slowly damped mode, then higher
order contributions can be neglected. The central frequency of the waveform
is given by f0 ≡ f220 and the quality factor is Q ≡ Q220 . The waveform forthe fundamental mode at large distances can be expressed as an exponential
damped sinusoidal
h22
ring (t) =Aring M −πf0 t/Qecos (2πf0 t) ,r

(3.36)

where the amplitude depends on the mass of the black hole and its distanceto the detector, and the frequency and damping factor of the sinusoidal wave

3.2 ringdown and quasi-normal modes51

are related to the physical parameters of the hole via 3.34 and 3.35. The peak
amplitude Aring can be computed from the stress-energy tensor, assuming thata fraction ε of the black hole’s mass is radiated away in the form of gravitational
waves
r 

5
GM
Aring =
Q−1/2 F(Q)−1/2 g(a)−1/2 ,(3.37)
ε
2
c2
where
7
24Q2

(3.38)

g(a) = 1 − 0.63 (1 − a)3/10 .

(3.39)

F(Q) = 1 +Figure 9 displays the shape of the ringdown waveform h22
ring (t) for threedifferent values of the spin of the black hole. As the spin increases, so do
the frequency and the quality factor. Highly-spinning black holes spend asignificantly longer time in their perturbed state before reaching the equilibrium, therefore emitting more radiation, which increases their detectability.
In fact, it can be shown from numerical relativity simulations, that a particular configuration with spin —known as hang-up— provides the maximallyspinning black hole in the final state, and it can be as “loud” as three times a
non-spinning black hole [220].
In this chapter, two analytical approaches have been introduced that describe parts of the coalescence process of a black-hole binary. The adiabatic
nature of the quasi-circular inspiral stage calls for a post-Newtonian approach,whereas the perturbative nature of the ringdown oscillations towards the
final Kerr black hole after the merger motivates the introduction of the quasinormal modes formalism. In chapter 4 we will see how the strong-field regime
around the black holes’ merger calls for full numerical relativity methods as
the only feasible approach to solving the Einstein equations.

We refer to “loud” in a
gravitational-wave
detection sense

4

BLACK-HOLE BINARIES IN NUMERICAL RELATIVIT Y

Numerical relativity is a fascinating research field astride general relativity andcomputational physics. In the many situations where an analytical solution
to the Einstein equations cannot be found, numerical relativity resorts tonumerical methods and algorithms that are usually run on supercomputers.
A primary goal of numerical relativity is to study space-times whose exact
form is not known analytically. As described in chapter 3, this is precisely thecase for the merger of two black holes. Although much work has been done
within the effective-one-body approach, the analytical approximations thatare valid in the perturbative inspiral and ringdown stages do not perform
well in the strong-field regime, and full-GR approaches such as that given bynumerical relativity are required.
The field of numerical relativity has a long and riveting history, much as
the field of gravitational-wave detection does. The pioneering attempts to
evolve wormhole initial data in a computer by Hahn and Lindquist in the
1960s [139] were followed by the work of Smarr and York [243, 242, 241] inthe 1970s and gave rise to a large body of research as the computational powerof the machines increased in the 1990s. A particular combination of choice of
formulation for the Einstein equations and technicalities in the treatment of
the numerics finally led to successful simulations of the last orbits of the binary
black-hole coalescence by Pretorius [212] and later by two other groups [88, 52]
in 2005. Ever since, the field has entered a new phase, characterized by thestudy of previously-unexplored physical phenomena, such as the computationof the recoil velocities —kicks— of black-hole mergers, the determination of
the final spin after the coalescence and the development of new methods toextract the most accurate gravitational radiation emitted by the binary system.
The direct applicability of these results to improved methods on gravitationalwave astronomy opens a promising era of synergistic collaborations between
numerical relativity and gravitational-wave data analysis.
This chapter presents a review of the key components that have made thisgolden age of numerical relativity possible and describes the simulations thatwill be used to construct the hybrid model of chapter 5.
4.1 solving einstein equations numerically
The covariant nature of the Einstein equations, when written in the compact
formulation given by equation 1.1, is evident. Time and spatial coordinates
are indistinguishable, which is only natural given the geometrical meaningof the equations in a space-time thought of as 4-dimensional manifold. Ifthese equations need to be integrated numerically, however, they need tobe rewritten in a more appropriate form. One way to proceed is to split the
4-dimensional covariant structure of the equations in such a way that a time
evolution can be performed, exploiting the special nature of time. There are
a number of other ways of dealing with the full Einstein equations, suchas characteristic methods or the use of harmonic coordinates. But here wewill only present the method commonly known as the space-plus-time 3+1

53

54

black-hole binaries in numerical relativity

β
αn

n

Σt

µ

µ

µ

t

µ

Σ0

Figure 10: Foliation of the space-time in the 3+1 decomposition of the Einstein equations. Σ0 and Σt represent two hypersurfaces of the foliation. nµ is theorthogonal vector to Σ0 and tµ the tangent to the curves threading thefoliation. The relation between these vectors is given by the lapse function
and the shift vector as tµ = αnµ + βµ .

decomposition of space-time, whose development in the 1960s and 1970s wasof the most importance to establish the basis and formalism needed for thesubsequent emergence of numerical relativity. The next sections succinctly
describe this formalism.
4.1.1 The 3+1 Decomposition
The 3+1 decomposition was first introduced by Arnowitt, Deser and Misnerin 1962 [41], together with other important concepts regarding the canonicalformulation of general relativity, now commonly referred to as the ADM
formalism. Ever since, other researchers have studied the well-posedness of
the Cauchy problem, thus clarifying the physical meaning of the problem
and supplying practical algorithms to find its solution. As a result of the
3+1 decomposition, the Einstein equations are split into constraint equations
which are solved to provide initial data on some initial slice, and evolution
equations which permit us to evolve the data from slice to slice [266].
As depicted in figure 10, the globally hyperbolic 4-dimensional space-time(M, g µν ) can be foliated by a set of 3-dimensional space-like hypersurfaces
Σt that completely fill the space. The foliation can be identified with the levelsets of a parameter t, which is a universal time function.
Consider now the future-pointing, time-like unit normal vector field to thehypersurfaces, nµ , which corresponds by definition to the 4-velocity of theEulerian observers. This vector nµ , defined asnµ = −α g µν ∇ν t,(4.1)

allows us to introduce the 3-dimensional metric γµν induced onto the hypersurfaces by the 4-metric g µνγµν = gµν + nµ nν ,(4.2)

4.1 solving einstein equations numerically55

and its contravariant form γµν = gµα gνβ γαβ . The mixed tensor γµν =µδµν + n nν is a projector onto the slices. The α function of equation 4.1
α−2 ≡ −g µν ∇µ t ∇ν t,(4.3)

is the lapse function.
We can construct a time vector field tµ which is tangent to the time lines,
i.e. the lines of constant coordinates astµ = αnµ + βµ ,(4.4)

where the spatial vector βµ = (0, βi ) is the shift vector. From the above
definition we find that tµ nµ = −α, which implies tµ ∇µ = 1. The shift thencorresponds to the projection of tµ onto the spatial hypersurface. Explicitly,
the vector tµ can be decomposed into its normal and tangent components
relative to the hypersurfaces
α = −tµ nµ ,
β

i

(4.5)

= γij tj .

(4.6)

As a matter of fact, the lapse function α and shift vector βi are gauge functions that define how coordinates move forward in time from slice Σ0 to
Σt . This is a reflection of the covariant character of the Einstein equations,which assures that any coordinate system is as valid as any other one when it
comes to describing the geometry of the space-time. The lapse sets the proper
time between successive hypersurfaces and the shift vector determines therelative velocity between Eulerian observers and the lines of constant spatial
coordinates.
Equipped with the above-defined {α, βi , γij }, we can now rewrite the line
element of the metric in terms of its time and spatial components as
ds2 = −α2 dt2 + γij (dxi + βi dt)(dxj + βj dt).

(4.7)

In this coordinate basis, the normal vector nµ is of the form
1
(1, −βi ).
(4.8)
α
Given the 3-metric γij it is possible to define a covariant derivative andassociated 3-dimensional Riemann tensor in a manner analogous to the 4dimensional expression given in equation 1.6. The spatial Riemann tensor
accounts for the intrinsic curvature of the slice Σt . However, one should
not forget that the slice is embedded in a 4-dimensional space, hence possessing extrinsic curvature as well. It is intuitively clear that, whereas the
4-dimensional Riemann tensor is a space-time object containing time derivatives of the 4-metric gµν , the 3-dimensional Riemann is a spatial object, andcan be computed only from derivatives of the spatial metric alone. There mustcertainly be some information missing when projecting the full Riemanntensor into the 3-dimensional slices, and that missing information that tells
us how the hypersurfaces are embedded in the space-time is called extrinsic
curvature.
The extrinsic curvature Kµν of the slice is defined via the projection of thegradient of the surface normal field and can also be expressed in terms of theLie derivative along the normal vector nµnµ = (−α, 0, 0, 0),nµ =1Kµν ≡ −γαν ∇µ nν = − Ln γµν ,
2

(4.9)

The covariant
derivative compatible
with γij will bedenoted D

The intrinsic andextrinsic curvatures ofthe slices and thecurvature of the
background space are
not independent, butthey are related bymeans of theGauss-Codazzi
equations

56

black-hole binaries in numerical relativity

and is a symmetric and purely spatial tensor. The last equality clarifies themeaning of the extrinsic curvature; it provides information on the rate ofchange of the spatial metric γµν as we move along the vector nµ along the hypersurfaces Σ. The complete geometry of a slice embedded in a 4-dimensional
space-time is represented by the triplet {Σt , γ, K}, which will turn out to be an
initial data set for the space-time. Obviously, not every set of 3-dimensionalslices can fit into the 4-dimensional space-time. Certain integrability conditions, called the Gauss-Codazzi equations, represent the sufficient and necessary geometrical identities that must be locally satisfied in order for thefoliation Σ to be correctly embedded in the full space-time.
4.1.2 The ADM Equations
The ADM equations encode the same physical content as the Einstein equations 1.1 but split in space and time and given in terms of the variables
α,

βi ,

γij ,

Kij .

(4.10)As a result of this projection of the full equations onto the hypersurfaces Σ andthe normal field nµ , a set of four equations arises that can be classified in two
categories: evolution equations for the variables γij and Kij and constraints
that need to be satisfied on every hypersurface. In principle one can computethe quantities 4.10 for an initial slice Σ0 and evolve the data to the next slice Σtby means of the time derivatives of γij and Kij . This process will eventuallylead to the complete specification of the geometry of the space-time, providing
at the same time a suitable algorithm to construct and numerically evolve any
physically relevant data.
The stress-energy tensor that appears on the right hand side of 1.1 includes
all the relevant information related to the matter content of the space-time.
The component T µν of the tensor refers to the µ–th component of the 4momentum tensor across a surface with constant xν coordinate. The stressenergy tensor is a symmetric object and its different components can beassociated with well-known physical quantities. Thus, T 00 is the energy density, T 0i represents the flux of energy across the xi surface, equivalent to T i0 ,which is the density of the ith momentum component. In addition, T ij (i 6= j)
is the shear stress and T ii represents a pressure-like quantity, the normal stress.
According to our intention of splitting the Einstein equations by projecting
them, it will be necessary to operate on their right-hand side, too. For thispurpose the energy density ρ, the momentum density jµ and the spatial stressSµν are introduced and defined as follows
ρ ≡ nµ nν Tµν ,ρjµ ≡ γσµ n Tσρ ,Sµν ≡ γµσ γνρ T σρ .

(4.11)(4.12)(4.13)

Thus we can finally write the ADM equations. The evolution equations for the
spatial metric and the extrinsic curvature read∂t γij = −2αKij + Di βj + Dj βi(4.14)

4.1 solving einstein equations numerically∂t Kij = βk Dk Kij + Kik Dj βk + Kkj Di βk − Di Dj α+ α (3) Rij − 2Kik Kk
j + KKij
1− 8πα Sij − γij (S − ρ) .
2

57

(4.15)

In 4.15, (3) Rij is the associated Ricci tensor with the metric γij . In addition to these evolution equations, the Hamiltonian and momentum constraint
equations read(3)

R + K2 + Kij Kij = 16πρ(4.16)

Dj Kji − Di K = 8πji

(4.17)

where (3) R and K denote the trace of the Ricci tensor and extrinsic curvature
respectively. In vacuum, the energy and momentum densities are zero, i.e.
ρ = 0 and jµ = 0.
In the set of equations 4.14—4.17, the gauge functions α and βi can be
freely specified. One of the issues to address when running numerical relativity
simulations is the choice of a well-behaved gauge that leads to a successful
numerical evolution. As it turns out, regardless of the chosen gauge, the ADM
system of equations is only weakly hyperbolic and thus does not give rise to awell-posed initial value problem in the most general case. Correspondingly,
the growth of high-frequency components in numerical simulations cannot
be bounded and reliable numerical simulations are not possible. Next section
describes an alternative formulation that has proven to be better suited in thiscontext.
4.1.3

Only for particularcases —like for instancespherical symmetry—
can the ADM equations
be made well-posed

The BSSN Formulation

A large number of alternative, hyperbolic formulations of the Einstein equations have been proposed to solve the problem of the weak hyperbolicity of the
ADM formulation. A system that has been found to behave in a satisfactorymanner, at least in binary black hole evolutions, is the one developed by Baumgarte and Shapiro [54] based on that of Shibata and Nakamura [239] after
Nakamura, Oohara and Kojima presented a reformulation of the ADM equations introducing a conformal transformation [187]. This alternative evolution
system is nowadays broadly known as the BSSN formulation.
The BSSN system belongs to the class of conformal traceless formulations.
Besides the use of conformal transformations and a traceless decomposition of
the extrinsic curvature, possibly the most important aspect is the promotionof the conformal connection to an independent variable. The idea behind the
BSSN formulation is to evolve a conformal factor and the trace of the extrinsic
curvature separately. Modifying the equations by using the constraints is the
crucial step to make the system hyperbolic. An alternative version of the BSSN
equations exists that does not include the conformal traceless transformation but is equally strongly hyperbolic; this approach is known as the NOR
system [186].
In the traditional BSSN formulation, the conformal factor is written asΨ = eφ so that the conformal metric isγ̃ij = e−4φ γij ,φ=1ln det γij ,12(4.18)

In geometry, twoRiemannian metrics g
and h on a smoothmanifold M are called
conformally equivalent
if g = uh for somepositive function u on
M. The function u iscalled the conformal
factor.
Traceless refers to thefact that the extrinsic
curvature is split in its
trace-free and trace
parts, and it is theformer that isconformallytransformed.

58

black-hole binaries in numerical relativity

requiring as an auxiliary constraint that the determinant of the conformal
metric γ̃ij is unity. The trace-free part of the intrinsic curvature1
Aij ≡ Kij − γij K,
3(4.19)

is rescaled like the metric itself
Ãij = e−4φ Aij .

(4.20)

Indices of this conformally transformed object 4.20 will be raised and loweredwith the conformal metric 4.18. We can now find the evolution equations for
φ and K making use of 4.14 and 4.15 and taking their trace1d
φ = − αK,
(4.21)dt6d1
K = −γij Dj Di α + α Ãij Ãij + K2 + 4πα(ρ + S) , (4.22)
dt
3
where
∂
∂
∂
d
=
− Lβ =− βi idt∂t∂t∂x(4.23)and the Hamiltonian constraint 4.16 has been used to eliminate the Ricci
scalar from 4.22. Now the trace-free parts of the equations yielddγ̃ij = −2αÃij ,dtdÃij = e−4φ −(Di Dj α)T F + α(RTijF − STijF )dt
+ α KÃij − 2Ãil Ãlj .

(4.24)(4.25)The superscript T F in the last equation denotes the trace-free part of a tensor.
The Ricci tensor Rij can now be decomposed into two terms
Rij = R̃ij + Rφij ,(4.26)

wherelRφij = −2D̃i D̃j φ − 2γ̃ij D̃ D̃l φ + 4 D̃i φ− 4γ̃ij D̃l φ D̃l φ ,



D̃j φ


(4.27)and D̃i is the covariant derivative operator associated with γ̃ij . The operatorR̃ij can be brought into a manifestly elliptic form by introducing the conformal
connection functionsiΓ̃ i ≡ γ̃jk Γ̃jk
= −∂j γ̃ij ,The symmetrization
operator is denoted byparentheses around theindices that are to beinterchanged: A(µν)≡ 12 (Aµν + Aνµ )(4.28)

i are the connection coefficients associated with γ̃ . In terms of
where the Γ̃jkij
these objects, the Ricci tensor can be written as

1
R̃ij = − γ̃lm ∂l ∂m γ̃ij + γ̃k(i ∂j) Γ̃ k + Γ̃ k Γ̃(ij)k2k+ γ̃lm 2Γ̃lk(i Γ̃j)km + Γ̃im
Γ̃klj .

(4.29)

4.1 solving einstein equations numerically

We can now appreciate the convenience of introducing the connection functions 4.28, because this way only the Laplace operator γ̃lm ∂l ∂m γ̃ij remains
in the Ricci tensor 4.29. The other second derivatives have been absorbed
in the derivatives of Γ̃ i . One can get rid of these second derivatives if theconnection functions are now promoted to independent functions, and an
evolution equation for them can be derived by permuting a time derivativewith the space derivative in equation 4.28


2 ij
d i
ij
m(j
i)
l
l
ij
Γ̃ = −∂j 2αÃ − 2γ̃
∂m β + γ̃ ∂l β + β ∂l γ̃
. (4.30)dt
3Finally, eliminating the divergence of Ãij with the help of the momentum
constraint 4.17 yields


2 ij
∂ i
ij
i
jk
ij
ij
Γ̃ = − 2Ã ∂j α + 2α Γ̃jk Ã − γ̃ ∂j K − γ̃ Sj + 6Ã ∂j φ
∂t
3


2
− ∂j βl ∂l γ̃ij − 2γ̃m(j ∂m βi) + γ̃ij ∂l βl . (4.31)
3Equations 4.21 through 4.25 together with 4.31 form the BSSN system of
evolution equations, equivalent to 4.14 and 4.15 but for the set of variables
φ,K,γ̃ij ,

Ãij ,

Γ˜i .

(4.32)

Since the connection functions are evolved as independent functions, their
original definition 4.28 serves as a new constraint equation, in addition to 4.16
and 4.17. Obviously, not all the variables of this system are independent, inparticular the determinant of γ̃ij has to be unity and the trace of Ãij has tovanish. These conditions can either be used to reduce the number of evolved
quantities or, alternatively, all quantities can be evolved and the conditions canbe used as a numerical check. Precisely this question of how the constraints
are used in the evolution equations is known to be a subtlety in obtaining
numerically stable evolutions with the BSSN formulation.
4.1.4 A Recipe for Successful Simulations
The long history of numerical evolutions of non-trivial space-times tells astory full of exciting research milestones on theoretical and numerical methods, major breakthroughs, dead ends and fresh starts that, as a matter of fact,
eventually led to a happy end. A complete description of all relevant research
milestones falls beyond the scope of this dissertation. It would not be inaccurate to say that some of the progress in numerical relativity was achievedby trial and error. This section briefly presents some of the ingredients that,a posteriori resulted fundamental for obtaining numerical codes able to successfully evolve a binary of black holes through the late inspiral to the merger
and ringdown.
Choice of Gauge
Many years of intense research on numerical methods and stability wereneeded to enable the transition between a theoretical formulation of generalrelativity which was suitable for implementation in a computer and the desired
result of a well-behaved, stable simulation of the black holes’ merger. The

59

60

We mean good in thesense of being adequate
for numerical
implementations

black-hole binaries in numerical relativity

use of appropriate gauge conditions was crucial to reach the final goal. As
mentioned in section 4.1.1, the choice of the lapse function α and the shift
vector βi is left free, since the choice of coordinates in Einstein’s theory isarbitrary. Good gauge conditions should, when possible, be adapted to theunderlying symmetries of the problem, avoid the formation of numericalsingularities and be mathematically well behaved [26].
An apparently simple gauge condition, known as geodesic slicing, definedby the equations
α = 1,

βi = 0,(4.33)

is by no means a good choice of gauge to evolve a black-hole space-time.
In this gauge, any coordinate observer, starting from rest, will fall into theblack-hole singularity. A superior choice is the maximal slicing, derived by
imposing that the volume of the spatial hypersurfaces is maximal, which isequivalent to the condition
K = γij Kij =

∂K
= 0.
∂t

(4.34)

Possibly the main disadvantage of this gauge choice is that it leads to an elliptic
equation for the lapse α, whose numerical solution might be computationallyprohibitive.
The simulations used for the construction of the full coalescence waveformmodel presented in chapter 5 make use of a particular member of a familyof slicing conditions proposed by Bona et al. [68], resulting in a hyperbolic
equation for the lapse which is easy to solve numerically. The lapse α is chosen
to satisfy the evolution equation∂dα = α − βi ∂i α = −α2 f(α)K,dt∂t(4.35)

with f(α) a positive but otherwise arbitrary function of α. The particular
choice implemented in the numerical codes that have produced the simulations employed in this thesis is f(α) = 2/α, which leads to
α = 1 + log γ,(4.36)a very easy-to-implement condition that, additionally, is tremendously robust.
Regarding the conditions for the shift vector βi , the mainstream way toproceed in the context of the most recent BSSN black hole evolutions —inparticular in the ones used in this work— has been based on the Γ -freezing
condition∂ iΓ̃ = 0,∂t

The minimal distortion
condition minimizeschanges in the shape ofvolume elements during
an evolution(4.37)

proposed as a natural choice for the shift in the BSSN formulation, as it freezes
three of the independent degrees of freedom. The Γ -freezing condition isequivalent to the minimal distortion condition and introduces an elliptic
equation for the shift, which is again computationally disadvantageous. An
innovative solution for the numerical implementation of the shift condition
was introduced by Alcubierre et al. [27] by means of a Γ -driver condition,
inspired by the idea of transforming the elliptic equation for the shift into

4.1 solving einstein equations numerically61

a parabolic one by making ∂t βi or ∂2t βi proportional to the given elliptic
operator. The Γ -driver can be written as3∂t βi − βj ∂j βi = αBi4iji∂t B − β ∂j B = ∂t Γ̃ i − βj ∂j Γ̃ i − ηBi ,(4.38)(4.39)

where η —do not mistake it for the symmetric mass ratio for binary systems
introduced in equation 2.6— is a freely-specifiable damping coefficient whichhelps avoid strong oscillations in the shift.
Excision and Punctures
There is still more to the problem than the choice of a suitable set of gauge
conditions. For years, the field of numerical relativity struggled to maintainstable evolutions, the instabilities propagating through the computational gridand eventually ending the simulation. The existence of infinites is obviously
problematic for a numerical code, and it would be desirable to find a proper
way of dealing with the black-hole singularities. At least two techniques were
implemented to deal with problems associated with the existence of physicalsingularities in the solutions to the equations. The excision method consists ofremoving a portion of a space-time inside of the event horizon surrounding
the singularity of a black hole, simply not evolving it. Due to the principleof causality and properties of the event horizon, this procedure should intheory not affect the solutions of the equations outside of the event horizon.
Additionally, ingoing boundary conditions on a boundary surrounding the
singularity but inside the horizon are imposed. This approach presents someproblems; while physical effects cannot propagate from inside to outside,
coordinate effects could. Besides, as the black holes move, the location of theexcision region needs to be continually adjusted to move with the black hole.
The full development of this technique, together with suitable gauge conditions
to increase stability and to allow the excised regions to move along the grid
was carried out for several years. The efforts did eventually pay off, and the firststable, long-term evolution of the orbit and merger of two black holes using
this technique was published by Pretorius in 2005 [212]. This undoubtedly
constituted a major breakthrough in the field of numerical relativity.
The second method that proved successful —and the one on which thesimulations used in this dissertation are based— was developed by Brandt
and Brügmann [73] and is known as the puncture approach. This method isa generalization of the Brill-Lyndquist prescription for initial data [74]. Oneof the advantages of the puncture method is the absence of inner boundary
conditions; the solution is factored into an analytical part, which contains the
singularity of the black hole, and a numerically constructed part, which isthen singularity free. The procedure can be viewed as follows: we know thatthe simplest solution for a Schwarzschild space-time in isotropic coordinates
implies a conformal factor Ψ = eφ of the form
φ=1+

m
,
2r

(4.40)

Pretorius did not use
BSSN, but rather ageneralized harmonic
formulation of theEinstein equations, notbased on the 3+1
scheme, which we do
not discuss here

The Schwarzschildsolution describes thegravitational fieldoutside a spherical,
non-rotating mass,
which can be a black
hole

62

black-hole binaries in numerical relativity

on a conformally-flat space-time γij = Ψ4 ηij . A generalization for N black
holes for time-symmetric initial data can be made exploiting the linearity of
the Hamiltonian constraint
φ=1+

N
X
i=1

mi
.
2 |~r − ~ri |(4.41)

With this insight, the basic idea behind the puncture method is the explicit
separation of the singular part in the expression for the conformal factor
φ = φBL + u,

φBL =

N
X
i=1

In a corotating frame
the coordinate positionsof the black holes
remain fixed while thereference frame rotateswith some angularvelocitymi.
2 |~r − ~ri |(4.42)

Thus, we can solve this equation for u and we would need to impose appropriate boundary conditions at infinity and, in principle, also at the holes’ location.
The key observation, and what was proved by Brandt and Brügmann, is that
there is no need for special boundary conditions at the punctures; they can beignored when solving for u without further complications. For the purposeof this dissertation it suffices to say that this fact simplifies enormously thetreatment of the singularities in the numerical domain; for a full discussion of
the implications of the punctures method for the geometry of the space-time
we refer the reader to [140, 141, 26].
At this point it is important to remember the following caveat: the methods
described above assume that the conformal geometry of the space-time is flat.
This might not necessarily be compatible with the physical system under study.
In particular, the method does not generate a Kerr solution for a single spinning black hole, since the Kerr space-time is not a conformally-flat geometry.
Conformally flat initial data always contain a dynamical component, which
will be radiated away in the form of non-physical gravitational radiation as thesimulation progresses. This pulse of “junk” radiation will propagate away fromthe computational domain; however, appropriate boundary conditions at theexternal edge of the grid are crucial to avoid contamination of the physicalgravitational radiation content of the system.
One last ingredient separates us from the goal of successful evolutions ofblack-hole binaries in the BSSN framework: for many years the opinion heldby a subset of the community was that corotating coordinates were a goodsolution to deal with the binary black hole problem. The idea was that working
out how to work with a corotating grid seemed easier than working out how tomove the black holes. This was also motivated by the assumption that keeping
the holes fixed in the grid would lead to more stable dynamics. Elaborate
gauge conditions for the shift vector βi in order to prevent the holes frommoving were required as a result. The corotating scheme was known to present
fundamental problems, especially regarding the outer boundaries, for which
no physically meaningful conditions could be applied. Additionally, extractionof the gravitational-radiation content of the space-time was technically more
involved.
The solution to this puzzle might appear obvious in view of what has been
said so far: why not letting the punctures move across the computational grid
while keeping the coordinate frame fixed. But an old issue with the moving
puncture approach was the lack of knowledge about what to do with the conformal factor, which blows up at the puncture. The puncture method factored
out the singular piece, and in order to maintain that analytical singular piece4.1 solving einstein equations numerically63

Table 5: Some properties of currently-working NR evolution codes. The table reflects
the status of published results at the time of writing. The columns list thename of the code, the employed evolution system (GH stands for generalizedharmonic formulation), the numerical technique (FD-k stands for finite
differences using k-th order stencils in the bulk), the η choices for the Γ̃ driver shift 4.39, the approximate location of the outer boundary and thefinest grid–spacing. Several of these parameters —such as the stencil order,
the damping η or the details of the grid— are tunable; we report characteristic
values that have been employed in the codes to obtain published results. For
the SpEC code, rmax decreases during the run.

Code

Reference

SystemTechniqueMηrmax /M

AMSS-NCKU[91, 126]

BSSN

FD–6

2

128

BAM

[77]

BSSN

FD–6

2

773

CCATIE

[206]

BSSN

FD–4

1

819

Hahndol[158, 256]

BSSN

FD–4, 62

> 1000LazEv[269, 88]

BSSN

FD–4, 8

6

1281

Lean

[244]

BSSN

FD–4, 6

1

256

Llama

[207]

BSSN

FD–8

1

3600

MayaKranc

[153]

BSSN

FD–4, 6

2

317.4

PU

[213, 212]

GH

FD–2

n/a

∞

SpEC

[69, 232]

GH

Spectraln/a

450 → 230

UIUC

[114]

BSSN

FD–4

0.25

409.6

stable, there was a strong belief in favour of keeping the punctures fixed. The
breakthrough was to realize that evolving the singular conformal factor was
actually quite easy. In fact, this was what indeed solved the problem and allowed the black holes to evolve from the last inspiral orbits through the plunge.
Two independent groups arrived at this same conclusion independently, usingslightly different, but essentially equivalent methods. The results appeared in
2005, by Campanelli et al. [88] and Baker et al. [52], shortly after Pretorius’
letter, and confirmed that indeed the problem of numerically simulating thebinary black hole merger had finally been solved. Since then, a number ofnumerical relativity groups across the world have presented their own versions of successful simulations, giving rise to a handful of working codes thatcan be used to compare results, produce new physics and test general relativity predictions. A summary of such codes is presented in table 5, together
with references and brief annotations about their technical aspects. Table 10
provides further information about these codes.
In the moving punctures approach, employed by the majority of the codesof table 5, the singular part of the conformal factor is not factored out; instead, the dynamical conformal factor has a logarithmic singularity which
is directly evolved. Rather than dealing with the conformal factor directly—as implemented in the CCATIE code—, the approach followed in Campanelli
et al. —and also in the BAM simulation shown in figure 20— is to evolve an
auxiliary variable defined as χ = Ψ−4 = e−4φ .

Later it was realizedthat the fixed-puncture
approach isproblematic, even if theblack holes do not move
Their codes are labelled
LazEv and Hahndol
respectively in table 5
Pretorius’ code islabelled PU

black-hole binaries in numerical relativityr h `=2, m=2 (t)

64

0.4
0.3
0.2
0.1
0.0
−0.1
−0.2
−0.3
−0.4
1200

h+

h×

1300

1400

1500

1600

1700

1800

1900

2000

1300

1400

1500

1600

1700

1800

1900

2000

1300

1400

1500

1600

1700

1800

1900

2000

|h|

160
140

φ (t)

120
100
80
60
40
1200

ω (t)

0.5
0.4
0.3
0.2
0.1
0.0
1200

t/M

Figure 11: Typical numerical relativity waveform for the late inspiral, merger andringdown or a binary black-hole system. The upper panel shows the twopolarizations h+ and h× of the gravitational-wave strain as well as theamplitude in dimensionless units as a function of time in units of the total
mass of the system. The middle and lower panels show the phase of the
waveform φ(t) and its instantaneous frequency ω(t). The chirp is clearly
noticeable. The data corresponds to a non-spinning, equal-mass simulation
made with the BAM code.

M is the total ADM
mass of the space-timeContrary to the initial intuition of many numerical relativists, the moving
puncture approach leads to robust and stable simulations of black-hole spacetimes lasting for many hundreds of M, allowing to follow the inspiral, merger
and ringdown of a binary and accurately extract their gravitational radiation.
Figure 20 shows one example for an equal-mass, non-spinning system evolved
with the BAM code. The simulation starts ∼ 2000 M before the merger —thepulse of junk radiation is clearly visible, although it is not shown here—, when
the two black holes are separated by a distance of 12 M. The holes are given
initial data compatible with a quasi-circular orbit, making use of a method
inspired in post-Newtonian evolutions [157] that reduces the eccentricity toa small value of e < 0.002. The system is evolved using the BSSN equations
and the moving puncture approach described above. Gravitational radiation
is extracted at rext = 90 M by means of the Newman-Penrose scalar Ψ4
as described in [77]. The two polarizations of the gravitational-wave strainh+,× (t) are plotted in the upper panel, together with the amplitude, for thedominant mode l = 2, m = 2 of the radiation. The phase φ(t) and angularfrequency ω(t) of the wave are shown as well.

4.2 numerical simulations of non-precessing binaries

4.2

numerical simulations of non-precessing binaries

Having seen that numerical simulations of black-hole binaries are possible,
there is nothing to prevent us from further exploring the parameter spaceof physical configurations of the binary. Since the already-mentioned firstsuccessful numerical simulations of equal-mass, non-spinning binary blackhole mergers were published [212, 88, 52] the NR community has continued
exploring the parameter space of the BBH system. It is well known that astrophysically relevant black-hole binaries ought to have components with
non-negligible spins. Likewise, systems with very different mass ratios andeccentric orbits are thought to exist in our Universe.
Aware of the complexity of parameter spaces with multiple degrees of
freedom, most match-filtered searches for coalescing binaries have so far employed non-spinning templates, neglecting the effect of the spin by assuminga small, tolerable loss in SNR [10, 15, 16]. Dedicated searches for spinningbinaries have attempted to model an enlarged parameter space by using aphenomenological template family designed to capture the spin-induced
modulations of the gravitational waveform [11]. It would be desirable to understand the mapping between phenomenological and physical parameters, andto devise searches for spinning systems based on strictly physical parameters.
As a first step towards the general case, the phenomenological model constructed in chapter 5 shall be restricted to a subset of possible physical configurations, those in circular orbits and with spins aligned or anti-aligned with theangular momentum, as we justified in section 3.1. There are a number of NR
simulations reported for non-precessing systems for a variety of spin values
and mass ratios. Results with the BAM code are reported in [145] for the orbital
hang-up case and in [144] for anti-aligned spins. The CCATTIE simulations
are presented in [206, 221, 223]; a long spectral simulation with anti-aligned
spins can be found in [92]. In this section we summarize all these numerical
waveforms used in the construction of our phenomenological model.
4.2.1 NR waveforms and Codes
The NR waveforms employed in the construction of the hybrid model usedin this paper are summarized in Table 6. They have been produced with four
independent NR codes, BAM, CCATIE, Llama and SpEC. The first 3 codes use
the moving-puncture approach [52, 88] to solving the Einstein equations
in a decomposed 3+1 space-time while the last implements the generalizedharmonic formulation [213, 69]. BAM and CCATIE use computational domains
based on Cartesian coordinates, and the SpEC code use a sophisticated series ofspherical and cylindrical domains; in the wave zone, the outer computational
domains have the same angular resolution, thus the computational cost only
increases linearly as the radius of the outermost shell is increased. The Llama
code is based on finite differencing but the set-up of the numerical grid in
the outer wave zone, as in SpEC, is also based on spherical coordinates with aconstant angular separation. The large wave-zone enables accurate waveform
extraction at large distances, accurate extraction of higher angular modes of
the radiation, and it allows the outer boundary to be far enough away so that
it is causally disconnected from the sphere where the radiation is extracted. A
summary of the properties of all three codes is given in [143].

65

66

black-hole binaries in numerical relativity

Table 6: NR codes and configurations used for the construction and verification of
our hybrid waveforms and phenomenological model. The mass ratio q is
defined as m1 /m2 , assuming m1 > m2 ; χ1,2 are the dimensionless spins
defined in equation. 3.1; a positive value of χ1,2 means that the spin is alignedwith the orbital angular momentum L, and negative values are anti-aligned.
Set

Code

Mass ratios

Spins

GW extraction#1 BAM [77, 157] q ∈ {1 − 4}(0, 0)

R = 90M

#2

”

q=1

(a, a), a ∈ ±{0.25, 0.5, 0.75, 0.85}

”

#3

”

q = 2, 3, 4(a, a), a ∈ {±0.5, 0.75}

”

#4

”

q=3

(−0.75, 0.75)”#5 CCATIE [206]
#6
”

q=1
q=1

(a, a), a ∈ {0, 0.2, 0.4, 0.6}(a, −a), a ∈ {0, 0.2, 0.4, 0.6}

R = 160M
”

#7ab

q=1

(±0.6, a), a ∈ {±0.3, 0, −0.6}”(0, 0)

Null Infinity1(0, 0)

R = 225M”#8 Llama [207] q ∈ {1, 2}#9

SpEC [231]q=1

The BAM data-set #1 covers the parameter space of non-spinning systems
for several mass ratios during at least the last 5 orbits before merger (length∼ 1100 − 1450 M, where M is the total ADM mass of the space-time) [146,23, 22, 144]. Data-set #2 consists of moderately long simulations covering
at least the last 8 orbits before merger (length ∼ 1500 − 2200 M) for equalmass systems with equal spins, and are described in depth in [145, 144]. Dataset #3 consists of unequal-mass, unequal-spins simulations [25]. Data-set#4 is a simulation with unequal mass and unequal spins employed in theverification of our fitting mode [25]. For the sets #1–4, initial momenta for
quasi-circular orbits were computed for non-spinning cases according to
the procedures described in [157], leading to low-eccentricity (e ≈ 0.006)
inspiral evolutions. A number of different methods were used for the spinningcases [76, 145, 144], depending on which method gave the lowest eccentricity
for a given configuration. The GW radiation is calculated from the Weyltensor component Ψ4 (see e.g. [202]) and extracted at a sphere with radius
R = 90 M.
The CCATIE data-sets #5, #6 and #7ab correspond to the s–, u–, r– andt–sequences studied in [220]. They span the last ∼ 4 − 5 orbits before merger(length ∼ 500 − 1000 M) and are in fact not sufficiently long for being usedin the hybrid construction. They are still useful to independently verify the reliability of our phenomenological fit. Data-set #5 corresponds to the hang-up
configuration analogous to the BAM set #1; data-set #6 consists of configurations with (χ1 , χ2 ) = (a, −a), i.e. zero net spin; data-set #7a was analyzed
in [206] in the context of the study of the recoil velocity (“kick”) of the final
merged black hole. GW radiation is extracted at R = 160 M via the ReggeWheeler-Zerilli formalism for perturbations of a Schwarzschild black hole
[217, 268, 17, 18].
1 Only the GW radiation corresponding to the Llama q = 1 simulation has been extracted atfuture null-infinity using the Cauchy-characteristic method; the q = 2 waveform has been
extracted at finite radius and extrapolated to r → ∞.

4.2 numerical simulations of non-precessing binaries

Data-set #8 consists of two waveforms for non-spinning black holes withmass ratios q = 1, 2. The black holes are evolved with the Llama code according to the set-up reported in [207]. The outer boundary is placed at 3600M
and the initial separation is 11M, corresponding to 8 orbits in the inspiralphase followed by the merger and ringdown. The wave-extraction for theq = 1 configuration uses the Cauchy-characteristic method [218, 219], taking
boundary data from the numerical space-time for a subsequent characteristic
evolution of the metric to null-infinity, thereby obtaining waveforms that areunambiguously free of any systematic gauge effects. The only remaining sourceof error is then due to numerical discretization. The equal mass waveformusing this code was reported in [219], while the q = 2 waveform is new.
Data-set #9 consists of a long non-spinning, equal-mass simulation that
follows 16 orbits of the binary plus merger and ringdown of the merged blackhole (length ∼ 4300 M). It was computed using the SpEC code with negligible
initial orbital eccentricity (∼ 5 × 10−5 ). The GW radiation is extracted via
Ψ4 at R = 225 M in a similar manner to #1–4. A full description of thissimulation is given in [232]. The long duration of the waveform allows for its
use in the estimation of the errors associated to the length of the NR data.
In particular, since it contains physical information at lower frequencies, anearlier matching is possible that reduces the ambiguities introduced by PN.
4.2.2 Going from Ψ4 to h
The gravitational waveforms calculated using NR codes are typically reported
in terms of the Weyl tensor component Ψ4 , which is a complex function
encoding the two polarizations of the outgoing transverse radiation. Ψ4 isrelated to the two polarizations of the gravitational wave perturbation h+,×
(in the transverse-traceless gauge) via two time derivativesΨ4 =d2[h+ (t) − ih× (t)]dt2

(4.43)

Going from Ψ4 to h+,× thus involves two time integrations and requires us
to fix two integration constants appropriately. The frequency domain offers astraightforward way of calculating the strain
h = h+ − ih×(4.44)

from Ψ4 , since integration is replaced by division:h̃ NR (f) = −

Ψ̃NR
NR
4 (f)
= ANR (f) eiΦ (f) ,
2
2
4π f

(4.45)

where x̃(f) denotes the Fourier transform of x(t) as defined later in equation 5.2. Choosing the integration constants can be avoided by removing low
frequency components via a suitable high-pass filter. This turns out to be aconvenient, accurate and possibly less cumbersome method of calculating
the strain h from Ψ4 . An illustration is given in Fig. 12, where Ψ4 data corresponding to a Llama simulation has been double-integrated in the time andin the frequency domain to yield h. In the case of the time-domain integration, a fitting procedure is needed to determine the two integration constants[60]; however, yet a small overall drift in h(t) remains. When performing thedivision in the frequency domain, tanh-window functions are employed to67black-hole binaries in numerical relativity
100
10

AHMfL

1
0.1
TD integration
FD integration

0.01
0.001

0.010

0.015 0.020

0.030

0.050 0.070 0.100

0.150 0.200

Mf
0.4

0.2

r h+

68

0.0

-0.2

TD integration
FD integration

-0.4
-1400

-1200

-1000

-800

-600

-400

-200

0

tM
Figure 12: Comparison between the double integration in the time domain (solid line)
and the division in the frequency domain (dashed line). Plotted is a NR
simulation from data-set #8 of Table 6 with q = 1. The upper panel shows|h̃(f)|; we observe cleaner high-frequency components when performing
the FD division, in contrast
to the TD integration. The lower panel showsqh+ (t) and |h(t)| = h2+ (t) + h2× (t); the window function employed by
our inverse Fourier transform algorithm is responsible for the loss of thefirst cycle of the waveform. Nevertheless, an overall cleaner |h(t)| during
the rest of the inspiral is observed when using the FD division.

pass-filter the data before computing the Fourier transform. Both methods
involve a certain degree of fine-tuning in order to produce clean results; inthis paper we use the frequency-domain version of the integration process.
This chapter concludes the description of the theoretical tools needed for theconstruction of the waveform model for the full coalescence of non-precessing
binary black-hole systems introduced in chapter 5.

5

NEW WAVEFORM MODEL FOR BINARY BLACK HOLES

Coalescences of black-hole binaries are expected to be powerful sources ofgravitational waves. Analytical and numerical models of the radiation emitted by these systems are crucial for detection and parameter estimation inmatched-filter searches. The description and present status of both approaches
has been discussed in the preceding chapters. Here, both methods are brought
together in order to construct a model capable of describing the full binary
black-hole coalescence process.
We present a novel approach to the construction of phenomenological gravitational waveform template models for non-precessing spinning black-holebinaries. Our method is based on a frequency domain matching of postNewtonian inspiral waveforms with numerical relativity-based binary blackhole coalescence waveforms. After introducing the need for full waveformmodels, section 5.2 outlines the conceptual differences between the analyticaland numerical approaches to the binary black-hole problem. We quantify
the various possible sources of systematic errors that arise in matching postNewtonian and numerical relativity waveforms, and we use a matching criteria
based on minimizing these errors. In section 5.3, an analytical formula for thedominant mode of the gravitational radiation of non-precessing black-hole binaries is presented that captures the phenomenology of the hybrid waveforms.
This model constitutes one of the main achievements in this dissertation. Its
implementation in current searches for gravitational waves should allow crosschecks of other inspiral-merger-ringdown waveform families and improve
the reach of gravitational wave searches.
5.1

This chapter
summarizes the work
presented in [227]full models for binary black-hole coalescence

As a generalization of the classic Kepler problem in Newtonian gravity, thebinary black hole (BBH) system in general relativity is of great interest from afundamental physics viewpoint. Equally importantly, this system has received
a great deal of attention for its relevance in astrophysics and, in particular,
as one of the most promising sources of detectable gravitational radiation
for the present and future generation of gravitational-wave detectors, suchas LIGO [13], Virgo [19], GEO600 [136] or LISA [238]. The Kepler problem
can be solved exactly in Newtonian gravity and it leads to the well-known
elliptical orbits when the system is gravitationally bound. In contrast, ingeneral relativity, the BBH system is not stable; it emits gravitational waves
which carry energy away, thereby causing the black holes to inspiral inwards,
and to eventually coalesce. The emitted GWs are expected to carry important
information about this process, and it is one of the goals of gravitational-wave
astronomy to detect these signals and decode them.
No analytic solutions of Einstein’s equations of general relativity are known
for the full inspiral and merger of two black holes. Post-Newtonian methods
can be used to calculate an accurate approximation to the early inspiral phase,using an expansion in powers of v/c. As for the coalescence phase, startingwith [212, 88, 52], the late inspiral and merger has been calculated by large-

69

v is the orbital velocity;
c is the speed of light70

A summary of the
published “long”
waveforms is given in
the review [142], and acomplete catalog ofwaveforms is being
compiled at [189]; more
recent work issummarized in [152]This window could beeither in the time orfrequency domain

new waveform model for binary black holes

scale numerical solutions of the full Einstein field equations. Since the initial
breakthrough, there has been dramatic progress in numerical relativity simulations for GW astronomy, including many more orbits before merger, greater
accuracy, and a growing sampling of the black-hole-binary parameter space.
NR results are now accurate enough for GW astronomy applications overthe next few years [143], and have started playing a role in gravitational wave
searches [87, 49].
Given PN and NR results, it is promising to try and combine them toproduce “complete” inspiral-merger-ringdown waveforms. PN techniques intheir standard formulation become less accurate as the binary shrinks, andthe approximation breaks down completely somewhere prior to the merger.
NR waveforms, on the other hand, become more and more computationallyexpensive the larger the number of cycles that one wishes to simulate; thecurrent record is 16 orbits for the equal-mass non-spinning case [232]. We
therefore would hope to combine PN and NR results in the region between
the point where NR simulations start, and where PN breaks down. To dothis it is critical to verify that the PN and NR results are in good agreementin this region, and that there is a consistent PN-NR matching procedure.
Much work has been done in comparing PN and NR results over the last
5-15 orbits before merger, so far focusing on the equal-mass non-spinningcase [82, 53, 146, 135, 69, 70], the equal-mass non-precessing-spin case [145],and one unequal-mass precessing-spin case [89]; these studies suggest that asufficiently accurate combination of PN and NR results should be possible.
One topic that has not received much attention, however, is the systematic
errors that are introduced by different choices of matching procedure.
One of the aims of this chapter is to further understand and quantify thevarious systematic errors that arise in the matching procedure. There arethus far two kinds of approaches to the NR-PN matching problem, both ofwhich have yielded successful results. The first is the Effective-One-Body
approach [80, 83, 98, 103]. Originally motivated by similar techniques inquantum field theory, the idea is to map the two body problem into an effective
one-body system with an appropriate potential, and with the same energy
levels as the two-body system. It was shown [80] that the appropriate one-bodyproblem (for non-spinning black holes) is that of a single particle moving in a
deformed Schwarzschild space-time. It turns out that most parameters of thisone-body system can be found by using the appropriate PN calculations, andthe remaining parameters are calculated by calibrating to numerical relativity
simulations. This approach has been successful so far for non-spinning systems
where only a single parameter needs to be calibrated by NR simulations [84,85]. The spinning case is more complicated, and work is underway to extend
the parameter space described by the model [199].
A complementary approach is to perform a phenomenological matchingof the GW waveforms in a window where both PN and NR are expected to
be good approximations to the true waveform. The first step is to construct a
hybrid NR+PN waveform by matching the two waveforms within the matching window. The waveform is completely PN before this window, completely
NR afterwards, and it interpolates between the two in the matching window.
Once the hybrid waveform is constructed and we are confident about thematching procedure, the second step is to fit the hybrid waveform to a parameterized model containing a number of phenomenological coefficients andfinally to map them to the physical parameters of the system. The resulting5.2 matching post-newtonian and numerical relativity

model would thus be parameterized by the masses and spins of the two black
holes. Most of the work in this approach has thus far been in the time domain[24, 22, 23, 25].
In the work presented here we take a complementary approach; both theconstruction of the PN+NR hybrid waveform and the matching to a phenomenological model are carried out in the frequency domain. The reasonsfor this are twofold. Firstly, the errors in the matching procedure are technicallyeasier to estimate in the frequency domain. Secondly and more importantly,
in light of these potential errors, comparing results between two independent
methods is a valuable way of ensuring that the matching procedure is robust.
The main results of this chapter are, firstly, to construct hybrid waveforms
for binary black hole systems with aligned spins in the frequency domain. We
do this by combining 3.5PN waveforms in the stationary phase approximation
with a number of NR results. We show that this construction is internallyconsistent and it yields hybrids which are, for the most part, sufficiently accurate for the initial and advanced LIGO detectors. The difference between
the various PN approximants is a more significant source of error than the
numerical errors in the NR waveforms. Using these hybrid waveforms, weconstruct a phenomenological frequency-domain waveform model depending on three parameters that covers the space of aligned spins and moderatemass ratios. We show that the model fits the original hybrid waveforms withoverlaps better than 98% for Advanced LIGO, and for the most part, better
than 99% for essentially all black hole systems observable with Advanced
LIGO, i.e. for systems with total mass ranging from ∼ 10 to ∼ 400M .
The post-Newtonian waveform model and the numerical waveforms that
we employ have been already described in preceding chapters. Section 5.2
describes the fitting method and the various systematic errors that appear inthis procedure. It quantifies the reliability of the waveforms for specific GW
detector and signal-to-noise ratios. Section 5.3 fits these hybrid waveforms toan analytic model. It shows that the model provides a good representation of
the hybrid waveforms and can be used in GW searches in the appropriate parameter space. Finally, section 5.4 concludes with a summary and suggestions
for future work.
5.2 matching post-newtonian and numerical relativity
5.2.1 Basic Notions
The basic criteria for evaluating the goodness of fit for the hybrid waveform
requires a notion of distance between two GW signals h(t) and h0 (t). The
simplest notion is the distance in the least-squares sense over an interval
t1 6 t 6 t2
Z t2
2
0
δt1 ,t2 (h, h ) =h(t) − h0 (t) dt .
(5.1)
t1

We shall use this for the numerical relativity waveform hNR (t) and the postNewtonian waveform hPN (t), with the interval [t1 , t2 ] being chosen so thatboth waveforms are reasonably good approximations (in a sense to be quantified later). Thus, the PN waveform is taken up to t2 and the NR waveform istaken to start at t1 ; they overlap within the interval [t1 , t2 ].

71

Eccentricity would
need to be added, if
appropriate

We use the same
parameters as in [25]72

new waveform model for binary black holes

Let us consider the frequency domain equivalent. Our convention for the
Fourier transform of a signal x(t) isZ∞x̃(f) =
x(t) e2πift dt .
(5.2)
−∞

One needs to be careful in converting the time interval [t1 , t2 ] to a frequency
interval [f1 , f2 ]. In principle, the Fourier transform is “global” in time, andsignals which have compact support in time cannot have compact support
in frequency, and vice versa. However, for the binary black hole waveforms
that we are considering, the frequency always increases in time, so that we
can sensibly associate a frequency interval [f1 , f2 ] with a given time interval[t1 , t2 ]. For these waveforms, we can consider the above distance definitionin the frequency domain:0δf1 ,f2 (h, h ) =Z f2

h̃(f) − h̃0 (f)

2

df .

(5.3)

f1

We shall use such a norm (applied to the phase) for constructing the hybrid
waveform.
When evaluating the goodness of a hybrid waveform for a particular detector, we need to consider detector-specific inner products, which are convenient
to describe in the frequency domain. Let Sn (f) be the single-sided power
spectral density of the noise in a GW detector defined as 1
E ñ(f)ñ∗ (f0 ) = Sn (f) δ(f − f0 ) .
2

See e.g. Papoulis [201]See e.g. Helstrom [149]Here n(t) is the detector noise time series with ñ(f) its Fourier transform andE refers to the expectation value over an ensemble of independent realizationsof the noise, which is assumed to be a zero-mean, stationary, stochastic process.
This equation implies that data at different frequencies are independent, andis one of the reasons why working in the frequency domain is so useful in data
analysis. The time domain description of the noise is more complicated; n(t)and n(t + τ) are in general not independent; E[n(t)n(t + τ)] is generally
non-zero. For stationary noise this is a function C(τ) only of τ, and is related
to Sn (f) via a Fourier transform.
Given Sn (f), we use the following definition of an inner product between
two signals x(t) and y(t)Z∞
x̃(f)ỹ∗ (f)df(5.5)
hx|yi ≡ 4<
Sn (f)
0where x̃(f), ỹ(f) are the Fourier transforms of x(t), y(t) respectively. This inner product is appropriate for Gaussian noise and forms the basis for matchedfiltering. It can be used to define a suitable notion of distance between two
signals h(t) and h0 (t) as hδh|δhi, where δh(t) = h0 (t) − h(t).
Following [175], we define a one-parameter family of waveforms whichinterpolates between h(t) and h0 (t) as
h00 (t; λ) = h(t) + λδh(t) .

See e.g. Kendalls [247]

(5.4)

(5.6)

If we use an unbiased estimator for λ, the variance σ2λ of the estimator isbounded from below by the Cramer-Rao bound5.2 matching post-newtonian and numerical relativity

σ2λ > hδh|δhi−1 .

73

(5.7)

This can be a useful bound for large SNRs which is in fact what we are interested in here; it is easier to distinguish between two loud waveforms anddemands on the waveform model are correspondingly more stringent. Thus,
a useful condition for being able to distinguish between the two waveforms is
σλ < 1. Thus, if h(t) is the true waveform and h0 (t) our approximation to it,then we say that h0 (t) is a sufficiently accurate approximation if hδh|δhi 6 1.
It is clear that hδh|δhi ∝ ρ2 where ρ = hh|hi1/2 is the optimal SNR. Hence,as we just remarked, the two signals are easier to distinguish when the detector
is more sensitive, or when the signal amplitude is larger. It will be convenient
to normalize the norm of δh and write this distinguishability criterion as11hδh|δhi > 2 .
2
ρ
ρ

(5.8)

Thus, for a given detector, we choose a reasonable guess ρ0 for the largest
expected SNR and we compute the normalized distance between the two waveforms hδh|δhi/ρ2 . If this exceeds 1/ρ20 , then we consider that the detector isable to distinguish between the two waveforms.
If we are interested in the less stringent requirement of detection rather
than in strict distinguishability, then a sufficient condition is [175]1hδh|δhi < 2,
ρ2

(5.9)

where  is the maximum tolerated fractional loss in SNR. If we are willing toaccept e.g. a 10% loss in detection rate, then a suitable choice is  ≈ 0.03.
A useful way to describe the efficacy of approximate waveform models isthrough the concepts of effectualness and faithfulness introduced in [98]. Let
hλ (t) be the exact waveform with parameters λ and the approximate waveapp
form model be hλ (t). The ambiguity function is defined as the normalized
inner product maximized over extrinsic parametershhλ |hλ0 iA(λ, λ ) = max q
app app
t0 ,φ0hhλ |hλ ihhλ0 |hλ0 iThis corresponds tosources uniformly
distributed in spaceapp

0(5.10)

where t0 is the time offset between the two waveforms, and φ0 is the initial
phase. Performing a further maximization over the parameters λ0 of the model
waveforms, we define Â(λ) = maxλ0 A(λ, λ0 ). If Â(λ) exceeds a chosen
threshold, e.g. 0.97, then the waveform model happ is said to be effectual. In
order to be able to estimate parameters we also need the model to be faithful.
This means that the value of λ0 which maximizes A(λ, λ0 ) should not be biased
too far away from λ.
5.2.2

Issues in Matching PN with NR

It is useful at this stage to discuss some of the issues that arise in combining
PN and NR results. Since PN and NR are both used to address the BBH
problem, one can use start with the two black holes very far apart, evolve
them using appropriate PN equations of motion and compute the resulting
waveforms. As one gets close to the merger, terminate the PN evolution anduse this end-point to construct initial data for the full NR simulation which

Effectual models are
sufficient for detection

74

See e.g. [134, 211]

In fact it can be
persuasively argued
that the cosmologicalsetting is more relevant
to GW observations
than strict asymptotic
flatness

This has not yet been
quantified

new waveform model for binary black holes

then evolves the black holes through the merger and ringdown. However, theformalisms and methods employed in the two cases are radically different
and there are potential difficulties in carrying out this procedure.
PN is based on a perturbative expansion in powers of the small parameter = v/c, where v is the orbital velocity and c is the speed of light. In the usual
formulations, PN theory uses a point-particle description of the black holes,
and their parameters can be viewed as effective parameters which couple in
the appropriate manner with the external background gravitational field. The
goal of PN theory is to find a one-parameter sequence of solutions to thefield equations gµν to any specified order in . It has recently been shown
rigorously [192] that, in the cosmological setting with gravitating perfect fluids,
the one-parameter family of solutions exists and admits an expansion in nto any order. While similar results in the asymptotically-flat case are not yet
available, it is certainly reassuring to know that PN works as advertised in thisnon-trivial setting. The errors in PN waveforms are then due to the systematicdifferences between the true waveform and the asymptotic series expansionin n truncated at a finite order, and this error depends on which particular
PN expansion one chooses to use.
In contrast, numerical relativity is based on the 3+1 formulation of generalrelativity as an initial value problem, and one solves the resulting partial
differential equations numerically. The GW signal is typically extracted at large
distances from the source by calculating the outgoing transverse componentof the gravitational radiation encoded in the Weyl tensor component Ψ4 . For
a given physical configuration (choice of masses, spins, separation etc.), one
specifies the initial data consisting of the spatial metric and extrinsic curvatureof the initial spatial slice. The initial data should be chosen to be as compatible
as possible with the space-time computed in the PN formalism, and significant
progress has been made in this regard [156, 260]. The black holes here are
not point particles but rather black hole horizons. The parameters of theblack hole are often computed as integrals over the apparent horizon, and in
most cases the parameters used in constructing the initial data are also usefulapproximations to the true ones. There are however possible systematic errors.
For example, if we are using the quasi-local horizon definitions, an important
requirement is that the horizon should locally be approximately axisymmetric.
The methods for finding the approximate symmetry vectors have become
increasingly accurate and reliable [107, 159, 55, 147, 177]. However, it should be
kept in mind that the assumption of approximate axisymmetry is expected to
become increasingly worse closer to the merger. Furthermore, the very use of
apparent horizons is gauge dependent; using a different time coordinate willlead to a different set of apparent horizons and possibly also different valuesof the parameters. In the inspiral phase when the horizons are sufficiently
isolated this gauge issue is not expected to be a problem, but as we get closer
to the merger, the variation in the parameters due to gauge choices could
become significant [188].
Let us elaborate a little more on the spin. Most post-Newtonian treatments
are based on the equations of motion derived in [95, 200]. The starting pointis the spin tensor Sµν constructed from moments of the stress energy tensor
T µν . Since Sµν has potentially 6 non-zero independent components, thesystem for the 4 equations of motion ∇µ T µν = 0 is over-determined. One
thus imposes the so-called spin supplementary conditions such as Sµν pν =0 or Sµν uν = 0 with pµ being the 4-momentum and uν the 4-velocity.

5.2 matching post-newtonian and numerical relativity

These different conditions lead to physically different equations of motion
and trajectories [172]. On the other hand, for black holes in NR, a common
method for evaluating spin employs the formalism of quasi-local horizons[45]. The final result for the magnitude of the horizon angular momentum is
an integral over the apparent horizon S:I
1J=−Kµν φµ dSν ,(5.11)
8π S
where Kµν is the extrinsic curvature of the Cauchy slice, φµ is a suitableapproximate axial symmetry vector on S [107, 159, 55, 147, 177], and dSb isthe area element on the apparent horizon. The direction of the spin is harder
to find, but some approximate methods are available [90, 159]. There is yet no
detailed study of possible analogs of the spin supplementary conditions inthis formalism, or on the equations of motion for horizons with a given setof multipole moments. For a horizon with area A and spin magnitude J, themass is given by the Christodoulou formula
r
A
4πJ2
m=
+
.
(5.12)
16π
A
Hence, uncertainties in spin can also lead to uncertainties in the mass.
As long as we are dealing with just the numerical or PN waveforms bythemselves, small effects in the definitions of mass and spin are not importantfor most applications. In fact, we can treat them as just convenient parameterizations of the waveform without worrying about their detailed physicalinterpretation. However, when we wish to compare the results from frameworks as different as PN and NR this may no longer work. Depending on thedetails of the matching procedure, systematic differences between the various
definitions might need to be taken into account, or at the very least theyshould be quantified. One valid approach is to not assume a priori that the PN
and NR parameters are equal to each other but rather, for a given numerical
waveform, we search over PN waveforms in a particular PN approximant andfind the best fit values.
5.2.3 An Illustration for Non-Spinning Systems
Let us now move to a concrete case of constructing hybrid waveforms, considering the non-spinning Llama waveforms, i.e. data set #7 in Table 6. Recall
that this data set consists of two waveforms with non-spinning black holes
with mass ratios 1 : 1 —used in figures 16 and 17— and 1 : 2 —figures 13, 14,
15 and 18—. Since these waveforms are calculated using the Llama code withextraction at future null-infinity with the Cauchy-characteristic method forthe equal mass case, or well into the wave zone for the 1 : 2 case, we have ahigh degree of belief that systematic effects of waveform extraction are small.
Even for these waveforms, based on the discussion above, in principle weshould not rule out a small mismatch in the values of the spin (and perhaps
also eccentricity) between the NR and PN waveforms. For simplicity, let us
consider only the possibility that the symmetric mass ratio η could be different, and restrict ourselves to non-spinning black holes and zero eccentricity.
We would like to match the Llama waveforms with the frequency domain PN
waveforms discussed in section 3.1 with the values of the spins set to zero.
The total mass M sets the scale for the time (and frequency); in addition75

new waveform model for binary black holes

Log10DΦ0

0.020

-1.8

0.015
width D Mf

76

0.010

-0.8

0.005

0.009

0.012
0.015
MfL fitting window

0.018

Figure 13: Contour plot for the fitting error ∆φ0 in the (fL , ∆f) plane. Here η is kept
fixed to the NR value and we optimize over φ0 and t0 .

we have the extrinsic parameters for the time offset and initial phase t0 and
φ0 . Furthermore, we only consider the ` = m = 2 mode, so that the PN
waveform is of the form h̃PN (Mf; φ0 , t0 , ηPN ) in the frequency domain.
Fitting Errors
For a given NR waveform hNR (t) we consider a time window (t0 , t0 + ∆t)
or, alternatively, in the frequency domain the matching region consists of alower starting frequency fL and a width ∆f. We match the two waveforms ina least-squares sense by minimizing the phase difference in Fourier space
Z fL +∆fδ=min

t0 ,φ0 ,ηPN fL|δφ(f; ηNR , ηPN , t0 , φ0 )|2 Mdf ,δφ(f) ≡ φNR (f; ηNR ) − φPN (f; t0 , φ0 , ηPN ) .

(5.13)

We optimize δ over all allowed time and phase shifts, i.e. (t0 , φ0 ), and thePN intrinsic parameters λPN . Given the previous discussion on the possible differences between the intrinsic parameters λ between the PN and NR
frameworks, here we have distinguished between the intrinsic parameter η ofequation 2.6 appearing in hPN and hNR . Note that we are not only neglecting
spins and eccentricity but also assume MPN = MNR = M. Future analyses
should successively drop these simplifications.
Let us now consider the choice of the optimal matching window (fL , fL +∆f), and the best fit values of (φ0 , t0 , ηPN ). For each window, the least squares
procedure gives a best fit value ηPN = η(fL , ∆f) and 1-σ error estimates∆η, ∆φ0 , ∆t0 . Our principle for choosing (fL , ∆f) is to pick the one forwhich the quality of fit between the NR and PN waveforms is the best, i.e. tominimize the fitting errors.
We first fix ηPN = ηNR , choosing the 1 : 2 waveform, and consider fittingfor (φ0 , t0 ). The result for ∆φ0 is shown in figure 13 as a contour plot in the(fL , ∆f) plane. There are clearly multiple best-fit islands but we already seethat the optimal window choice turns out to be a long frequency width starting5.2 matching post-newtonian and numerical relativity

Log10DΗ

0.025

-3.47

width D Mf

0.020
0.015

-2.6

0.010

0.009

0.01
0.011
MfL fitting window

0.012

Log10DΦ0

0.025

-1.28

width D Mf

0.020
0.015

-0.4

0.010

0.009

0.01
0.011
MfL fitting window

0.012

Log10 D t0

0.025

-0.84

width D Mf

0.020
0.015

0.1

0.010

0.009

0.01
0.011
MfL fitting window

0.012

Figure 14: Dependence of the fitting errors in η, φ0 and t0 on the frequency window(fL , ∆f). Note that there is a clear choice of (fL , ∆f) which optimizes thefit between the NR waveform and the PN waveforms with different η. At
the best fit point, the accuracy in η by this fitting procedure is better than
10−3 .

77

new waveform model for binary black holes® NR data
0.24
mass-ratio 1:2
0.22
0.20
Η

78

0.18
0.025

0.16
D Mf

0.14
0.12

0.006

0.009

0.01
0.011
MfL fitting window

0.012

Figure 15: Best fit value of η as a function of the start frequency fL of the matchingwindow for the waveform which corresponds nominally to a mass ratio
1 : 2, i.e. ηNR = 2/9 = 0.222 . . .; this is shown by a horizontal dashed line.
The vertical dashed line at MfL = 0.009 is the start frequency of the NR
waveform. A rectangle highlights the region of minimal fitting errors fromfigure 14. We see that the best determined values of η are clearly smaller
than ηNR .

at low frequencies, or a relatively short window starting closer to the merger.
Regarding the increasing error PN that most likely introduces towards higher
frequencies, we prefer using an early and long matching window. Though we
do not show it here, the result is similar for the time offset t0 .
It is more interesting instead to generalize this and allow all three parameters(ηPN , φ0 , t0 ) to vary. The main result is displayed in figure 14, which shows
contour plots of the fitting errors ∆η, ∆φ0 and ∆t0 in the (fL , ∆f) plane.
There are now clear and consistent minima for all errors and thus a clear best
choice for fL and ∆f. At this optimal choice, we see that we can fit η, φ0 andt0 to better than 10−3 , 0.06 and 0.15M, respectively. Apart from the error∆η, the actual best fit value η is also of great interest. Figure 15 shows thevalue of η as a function of the start frequency of the matching window fLand ∆f. The x-axis on this plot is the start point of the fitting window fL , andthe color bar indicates ∆f. The most trustworthy values correspond to the
optimal choice of (fL , ∆f) obtained in figure 14; we indicate the union of all
three minimal-error islands as a rectangle in figure 15.
To summarize, from figures 14 and 15 we deduce that, if we were to ignoreηNR (the value that the numerical simulation nominally assumes) and simply
try to find the best fit with the PN waveforms described in section 3.1, thenwe can clearly estimate the best matching region (fL , fL + ∆f) and a best fit
value ηPN = η ± ∆η. This procedure illustrates a trade-off between trying tomatch at early frequencies, where our PN model is more reliable and having asufficiently long fitting window, in which a considerable frequency evolution
leads to an accurate estimate of the fitting parameters.

5.2 matching post-newtonian and numerical relativity

79

Accuracy of the Hybrid Waveform
Later we shall show a phenomenological fit for the hybrid waveform and weshall claim that the fit reproduces the hybrid waveform sufficiently accurately,
but here we first ask whether the hybrid waveform is itself sufficiently accurate
subject to various errors. The basic criteria for evaluating this is the notion ofa distance between two signals whose difference is δh, as given in equation 5.8.
For two signals h and h0 , we shall consider the normalized distance squaredhδh|δhi/ρ2 , where ρ is calculated from our best model. Now the total mass
M becomes important. Previously, when we looked at the least square fits inequation 5.13, the total mass appeared just as a scale factor. However, in the
inner product equation 5.5, the power spectral density Sn (f) sets a frequency
scale, and the value for hδh|δhi becomes mass-dependent. We shall consider
two design noise curves, Initial and Advanced LIGO [3, 20]. We are thenaddressing the question of how different our hybrids would be if we were touse a slightly different result on either the NR or PN side.
On the NR side, we first consider data computed at different resolutions.
The Llama waveforms for the equal-mass case have been computed at low,
medium and high resolutions corresponding to spacing h = 0.96, 0.80 and
0.64 on the wave extraction grid. We combine them with the TaylorF2 modelfrom section 3.1 by using the optimal matching window discussed aroundfigure 13 and ηPN = ηNR . The result is shown in figure 16. Hybrids constructedwith medium- and high-resolution waveforms would be indistinguishable
even with Advanced LIGO at a SNR of 80 over the considered mass range.
Thus, we conclude that the numerical errors related to a finite resolution are
not relevant in the hybrid construction process.
The uncertainties increase when comparing NR data produced by different
codes. Similar to the analysis of different resolutions we calculate the distanceof hybrid waveforms for non-spinning black holes with mass ratio 1 : 1 and1 : 2. Results from data set #1 and #8 (see Table 6) were used, and the distanceplot in figure 17 shows that the 1 : 2 waveform would be distinguishable forAdvanced LIGO at SNR 20 for a total masses between ∼ 30M and ∼ 65M .
Note that these errors are dominated by our matching to PN which possibly
yields different fit parameters for the PN model and therefore amplifies small
differences in the NR data. Towards higher masses, the influence of thismatching decreases as well as the distance of both waveform. However, as
we shall show next, all these errors are still small compared to the intrinsic
uncertainties introduced by PN and they do not matter for Initial LIGO. If wecare only about detection with a minimal match  = 0.03 [see equation 5.9],
we have even less to worry about.
The errors on the PN side turn out to be much more important. figure 18
illustrates the effect of using different PN approximants combined with thesame Llama 1 : 2 simulation. The dashed curve shows the difference in the
hybrid waveforms when we match 3PN or 3.5PN phase following the TaylorF2
frequency domain approximants described in section 3.1 . We see that thedifference between the 3PN and 3.5PN hybrids becomes significant evenfor Initial LIGO at SNR of 8 between a total mass of ∼ 5M and ∼ 35M .
Similarly, the differences between the F2 and Taylor T1 & T4 approximants
are also significant. For detection with  = 0.03 (see equation 5.9), we needto look at the horizontal line with hδh|δhi/ρ2 = 0.06 in figure 18. For all thecurves except the hybrid constructed with the SpEC waveform, there is a small

3PN amplitude, 3.5PN
TaylorF2 phasecombined with highest
resolution NR
waveform

The amplitude is taken
at 3PN order in bothcasesnew waveform model for binary black holes

0.01

iniLIGO  advLIGO
SNR 20Ρ-2 ÈÈD hÈÈ20.001

high-low
SNR 80

10

-4

10-5

high-med.

0

20

40

60

80

100

MM

Figure 16: Distinguishability of different hybrid waveforms constructed from Llama
equal-mass waveforms at different resolutions. The two sets of curves arefor Initial and Advanced LIGO, and we consider the difference between
the high-medium resolution waveforms, and the high-low waveform resolutions. The horizontal lines are the lines of constant SNR (in fact it is
1/SNR2 ). If the distance measure goes above these lines, then the waveforms can be distinguished from each other.

0.010
0.005

iniLIGO advLIGO
SNR 200.002
Ρ-2 ÈÈD hÈÈ2

80

1:2

0.001
1:1

5 ´ 10-4

SNR 802 ´ 10-4
1 ´ 10-4
0

20

40

60

80

100

MM

Figure 17: Distinguishability of different hybrid waveforms constructed from NR
waveforms produced with either BAM or Llama. The solid lines indicate thenormalized distance in the equal-mass case, dashed lines show the case of
mass-ratio 1 : 2. The highest available resolution was always used.

5.2 matching post-newtonian and numerical relativity

0.1

F2 3-3.5 PN

Ρ-2 ÈÈD hÈÈ2

SNR 8

0.01
0.001

F2-T4
F2-T1

early
match

10-4
10-5
20

40
MM

60

80

Figure 18: Initial LIGO’s ability to distinguish hybrid waveforms constructed fromdifferent PN approximants. This plot shows that the hybrids are not sufficient for detection at the  = 0.03 level [equation 5.9] only for a small
range of masses. “Early match” is a reference for matching 3PN or 3.5PN
F2 at early frequencies to the long equal mass SpEC waveform.

range of masses for which the difference between the hybrids would mattereven for detection.
As a reference, we employ the same procedure for the longest equal mass
waveform available, i.e. the data-set #9 based on the SpEC code which covers∼ 16 orbits before merger. For this longer waveform, we again match the
TaylorF2 phase at 3PN and 3.5PN order. Figure 18 shows that the hybrid
remains accurate for a larger range of total masses; this is mainly because
the waveform is considerably longer than the Llama waveforms used above,hence it allows us to use an earlier and longer matching window.
Having carried out this study of errors for non-spinning waveforms, we can
now draw some conclusions for the aligned-spin case. In principle, the procedure outlined here remains valid; we should search over not only {η, t0 , φ0 },but now also over the spins {χ1 , χ2 }. We would not expect the results to bebetter than shown here for non-spinning waveforms because (i) we are adding
two more parameters and (ii) the waveforms #1-4 are expected to have morewave-extraction systematic errors than the Llama results considered here.
Most importantly, as we have just seen, the intrinsic errors in PN are moresignificant whereas the numerical accuracy is not the bottleneck. The intrinsic
parameter biases in PN also show up when different PN models are comparedwith each other. An extensive comparison of different PN models is madein [86]; this paper quantifies the mutual effectualness and faithfulness of thedifferent PN models and shows that errors of ∼ 20% are not uncommon for
Advanced LIGO. The less than 10% error in η shown in figure 15 are thus entirely consistent with the differences between different PN models. To address
this, one needs either improved PN models or a greater variety of longer NR
waveforms such as the long SpEC simulation.
As a simplification, in what follows below we will choose the matchingwindow based on maximizing over the extrinsic parameters (t0 , φ0 ) motivated by figure 13. In that figure, we observe the best fit region extending

81

82

new waveform model for binary black holes

diagonally from M∆f ≈ 0.013 on the y-axis, to the bottom right corner. It
turns out that for this diagonal, the upper frequency of the window does notvary much, 0.020 . MfL + M∆f . 0.024, and we shall use this fact below
for constructing hybrid waveforms for aligned spinning systems.
5.2.4 Construction of Hybrid Waveforms for Aligned-Spin Systems

We make a choicecompatible with that forthe phase

Let us now proceed to the construction of a hybrid waveform model for nonprecessing, spinning systems with comparable mass. Again, the waveforms
described in section 3.1 will be the basis for our model at low frequencies
corresponding to the inspiral stage. On the other hand, the NR simulations
described as data-sets #1–3 in Table 6 contain physical information for frequencies above Mf ≈ 0.008. We will refer to figure 13 to justify our choiceof an overlapping window at Mf ∈ (0.01, 0.02). Once this interval is fixed,
we use the freedom in t0 and φ0 to align the PN and NR phases; both transformations keep the overlap invariant and are therefore irrelevant from thepoint of view of template construction.
Having identified the appropriate overlapping window, we now carry out
the following matching procedure for all NR simulations of data-sets #1–3: PN
and NR phases are aligned in the interval Mf ∈ (0.01, 0.02) by adjusting t0
and φ0 ; the middle point is taken as matching point between PN and NR and
we construct a hybrid phase consisting of TaylorF2 at low frequencies and NR
data at high ones. An analogous procedure is applied to the amplitude, but
in this case there is no freedom for adjusting any parameter. Hence, we usean educated guess for the matching frequency and find the frequency which
minimizes ANR (fmatch ) − APN (fmatch ). Due to the existence of a common
region where PN and NR overlap, such a point can always be found. The hybrid
amplitude consists of PN before and NR after fmatch . Small wiggles in the NR
amplitude, due to the Fourier transform, do not affect the phenomenologicalfit significantly. The most important ingredient for arriving at an effectualmodel is the phase.
Figure 19 illustrates the above-described hybrid construction method formatching PN and NR data in the frequency domain. The procedure introduces
no resizing of neither data and allows for the construction of waveforms containing all the information from the TaylorF2 approximant at low frequencies
and input from the NR simulations for the late inspiral, merger and ringdown.
The matching procedure has been applied to NR data-sets #1–3. The resulting
hybrid PN-NR data cover a part of the parameter space corresponding toequal-valued, (anti-)aligned spins for 0.16 6 η 6 0.25 and constitute the“target” waveforms to be fitted by the analytical phenomenological model
described in section 5.3.
A fundamental check to assess the validity of our matching procedureis the verification that the hybrid waveforms do not contain irregularities
arising from the way PN and NR are stitched together. We show that thisis indeed not the case by performing the inverse Fourier transform of our
hybrids and comparing them with the numerical data they were created fromin the vicinity of the merger. The results of the perfect agreement in the time
domain can be seen in figure 20.

5.2 matching post-newtonian and numerical relativity

100

A22 HMfL

10

1

0.1

0.005

0.010

0.020

0.050

0.100

0.050

0.100

Mf
1500

post-Newtonian
NumRelHybrid

FHMfL

1000

Matching point

500

0
0.005

0.010

0.020

Mf
Figure 19: Illustration of the method for constructing PN-NR hybrid waveforms in
the frequency domain. The data corresponds to an equal-mass binary withaligned spins χ1 = χ2 = −0.25. The left panel shows the amplitude andthe right displays the phase of the dominant ` = 2, m = 2 mode of the GW
complex strain h̃(f). The green dotted lines correspond to the TaylorF2
PN approximant and the red dot-dashed curve is the NR data. The hybrid
waveform is depicted in solid black and the matching points for amplitude
and phase are indicated with a dashed line.

83

84

new waveform model for binary black holes

0.4

HybridNR data

0.2
0.0
-0.2
-0.4
-600

-500

-400

-300 -200
tM

-100

0

100

-600

-500

-400

-100

0

100

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-600

-300 -200
tM

-500

-400

-300 -200
tM

-100

0

100

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3

Figure 20: Comparison of hybrid waveforms with the numerical data used in theirconstruction in the time domain. The three waveforms correspond toequal-mass simulations of spins χ1 = χ2 = 0.85, 0 and −0.75 in theupper, mid and bottom panel respectively. We plot the final cycles of thehybrid waveforms in the time domain, with the corresponding numericalsimulation on top of them. The agreement in the time domain confirmsthat our matching procedure does not introduce irregularities in the final
hybrid waveform. Additionally, the accumulation of orbits in the hang-up
configuration displayed in the upper panel is also clearly visible.

5.3 phenomenological model

5.3

phenomenological model

In this section we present the phenomenological model developed in order
to fit the hybrid PN-NR waveforms of section 5.2 to an analytical formula. A
geometric description of the procedure for constructing phenomenological
waveforms parameterized by just the physical parameters is detailed in [23],and here we just summarize it. Let M be the space of intrinsic physical parameters that we are interested in. In the present case, this is the four-dimensional
space of the component masses and spins λ = {M, η, χ1 , χ2 }. For each point
λ in M, let h(t; λ) be the true physical waveform that we wish to approximate; in particular we consider only the dominant ` = m = 2 mode in thispaper. We start with some known signals in this parameter space at N points
λ1 , λ2 , . . . , λN . We take these known signals to be the hybrid waveforms
whose construction we described earlier. Here the NR waveforms are the BAM
waveforms of data sets #1-3 summarized in Table 6, and the PN model is the
3.5PN frequency domain model for aligned spins described in section 3.1.
Given the finite set of hybrid waveforms constructed from these ingredients,
we wish to propose a phenomenological model hphen (t; λ) that interpolates
between the hybrid waveforms with sufficient accuracy. In constructing thisphenomenological model, it is convenient to work not with the physical parameters λ, but rather with a larger set of phenomenological parameters λ̃,e is the space of phenomenological pawhich we shall shortly describe. If M
e denoted λ̃(λ),
rameters, then we need to find a one-to-one mapping M → M
e corresponding to the physical parameters. As
and thus the subspace of M
the end result of this construction, for every physical parameter λ, we willknow the corresponding phenomenological parameter λ̃(λ) and thus thecorresponding phenomenological waveform hphen (t; λ̃(λ)).
Following the construction procedure of section 5.2.4, we split our waveforms in amplitude and phase, both of which shall be fitted to a phenomenological modelh̃phen (f) = Aphen (f) eiΦphen (f) .

(5.14)For both the amplitude and the phase of the dominant mode of the GW
radiation, we make use of the insight from PN and perturbation theory for thedescription of the inspiral and ringdown of the BBH coalescence respectively,
and introduce a phenomenological model to complete the description of the
waveforms in the merger.
5.3.1 Phase Model
The PN approach for the GW radiation based on the stationary phase approximation, introduced in equation 3.19 of section 3.1, gives an adequate
representation of the phase of the dominant mode during the adiabatic inspiral stage of the BBH coalescence ψ22
SPA (f). As the system transitions towards
the merger phase, it is expected that further terms in the expansion are required to capture the features of the evolution. With this ansatz in mind, we
propose a pre-merger phase ψ22
PM (f) of the form
ψ22
PM (f) =1
α1 f−1/3 + α2 + α3 f + α4 f2/3 ,η(5.15)

85

86

new waveform model for binary black holes

where the αk coefficients are inspired by the SPA phase, redefined and phenomenologically fitted to agree with the hybrid waveforms in the region
between the frequencies fISCO of equation 2.7 and fRD of equation 3.34. As
for the post-merger phase, the Teukolsky equation [252] describes the ringdown of a slightly distorted spinning black hole. The metric perturbation for
the fundamental mode at large distances can be expressed as an exponential
damped sinusoidal
h22
ring (t) =Aring M −πfRD t/Qecos (2πfRD t) ,r

(5.16)

where M is the mass of the ringing black-hole, r the distance from the source,
and Q and fRD correspond, respectively, to the quality factor of the ringing
down and the central frequency of the quasi-normal mode. These can beapproximated by the fit formulas given by equations 3.34 and 3.35. The spin of
the final black hole after the binary has merged can be inferred from the spinsof the two black holes. In our case, we use the fit presented in [223], whichmaps the mass-ratio and spins of the binary to the total spin a of the final
black hole. This analytical treatment of the ringdown motivates a linear ansatz
for the post-merger phase ψ22
RD (f) of the form
ψ22
RD (f) = β1 f + β2 .

(5.17)The transition between the different regimes is smoothened by means oftanh-window functions
1
4(f − f0 )±wf0 =1 ± tanh(5.18)
4σto produce the final phenomenological phase−+ − 2222 +Φphen (f) = 2ψ22
SPA wf1 + wf1 wf2 ψPM + 2ψRD wf2 ,(5.19)

with f1 = 0.93fISCO , f2 = 1.1fRD and σ = 0.015. We choose these particulartransition points after having found them to provide the best match between
the hybrids and the phenomenological model.
5.3.2 Amplitude Model
In a similar manner to the phase, we approach the problem of fitting theamplitude of the GW wave by noting that the PN amplitude obtained fromthe SPA expression could be formally re-expanded as!
5
X
ÃPN (f) = CΩ−7/6 1 +γk Ωk/3 ,(5.20)
k=2

where Ω = πMf. We introduce a higher-order term to model the pre-merger
amplitude ÃPM (f)ÃPM (f) = ÃPN (f) + γ1 f5/6 ,(5.21)

where the γ1 coefficient is introduced to model the amplitude in the premerger regime. The ansatz for the amplitude during the ringdown isÃRD (f) = δ1 L (f, fRD (a, M), δ2 Q(a)) f−7/6 ,(5.22)

5.3 phenomenological model

50.0

AHMfL

10.0

HybridPN Amplitude
Modified PN
Lorentzian
Phenom Fit

5.0

1.0
0.5

0.010

0.015

0.020

0.030

0.050

0.070

0.100

Mf
Figure 21: Fitting procedure for the amplitude. The γ1 term of equation 5.21 is introduced to follow the behavior of the amplitude in the pre-merger regime
whereas the Lorentzian curve correctly describes the post-merger. The twopieces are glued together in a smooth manner using tanh-windows.

where only the width and overall magnitude of the Lorentzian function
L(f, f0 , σ) ≡σ2(f − f0 )2 + σ2 /4(5.23)

are fitted to the hybrid data. The factor f−7/6 is introduced to correct theLorentzian at high frequencies, since the hybrid data shows a faster fall-off.
The phenomenological amplitude is constructed from these two pieces in amanner analogous to the phasehi+Ãphen (f) = 2 ÃPM (f)w−
+
Ã
(f)w
(5.24)
RDf0
f0 ,with f0 = 0.98fRD and σ = 0.015. Figure 21 demonstrates how this phenomenological ansatz fits the hybrid amplitude in a smooth manner throughthe late inspiral, merger and ringdown.
5.3.3 Mapping the Phenomenological Coefficients
Our models for the amplitude and phase involve 9 phenomenological parameters {α1 , α2 , α3 , α4 , β1 , β2 , γ1 , δ1 , δ2 } defined in equations 5.15, 5.17, 5.21
e from the physical toand 5.22. We now need to find the mapping M → M
these phenomenological parameters. Following [25] we construct the quantity
χ≡

1+δ
1−δ
χ1 +
χ2
2
2

with δ ≡m1 − m 2,
M(5.25)

that encodes the BH spins weighted by their relative masses. Thus, our phenomenological waveforms are parameterized only by the symmetric mass
ratio η and the spin parameter χ, as well as by the total mass of the system M
through a trivial rescaling. Figure 22 shows the mapping of αk , βk , γk andδk to surfaces in the (η, χ)–plane.

87

88

new waveform model for binary black holes

Α2

Α1

-1
-2
-3

40
20
0
0.15

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0 Χ

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0.2

1

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0.2

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0.2
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0.25 -1

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1

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300
200
100
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1

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1.2
1.0
0.8
0.6

1
0 Χ

0.15
0.2

Η
0.25 -1

Η
0.25 -1

Figure 22: Map of the phenomenological parameters to the physical parameters of
the binary η and χ.

The 9 phenomenological coefficients introduced in our model, denoted
generically by Λk , are expressed in terms of the physical parameters of thebinary as
X
(ij)
Λk =
ζk ηi χj ,(5.26)
i+j∈{1,2}

which yields 5 coefficients ζ(ij) for each of the 9 parameters, as given in table 7.
We evaluate the goodness of fit between the phenomenological model andthe hybrid waveforms in terms of the overlap, i.e. the ambiguity function
A(λ, λ0 ) defined in equation 5.10. In evaluating the overlap, we maximize overthe extrinsic parameters t0 , φ0 as indicated in equation 5.10, but in this paper
we do not perform the additional maximization over the model parameters
λ0 . Thus, the results in this section can be viewed as a lower bound on theeffectualness. We shall study the effectualness and faithfulness in greater detailin a forthcoming paper.
Figures 23 and 24 illustrate this fact using the design curve of the Advanced
LIGO detector. The first plot shows the overlap between hybrid waveforms
constructed in section 5.2.4 and their corresponding phenomenological fit.
The match approaches unity by construction at low masses and degradeswith increasing total mass. Nevertheless, for none of the hybrid waveforms
employed in the construction of our model does the overlap fall below a valueof 0.98, thus reflecting the fact that the phenomenological waveforms effectually represent the target signals. In figure 24, and as a further test to assess
the robustness of our model for systems with unequal-spin configurations,
we compute the overlap between the phenomenological waveforms and the
NR data-sets #4-7 that were not used in the construction of the model. At
low masses there is no contribution of the short NR waveforms, therefore
the overlaps can not be computed; however, at the masses for which the NR
part falls within the Advanced LIGO band we obtain overlaps > 0.97, which5.4 summary and future work

Table 7: Coefficients to map the 9 free parameters Λk of our phenomenological modelto the physical parameters of the BBH binary.

Λk

ζ(01)

ζ(02)

ζ(11)

ζ(10)

ζ(20)

α1

−1.68

0.77

0.11

16.53

−47.74α2−32.35−8.63−31.87−182.27480.58α3790.19−44.11751.892018.15−2847.51

α4−14.44

−2.01

33.46

163.11−1753.92β1−347.25−37.21−920.76−930.652784.18β26930.34−194.52

18935.20

14960.90−47393.80

γ1

−1.85

−3.13

−64.08−60.89686.61

δ1

0.06

0.03

0.37

0.10

0.83

δ2

1.01

0.44

5.00

5.43

−11.43indicate that our phenomenological model can reliably be extrapolated tophysical configurations with unequal spins, where the spins of the black holes
are encoded in the single parameter χ.
5.4 summary and future work
The aim of the central part of this dissertation has been to construct an analytical model for the inspiral and coalescence of binary black hole systems withaligned spins and comparable masses in circular orbits. Since this requires
merging post-Newtonian and numerical relativity waveforms, one of the main
themes has been to quantify the internal consistency of hybrid waveforms.
This is important because even if one succeeds in finding a useful fit for afamily of hybrid waveforms, one still needs to show that the hybrid one started
with is a sufficiently good approximation to the true physical waveforms. We
investigated the systematics of constructing hybrid waveforms for accurate
non-spinning waveforms based on the Llama code and we saw that numericalerrors are not significant. This suggests that in order to improve the accuracyof hybrid waveforms, it would be useful for numerical relativists to calculate
longer waveforms so that the matching with PN can be done earlier in theinspiral phase.
With the hybrid waveforms for non-precessing systems at hand, we constructed an analytical model for the waveform which has an overlap of better
than 98% for Advanced LIGO with the hybrid waveforms for systems witha total mass ranging from ∼ 10 to ∼ 400M . In the future we will study
in greater detail the effectualness and faithfulness of this waveform model,
thereby quantifying more precisely its performance for detection and parameter estimation. We will also quantify the behavior of these templates in real
non-Gaussian detector noise, and use them in real searches for gravitational
wave signals. Eventually, work is underway in extending the model to include
precessing spins. Our phenomenological model for can be readily applied toexistent GW detection efforts within the LIGO/Virgo Scientific Collaborations. Ongoing searches are already making use of IMR waveforms, such as

89

new waveform model for binary black holes

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Figure 23: Overlaps between the hybrid waveforms constructed according to theprocedure described in section 5.2.4 and the proposed phenomenologicalfit for Advanced LIGO. The labels indicate the values of (η, χ) for someconfigurations. We plot O(λ) = A(λ, λ), i.e. we compute the ambiguity
function (equation 5.10) without maximizing over the parameters of themodel waveform; this is a lower bound on the effectualness.

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Figure 24: Overlaps between the NR data-sets #4–7ab and the predicted phenomenological waveforms from our model for advanced LIGO. The labels indicate
the values of (η, χ) for some configurations. Note that the short durationof the NR data prevents us from computing overlaps at lower masses.

5.4 summary and future work

the EOBNR family and the phenomenological family of [24, 23, 22, 25], inthe form of software injections and as filter approximants. Our newly developed frequency-domain matching procedure should serve to cross-check thevalidity of these alternative approaches and to complement them.
This chapter concludes the part of this dissertation devoted to theoreticalmodelling of gravitational-wave sources. The next chapter focus on applications for real gravitational-wave searches, transfer of numerical relativity
results into current data analysis efforts and implications for gravitationalwave astronomy.

91Part III
APPLICATIONS6

REAL SEARCHES FOR GRAVITATIONAL WAVESIn November 2005 the three first-generation LIGO detectors described in
section 1.4 reached their design sensitivity and began a two-year period of observations which concluded in October 2007 [13]. Although the astrophysicalestimates for rates of coalescing binaries presented in section 2.4.3 depend
on a number of assumptions and unknown model parameters, and are still
uncertain at present, searches for these signals in the real output of the interferometers constitute a promising avenue towards detection or, at worst,
upper limit estimation.
The inspiral group of the LIGO Scientific Collaboration is engaged in thechallenge of analyzing the LIGO data and searching for coalescing binaries.
Results from searches for gravitational waves associated to neutron-star, blackhole and mixed binaries using data from previous science runs with everincreasing sensitivity are reported in [5, 4, 6, 7, 10]. The strategy followed toanalyze the S5 data has consisted in splitting up the parameter space of thebinaries to be searched for. A search for signals from binaries with component
masses greater than or equal to 1 solar mass (M ) and total mass ranging
from 2 to 35 M is carried out using inspiral templates from post-Newtonian
theory; a partially-overlapping search for systems with component masses
between 1−99 M and total masses between 25−100 M is the first effort toincorporate a filter family of waveforms modelling the three stages —inspiral,merger and ringdown— of the full binary black-hole coalescence.
This chapter describes the data analysis strategy carried out in these two detection efforts and presents the outcome of the low-mass search, which yielded
no plausible detection of gravitational-wave signals but placed a stringent
limit on the merger rate of binaries in its correspondent mass range.
6.1 the data analysis strategy for coalescing binaries
This section explains the infrastructure developed within the inspiral groupof the LIGO Scientific Collaboration to search for signals from binary coalescences in the output of the LIGO detector. The detection algorithm has
been applied to searches from the third LIGO science run [10] onward. The
code, often referred to as the inspiral pipeline performs a series of hierarchicaloperations in order to search for real signals buried in the detector noise. The
most relevant steps are briefly described in the next sections.
6.1.1 The Optimal Filter
The optimal detection method for modulated sinusoidal signals of known
form buried into stationary Gaussian noise is known to be the matched filter
algorithm [148]. In the case of coalescing black-hole or neutron-star binaries,
the signal might not be precisely known, but it is parameterized by a numberof physical characteristics of the binary, such as the masses, initial phase, timeof arrival, distance, sky location, orientation and, eventually, spins. The exact
parameters are not know a priori, but the output of the detector can be filtered95

This period is known as
the fifth LIGO sciencerun, or S5

These two searches arecommonly referred toas low-mass andhigh-mass search

96real searches for gravitational waves

with a family of templates that discretely cover the parameter space to be
searched for. In the most realistic scenario, the actual noise of the detector
is not perfectly described by a stationary Gaussian process and additional
methods need to be employed in order to make the matched filter process
more robust again non-Gaussian features.
Let us for the moment assume that the output signal of the detector is fairly
well represented by stationary Gaussian noise plus a gravitational-wave signals(t) = n(t) + h(t)(6.1)

The one-sided power spectral density Sn (|f|) of the Gaussian noise processn(t) is given by equation 5.4. The typical sensitivity curves of the LIGO and
Virgo detectors over the course of the S5/VSR1 data-taking period can beseen in figure 4. The matched filter output of a data stream s(t) with a filter
template of the form h(t) is the complex quantityZ∞s̃(f)h̃∗ (f) 2πiftz(t) = 4
e
df.
(6.2)
Sn (f)
0
The inner product in the space of template waveforms is given by equation 5.5.
To construct a well-defined signal-to-noise ratio (SNR), a normalization for
the template has to be calculatedZ∞h̃(f)h̃∗ (f)2
σ = hh|hi = 4
df.
(6.3)
Sn (f)
0This lower frequency is
associated to theseismic noise of theinterferometer

In all the above integral expressions, the limits of the integration are usuallyreplaced by flow and ffinal , with flow given by the corresponding lower cut-off
frequency of the detector —40 Hz for initial LIGO, 30 Hz for initial Virgo and
10 Hz for the advanced detectors— and ffinal determined by the ending frequency of the template, such as the fISCO or any other characteristic frequency
related to the post-Newtonian dynamics, or possibly the Nyquist frequency
in the more general case of a full inspiral-merger-ringdown template. We will
discuss the choice of this frequency when describing the differences between
the two searches presented in sections 6.2 and 6.3.
Thus the real quantity
ρ(t) =

It is optimal in the
Neyman-Pearson sense;
it yields maximum
detection probability fora given false-alarm rate|z(t)|σ(6.4)

is the signal-to-noise ratio (SNR) of the matched filter, which acts as the
optimal detection statistic in the presence of stationary, Gaussian noise. To
determine whether a signal is present in a given segment of detector data, athreshold in SNR ρ∗ is chosen such that> ρ∗
assume signal is present
if ρ
(6.5)
∗
<ρ
assume signal is absent.
This classification allows for the events for which the SNR crosses thethreshold to be listed as triggers and ranked according to their significance.
The probability that the SNR exceed some given value falls exponentially withincreasing threshold∗2 /2p(ρ > ρ∗ ) = e−ρ

(6.6)

6.1 the data analysis strategy for coalescing binaries

and it follows that high SNR values have low probability of having been
produced by noise and thus are a reasonably good indicator that a real signal
is present.
Nevertheless, not every trigger is necessarily connected to a true gravitationalwave event. The probability that ρ > ρ∗ and yet no signal is present is called
false-alarm probability and that of ρ < ρ∗ when a signal is present is the falsedismissal probability. One of the goals pursued when tuning gravitationalwave searches is to choose the value of ρ∗ carefully so that these probabilities
are minimal. The threshold is set by the maximal false-alarm rate tolerated ina given search, which in turn is decided by the expected event rate. On a practical level, the presence of non-Gaussian and non-stationary features in thedetector noise implies that further methods —such as signal-based-vetoes—
are needed to reduce the rate of false alarms and false dismissal.
6.1.2

Template Construction

Ideally, the gravitational signal to be measured and the template waveform
used for filtering should have the same functional form. In practice, however,theoretical uncertainties, the use of approximations to solve the Einstein equations, and the fact that the parameter space of the signal is continuous but the
search is done with a finite set of templates prevent the signal-to-noise ratio
from reaching its maximum. The real signals lie outside the submanifold of
the search templates that lives in the full manifold of all possible detector outputs [197]. An important question is, thus, how to lie down a set of templates
so that the loss of SNR from the mismatch between signal and template does
not result in excessive missed detection candidates.
The metricgµν = hhµ |hν i(6.7)

allows to compute the mismatch in the space of template waveforms. Templates on this space can now be placed, ensuring that the furthest distancefrom any point to a template is less than a tolerance value . The loss of SNR
due to the discretization of the parameter space is thus bound by .
Due to the fact that the metric 6.7 takes a more convenient form in a particular system of coordinates than in others, the space is usually parameterized
in terms of auxiliary variables related to M and η, the chirp mass 3.2 and symmetric mass ratio 2.6 of the binary. An hexagonal placement algorithm [93] isemployed when laying out the templates, for it provides the highest coverage
efficiency. The searches described in sections 6.2 and 6.3 set a value of 0.97
for the fitting factor between elements of the template bank, which ensuresthat no more than 10% of the signals are loss due to mismatch.
6.1.3 Signal-Based Vetoes
In the more realistic case of non-stationary, non-Gaussian detector noise,
spurious signals might cause events with unusually high SNR. The presenceof environmental or instrumental sources of disturbances in the detector cangive rise to loud events that the analysis pipelines might easily confuse withreal signals and erroneously classify as triggers. The noise of the detector often
presents “glitchy” features that might arise from a number of sources, such as

97

98

The commonly usedvetoes are theχ2 –discriminator
test [30] and the r2 vetoreal searches for gravitational waves

environmental disturbances or technicalities associated to the instrumentation. A matched filter algorithm will respond to these glitches by giving a large
SNR response, although the features of the glitch might bear no resemblance
to a true gravitational-wave signal.
In order to overcome this problem, which could have undesired effects
in the reliability of the search algorithm, several signal-based vetoes can be
implemented, that aim at differentiating between glitches and real signals.
The χ2 –test responds to the intention of constructing a detection statistic
capable of indicating if the filter template and the signal match sufficiently well,and discard spurious triggers if they do not. An orthogonal decomposition of
the template h̃(f) can be made so that the complete set of p templates h̃i (f)satisfieshh̃i (f)|h̃j (f)i =

δij
p

p
X
hh̃i (f)|h̃i (f)i = 1

(6.8)
(6.9)

i=1

If the signal perfectly matched the template h̃(f), then a SNR of ρ/p would
be expected for each projection h̃i (f). Under that expectation, the following
quantity can be constructed
χ2 =


p 
X
ρ 2
,
ρi −
p

(6.10)

i=1

where ρi is the SNR associated to the ith bin of the orthonormal set, ρ is thetotal SNR and p is the number of χ2 bins. Figure 25 illustrates how the test
works in the case of a simulated inspiral signal and a spurious glitch. Althoughboth events present a similar SNR value, the decomposition in bins —p = 4
in this case— allows for discriminating between the two. While the chirppresents a value of χ2 = 1.296, the spurious signal has χ2 = 68.4 [30], hence
it is clear that thresholding in χ2 separates both events.
In Gaussian noise this test is χ2 –distributed with an expectation value2hχ i = p − 1, which does not coincide with the actual expectation value
when signals are present, due to the use of a discrete template bank to searchfor gravitational waves. If δ is the mismatch between the template and thesignal, then the expectation value of χ2 in the presence of signal without noiseat the time of the maximum SNR is hχ2 isignal = δρ2 . This quadratical scaleof the expectation for χ2 with the SNR introduces the possibility that a loud
real signal also presents a large χ2 . The final threshold is hence done on anormalized quantity defined as
ξ2 =

χ2
.
p + δρ2(6.11)

The condition ξ2 6 ξ∗2 is required for a trigger to pass the χ2 –test. The
parameters δ, p and ξ∗2 can be tuned. In practice, δ accounts not only forthe discrete template bank but also for possible inaccuracies of the waveformmodel, and is chosen so that no simulated signals are rejected.
The r2 –test is another reliable signal-processing technique to discriminate
between real and spurious triggers. The veto evaluates the time-dependent
quantity r2 = χ2 /p during a certain amount of seconds before the inferred6.1 the data analysis strategy for coalescing binaries
Simulated Chirp (SNR = 9.2)
5

Spurious Event (SNR = 8.7)
5

z3(t)

z4(t)

High freq filter

0

0

−5

−5

5

5

0

0

−5

−5

TIME

z1(t)

z2(t)

TIME
5

5

0

0

−5

−5

5

5

0

0

−5

−5

t = t0t = t0Figure 25: Diagram demonstrating how the χ2 –test discriminates between true andspurious signals. For the simulated chirp, all filters in the different frequency bands peak at the same time offset t0 which maximizes the SNR.
At this instant in time, all of the contributions zj —corresponding to theρi defined in the text— are approximately the same value. However when
the filter was triggered by the transient burst, the filters in the different
frequency bands peak at different times. Figure credit: B. Allen [30].

merger time of the trigger. A true signal spends less time above a given threshold than a spurious glitch, so that the condition r2 6 r∗2 acts as a powerful
discriminator among the two.
6.1.4 Coincidence Test and Background Estimation
Even when signal-based vetoes are applied to the candidates that exceed agiven SNR threshold, the fundamentally noisy nature of the detectors’ output
gives rise to a substantial number of accidental triggers. In order to increase
confidence in these detection candidates, coincidence at two or more detectors
is required. Actually, for a gravitational-wave detection to be reliably claimed
and verified, it will have to be found in various detectors and also by othermeans —associated neutrino or electromagnetic observations—. For events
to be considered coincident, the time of coalescence and masses of the triggers
recorded in different detectors are required to agree within a certain tolerance.
The tuning of the diverse parameters that control those thresholds is crucial toensure a reasonably large number of events that allows for reliable statistics but
without this number being so large that the software infrastructure overflows.
The coincident algorithm implemented in the low- and high-mass searches
described below creates a time-sorted list of triggers from the multiple detectors whose data are being analyzed. For each single trigger found at a timeti , all triggers occurred within a time t ∈ (ti , ti + T ), where T is some time

99

100

A triple trigger is also adouble trigger for every
pair of detectors thatcan be formed from thetriplet
These are called
time-shifted triggers, incontrast to the potential
candidates, zero-lag
triggers

real searches for gravitational waves

window, are found. This step is repeated over all single triggers until a listof coincident triggers is found. Finally, the algorithm loops over the coincident triggers and removes coincident triggers that are subsets of higher ordercoincident triggers.
Again due to the unpredictability of the detectors, purely accidental coincidences might still occur, giving rise to an undetermined number of background triggers that need to be separated from the potential real gravitationalwave signals. The estimation of this background is carried out by a time-slides
procedure. Most certainly, if the outputs of multiple detectors were to be
shifted by a number of seconds with respect to each other and a search was
performed in the time-shifted data, none of the recorded coincident triggers
could ever be claimed to be a real gravitational wave. Thus, time slides provide
an accurate estimation of the background due to accidental coincidences,provided that the outputs of the detectors are uncorrelated. Unfortunately,
this is not the case for the pair of interferometers H1H2, since they are situatedat the same physical location and therefore share common sources of noise.
6.1.5 Detection Statistic: False-Alarm Rate
In purely Gaussian, stationary noise, the SNR as defined in section 6.1.1 provides a way of measuring the false-alarm rate that is independent of the filter.
In real data, this is not true, and a better statistic needs to be devised. This is thereason why previous searches for gravitational waves from inspiralling binaries [5, 4, 6, 7, 10] have made use of an improved detection statistic constructedwith input not only from SNR, but also from χ2 . This statistic, known as effective SNR, improves separation of signals from background and is definedasρ2ρ2eff = r
  2 ,ρ
χ
1 + ρ0χdof

p is the number of binsused in the χ2 veto(6.12)

where ρ0 is a free parameter and χdof = 2p − 2 is the number of degrees of
freedom of the χ2 veto in Gaussian noise. Values of ρ0 = 250 and ρ0 = 50 are
commonly used in the low- and high-mass searches respectively. The effective
SNRs for the single-detector triggers that form a N-detector coincident trigger
are then added to yield the combined effective SNR as final detection statistic
ρ2c =

N
X

ρ2eff,i .

(6.13)

i=1

Past searches explored asmaller part of the mass
parameter space. For
an extended parameter
space FAR provides
superior performance

Yet superior efficiency is obtained if a different statistic is used, namely
the false-alarm-rate (FAR) or its inverse (IFAR) statistic. If an estimation
for the background triggers exists —which is our case, since we employ thetime-slides method of section 6.1.4—, then the time-shifted triggers provide
a measure of the false-alarm rate for every zero-lag coincident trigger. The
statistic is calculated by counting the number of time-shifted, i.e. non-real,
triggers with a combined SNR 6.13 greater than or equal to that of the zero-lag,i.e. potentially real, coincident triggers. Since the time-shifted, backgroundtriggers were generated by multiple time slides, we have to divide among
the number of time slides that were performed. For this number to be ameaningful rate that can be easily compared with results from other searches,6.2 the low-mass search in s5 ligo data101

we normalize by a time of observation of one year. This provides a number—the false-alarm rate, or FAR— as a function of effective SNR for each zero-lag
trigger, that ranks its significance with respect to the background.
hibgzero−lag
N
ρ
>
ρ
eff
eff
1 yr
FAR(ρeff ) =.
(6.14)
Tanalyzed
Nslides
If using the inverse false-alarm statistic, or IFAR, the inverse of FAR is calculated. The meaning of this statistic is as follows: for some given effective SNR,
we expect a certain number of zero-lag triggers. For instance, at a normalized
IFAR of 0.1 —i.e. a normalized FAR of 10—, we expect 10 zero-lag triggers
per year with an effective SNR greater than or equal to the effective SNR that
corresponds to that IFAR. For an IFAR of 1, we expect 1 foreground trigger,
and so on. Plotting the number of triggers versus their IFAR should yield afunction of the form 1/x, allowing us to conclude whether the actual zero-lag
triggers are consistent with their expected behaviour or if, on the contrary, an
unusual result —and, hence, a candidate— is found.
The subtlety here, and the reason why FAR supersedes combined SNR, isthat different types of triggers present different backgrounds. Firstly, a triple
coincidence trigger is much more unlikely to occur than a double, hence thebackground with which it can be compared is very different in these twosituations. In particular, for H1H2 triggers we are not even in the positionof providing a reliable background estimate, due to already-mentioned correlated sources of noise. All these differences are taken into account when
computing FAR, since each trigger is compared with the background triggers
in its category only.
Perhaps more importantly, FAR also allows to differentiate triggers according to their mass. Due to the extended mass range surveyed by the searches
presented in this chapter, a large number of templates are employed for filtering. Higher-mass templates correspond to binaries that merge at lower
frequencies within the LIGO/Virgo bands, hence spanning a shorter duration than templates corresponding to lower-mass binaries. As a consequence,
they are more likely to be confused with noise glitches and present a higher
sensitivity to non-stationary noise transients in the detector. If we ranked
the associated candidates according to their combined effective SNR 6.13,
they would dominate the statistic, possibly shadowing potentially interesting
lower-mass triggers. In short, the FAR is not only dependent on the type of
trigger —double, triple— under consideration, but also on the chirp mass. All
this results in a biased false-alarm rate that we can correct by splitting computation of the FAR in several mass-dependent populations. A subsequent
combination method among all different kinds of triggers allows to quote a
final, single number for the FAR or IFAR of any potential candidate. This is
the final ranking statistic employed in the searches described in the following
sections.
6.2 the low-mass search in s5 ligo data
In this section we present a search for gravitational-waves from binaries withtotal mass between 2 and 35 M and a minimum component mass of 1 M
in LIGO observations between November 14, 2006 and May 18, 2007 [16]. The
mass parameter space surveyed by this search can be visualized in figure 26 in

In our searches, we
separate themaccording to their chirp
mass M102

real searches for gravitational waves
100

low mass

m2(M )

high mass

35
25

1

1

25

35

m1(M )

100

Figure 26: Parameter space of masses surveyed by the low- and high-mass searches
for coalescing compact binaries in S5 LIGO data.

red. Prior results from a search for systems with the same mass distribution
in data taken from November 4, 2005 to November 14, 2006 were reported
in [15]. No gravitational-wave signals were observed during this search and so
upper limits on rates for coalescences of compact binaries are reported, using
the results of [15] as prior rate distributions. The final results quoted here arethus derived from LIGO observations in the period November 4, 2005 to May
18, 2007.
6.2.1 Description of the Search

These segments arereferred to as “months”

The author performed
the analysis of “month4”, from March 6 toApril 3, 2007

The data-analysis pipeline used in this search consists fundamentally of themain stages described in section 6.1, thus this section only describes the most
specific characteristics of this concrete search, referring to [10, 15] for extended
details. The most substantial change in this analysis is a modification to theway in which the significance of candidate events is compared to instrumental
noise background. In previous searches, the noise background was computed
using the entire observation period. Using this method, the non-stationarityof the noise could lead to candidates being compared to a background that
does not correctly represent the state of the detector at a given time. Thisis especially noticeable for candidates found at the end of the observation
period, when the sensitivity of the detector is likely to have improved. In the
search presented here, the observation period is instead split into six fourweek segments and one 18 day segment and the instrumental backgroundis measured independently in each month, as the detector behavior varied
over the course of the S5 run. Candidate triggers are therefore compared to a
background that better reflects the instrumental behaviour at the particulartime of the trigger. Each month was searched independently for gravitationalwave candidates and in the absence of detections, the results from the months
are combined —together with the results from [15]— to set an upper limit on
the CBC rate.

6.2 the low-mass search in s5 ligo data

The search for gravitational waves is done at times when at least two of the
LIGO detectors were operational, which comprises a total of 0.28 yr when
all three detectors were operational —H1H2L1 coincident data—, 0.10 yr of
H1H2 coincident data, 0.02 yr of H1L1 coincident data, and 0.01 yr of H2L1
coincident data. Due to above-mentioned noise correlations between the
co-located H1 and H2 detectors, the estimation of instrumental backgroundusing time-shifted data fails. Therefore no search is done at times when only
the H1H2 detectors are operating. Approximately 10% of data is designatedplayground.
Inspiralling low-mass binaries targeted in this search radiate at frequencies
that sweep across the sensitive band of the LIGO detectors and their merge
happens at the end of the LIGO band. An appropriate choice for the matchfiltered search is the use of PN templates terminated at fISCO . This method issuboptimal if a true signal differs from our template family due to unforeseen
physical effects. Matter effects in BNS and BHNS are not included in our templates, but are expected to be important only at higher frequencies [240, 167].
We construct template banks [93] of restricted second order PN waveforms in
the frequency domain [253, 229, 108] such that no more than 3% of the SNR
is lost due to the discreteness of the bank [198]. A “trigger” is generated if thematched-filter SNR of the strain data filtered against the template exceeds
a threshold of 5.5 [31]. The triggers are subject to a coincident test such as
the one described in 6.1.4, rejecting those that do not appear in at least twoof the three LIGO detectors [224]. There are in principle four possible typesof coincidence for three simultaneous detectors: H1H2L1 triple coincident
triggers and three different double coincident types: H1H2, H1L1 and H2L1.
We discard H1H2 double coincident triggers, due to the problems estimating
the background for these triggers and discard H2L1 triggers when the H1
detector is operating nominally, since the 4 km H1 detector is more sensitive
than the 2 km H2 detector.
Coincident triggers are subjected to consistency checks using the signalbased vetoes described in 6.1.3 [8, 30, 225]. All triggers occurred at timesof poor detector data quality are flagged using environmental and auxiliary
data and vetoed [15]. Depending on the severity of the instrumental artifact,
we apply two categories of data-quality vetoes, one being more severe thanthe other The triggers that survive these vetoes an effective SNR statistic,
computed from the trigger’s matched-filter SNR and the value of the χ2 signalbased veto for that trigger as indicated by equation 6.12. After discarding
playground data and times in both veto categories, a total of 0.21 yr of triple
coincident H1H2L1 data, 0.02 yr of H1L1 coincident data, and 0.01 yr of H2L1
coincident data remain. In the absence of a detection, these data are used tocompute upper limits on the rate of coalescences of neutron-star, black-holeand mixed binaries.
As explained in section 6.1.4, the rate of instrumental noise artifacts is measured by time-shifting data from the Livingston and Hanford observatories.
The data are offset by more than the light-travel time between observatories,
thus triggers which survive the pipeline are due to noise alone. We performed
100 such time-shifts to obtain a good estimate of the noise background in
our search. It is important to recall here what we said in section 6.1.5: binaries of higher masses merger at a lower frequency and thus contain fewer
gravitational-wave cycles in the sensitive band of our detectors; this meansthat our signal-based vetoes are not as powerful as for long, lower-mass sys-103

The 4 and 2 km
Hanford and the 4 km
Livingston detector are
denoted H1, H2 and L1
respectively

The playground dataconsists of 600 of every
6370 s of data that areused to check and tune
the pipeline

These templates do notcontain the amplitude
corrections discussed inchapter 3

H1 and H2 data are
kept fixed with respect
to each other104

real searches for gravitational waves

Figure 27: Combined and un-combined inverse false-alarm rates for triple H1H2L1and double H1L1 triggers in month 4 of the 2nd year S5 low-mass search.
The upper panel shows the IFAR of the triggers separated in three chirp
mass categories. The lower panel shows the combined results. No candidates stand significantly above the expected background, depicted by adashed line. The shadowed areas denote regions at one and two standard
deviations of the expectation value.

These regions aredefined by the chirp
mass: Mlow 6 3.48 6
Mmid 6 7.40 6 Mhightems. High-mass templates are therefore more sensitive to non-stationary
noise transients and hence our false-alarm rate for them is larger. In orderto account for this mass-dependent behavior we compute the backgroundfor three different mass regions and compare foreground and backgroundwithin each of these ranges. Specifically, in each region we count the numberof background triggers with effective SNR greater than or equal to a given
foreground trigger; dividing this number by the amount of background time6.2 the low-mass search in s5 ligo data

105

analyzed gives us the false-alarm rate for that trigger. This allows us to define
a single detection statistic for every trigger in each of the mass categories.
The false-alarm rate can then be directly compared to obtain a ranking of thesignificance of the triggers, regardless of their mass [15].
The results of the IFAR calculation for month 4 of this search are shownin figure 27. In the upper panel we see the triple H1H2L1 and double H1L1
triggers with their un-combined IFAR values plotted on top of the expectedbackground. The three different mass regions are delimited by the chirp massof a 8 − 8 M and 17 − 17 M binary. In the lower plot, all results are
combined to produce final IFAR values. No candidate stands significantly
above the background for this month. We proceed to the discussion of the
final results of the full seven-month search, namely the calculation of upper
limits on the rate of binary coalescences.
6.2.2 Search Results
The seven months of data were analyzed separately using the procedure described above. No gravitational-wave candidates were observed with a FAR
significantly above those expected from the noise background. The loudest
trigger in this search was a triple coincident trigger with a FAR of 6 per year.
This is consistent with the expected background, since we searched 0.21 yrof data. The second and third loudest triggers had FAR values of 10 and 11
per year respectively. Although we did not have any detection candidates, we
exercised our follow-up procedures by examining any triggers with a FAR ofless than 50 per year. This exercise prepares us for future detections and often
identifies areas where our search pipeline can be improved to exclude noisetransients.
In the absence of detection candidates, we use our observations to set an
upper limit on the CBC rate. We follow the procedure described in [72, 71, 62]
and use the results reported in [15] as prior information on the rates. We
present five different classes of upper limits. The first three limits are placedon binaries of neutron stars and/or black holes assuming canonical mass
distributions systems. We also present upper limits as a function of the total
mass of the binary and, for BHNS binaries, as a function of the black holemass. We combine the results from each of the seven months, along with theprior results from the first year analysis, in a Bayesian manner, using the sameprocedure as described in [15].
We first calculate upper limits on BNS, BBH and BHNS systems assuming
the objects have no spin, and summarize the results Tables 8 and 9. The rate ofbinary coalescences in a galaxy is expected to be proportional to the blue lightluminosity of the galaxy [169]. Therefore, we place limits on the rate per L10
per year, where L10 is 1010 times the blue solar luminosity. To calculate thesearch sensitivity, the analysis was repeated numerous times adding simulated
signals with a range of masses, distance and other astrophysical parameters
to the data. Table 9 shows the sensitivity of the LIGO detectors to coalescing
binaries quoted in terms of the horizon distance i.e., the distance at which an
optimally oriented and located binary would produce an SNR of 8. Similar
information is graphically shown in figure 28 for month 4 of this search. The
plot shows the comparable reach of the H1 and L1 detectors in Megaparsecs,
which is about twice as large as that of H2. The horizon distance is computed
assuming signals terminating before the merger, and it reaches its maximum

BNS:
m1 = m 2 =(1.35 ± 0.04) M
BBH:
m1,2 =(5 ± 1) M M
BHNS:
m1 = (5 ± 1) M ,m2 =(1.35 ± 0.04) M

The Milky Way contains∼ 1.7 L10 [163]106

real searches for gravitational waves

Table 8: Detailed results from the BNS search. The observation time is the time usedin the upper limit analysis. The cumulative luminosity is the luminosity towhich the search is sensitive above the loudest event for each coincidence
time. The errors in this table are listed as one-sigma logarithmic error bars
(expressed as percentages) in luminosity associated with each source error.
BNS Search
Coincidence time
Observation time (yr)

H1H2L1

H1L1

H2L1

0.21

0.02

0.01

Cumulative luminosity (L10 )

490

410

110

Calibration error

23%

23%

26%

Monte Carlo error

3%

7%

10%

Waveform error

31%

32%

31%

Galaxy distance error

16%

16%

3%

Galaxy magnitude error

19%

19%

17%

Table 9: Overview of results from BNS, BBH and BHNS searches. Dhorizon is the horizon distance averaged over the time of the search. The cumulative luminosity
is the luminosity to which the search is sensitive above the loudest eventfor times when all three LIGO detectors were operational. The first set ofupper limits are those obtained for binaries with non-spinning components.
The second set of upper limits are produced using black holes with a spin
uniformly distributed between zero and the maximal value of Gm2 /c.
BNS, BBH and BHNS Searches
Component masses (M )Dhorizon (Mpc)
Cumulative luminosity (L10 )Non-spinning upper limit yr−1 L−1 10Spinning upper limit yr−1 L−110

1.35/1.35

5.0/5.0

5.0/1.35

∼ 30

∼ 100

∼ 60

490

11000

2100

1.4 × 10−2

7.3 × 10−4

3.6 × 10−3...

9.0 × 10−4

4.4 × 10−3for binaries with total mass ∼ 30 M . If the merger and ringdown were taken
into account, the peak would shift towards larger masses.
There are a number of uncertainties which affect the upper limit calculation,
including Monte Carlo statistics, detector calibration, distances and luminosities of galaxies listed in the galaxy catalog [169] and differences between the
PN templates used to evaluate efficiency of the search and the actual waveforms. The effect of these errors on the cumulative luminosity are summarized
for the BNS search in Table 8. We marginalize over all of the uncertainties [72]
to obtain a posterior distribution on the rate of binary coalescences.
In figure 29, we show the derived distribution of the rate of BNS coalescences. The distribution is peaked at zero rate because there are no detection
candidates. We include the distribution for all searches previous to this one(which is our prior). In addition, we present the result that would be obtained
from each month, were it analyzed independently of the others and of theprevious searches. This provides an illustration of the amount that each monthcontributes to the final upper limit result and demonstrates the improvement6.2 the low-mass search in s5 ligo data

Figure 28: Horizon distance in Mpc as a function of the binary’s total mass for thethree interferometers in operation during month 4 of the 2nd year S5
low-mass search. The horizon distance is defined as the distance at which a
detector measures ρ = 8 for an optimally- oriented and optimally-locatedbinary, i.e. an overhead, face- on orbit.

Non-normalized Probability

1.0Month 0Month 5S5 first year
Combined posterior

0.8

0.6

0.4

0.2

0.0
0.00

0.01

0.02

Rate / yr /L10

0.03

0.04

Figure 29: The posterior distribution for the rate of BNS coalescences. The dashedblack curve shows the rate computed in [15]. The solid black curve shows
the result of this search using the previous analysis as a prior. The figure also
shows the rate distributions for two of the individual months computed
using a uniform prior. The improvement from month 0 to month 5 is due
to increasing detector sensitivity during this search.

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Figure 30: The marginalized 90% rate upper limits as a function of mass. The upper
plot shows limits for BBH systems as a function of the total mass of thesystem. The lower plot shows limits for BHNS systems as a function of
the black hole mass, assuming a fixed neutron star mass of 1.35M . Herethe upper limits are calculated using only H1H2L1 data since the relatively
small amount of H1L1 and H2L1 data makes it difficult to evaluate thecumulative luminosity in the individual mass bins.

in sensitivity of the detectors during the search. The upper limit is finallyobtained by integrating the distribution from zero to R90% so that 90% of theprobability is contained in the interval. The results obtained in this way are−4 yr−1 L−1 and
R90%,BNS = 1.4 × 10−2 yr−1 L−1
10 , R90%,BBH = 7.3 × 10
10
−1
−3
−1
R90%,BHNS = 3.6 × 10 yr L10 .
Additionally we calculate the upper limit for BBH systems as a function of
the total mass of the binary, assuming a uniform distribution of the component
masses. For BHNS systems, we construct an upper limit as a function of theblack hole mass, assuming a fixed neutron star mass of mNS = 1.35 M .
These upper limits are shown in Fig 30.
Finally, we present upper limits on coalescence rates where the spin of thecomponents of the binary is taken into account. Astrophysical observationsof neutron stars indicate that their spins will not be large enough to have asignificant effect on the BNS waveform observed in the LIGO band [179, 40].
Theoretical considerations limit the magnitude of the spin S of a black hole tolie within the range 0 6 S 6 Gm2 /c. However, the astrophysical distributionof black hole spins, and spin orientations, is not well constrained. Therefore,
we provide a sample upper limit for spinning systems using a spin magnitude
and orientation distributed uniformly within the allowed values. This gives6.3 the high-mass search in s5 ligo data

upper limits on the rate of BBH and BHNS systems of R90%,BBH = 9.0 ×−3 , yr−1 L−1 . These rates are
10−4 yr−1 L−1
10 and R90%,BHNS = 4.4 × 10
10
about 20% larger than the non-spinning rates.
6.2.3

Discussion of the Low-Mass Search

By combining the results of this search with our previous results, we set a
new upper limit on the rate of coalescences in the local universe which isapproximately a factor of 3 lower than that reported in [15]. This improvement
is significant, even though we searched only two thirds as much data asin [15]. It is due, in part, to improvements in detector sensitivity during S5
which increased the horizon distance. Moreover, the shorter analysis timeand improved stationarity of the data, led to many of the months having aless significant loudest event than in the previous search. Both of these effects
increased the luminosity to which the search was sensitive, thereby improving
the upper limit.
Astrophysical estimates for CBC rates depend on a number of assumptions
and unknown model parameters, and are still uncertain at present. In thesimplest models, the coalescence rates should be proportional to the stellar
birth rate in nearby spiral galaxies, which can be estimated from their blue
luminosity [169]. The optimistic, upper end of the plausible rate range for BNS−5 yr−1 L−1 for BBH and BHNS
is 5 × 10−4 yr−1 L−1
10 [162, 161] and 6 × 10
10
[196, 195]. The upper limits reported here are ∼ 1–2 orders of magnitude
above the optimistic expected rates. With the next run that started in summer
2009, the improved Enhanced LIGO and Virgo are expected to bring us close
to the optimistic rates. The most confident BNS rate predictions are basedon extrapolations from observed binary pulsars in our Galaxy; these yieldrealistic BNS rates of 5×10−5 yr−1 L−1
10 [162, 161]. Rate estimates for BBH and
BHNS are less well constrained, but realistic estimates are 2 × 10−6 yr−1 L−1
10
for BHNS [196] and 4 × 10−7 yr−1 L−1
10 for BBH [195]. Thus, the expected
rates are ∼ 2–3 orders of magnitude lower than the limits presented here.
The Advanced LIGO and Virgo detectors, currently under construction, will
increase our horizon distance by an order of magnitude or more, allowing us
to explore the validity of the astrophysical predictions.
6.3 the high-mass search in s5 ligo data
The upper limits obtained in section 6.2 constitute the most up-to-date results
for the rate of coalescences of low-mass binaries obtained via gravitationalwave measurements. The use of post-Newtonian templates is perfectly justified
in such a search, since binaries with masses within the surveyed range are welldescribed by inspiral-only models; hence, matched-filter searches employing
inspiral-only approximants are a reasonable choice. Searches for more massive
systems require instead filters incorporating the last stages of the coalescence.
Breakthroughs in numerical relativity have enabled the development of fullwaveform models that can be applied to such searches.
This section describes a search for signals from binary black hole coalescences in LIGO S5 data when both LIGO sites were operating and collecting
coincident data. The search covers systems with total mass 25 M 6 M 6100 M and component masses 1 M 6 m1 , m2 6 99 M with negligible
black hole spins. For black holes with this total mass, the merger occurs in

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the LIGO detectors’ most sensitive frequency region. This is the first analysis
that incorporates a template family of waveforms modeling the three stages—inspiral, merger and ringdown— of the full coalescence process. At the time
when this dissertation was printed out, the final results had not yet been
made public. Hence, this section summarizes the search strategy, describes
the main novelties involved in a search for higher-mass systems and presents
preliminary results corresponding to the playground data analyzed by theauthor. The definitive results of the search will be published in [2].
6.3.1 Motivation and Strategy for a High-Mass Search

Wolf-Rayet stars are
evolved, massive stars—over 20 M —, which
are losing mass rapidly
by means of a verystrong stellar windThis is the distance atwhich the detector
measures a SNR ρ = 8
for anoptimally-oriented andoptimally-locatedbinary, i.e. an overhead,face-on orbit.

The existence of binaries formed by compact objects of mass below 10−15 M
is well established through X-ray observations [193]; population-synthesis
models suggest chirp masses of ∼ 5–10 M for black-hole binaries that merge
within 10 Gyr [79, 194]. Nevertheless, it has also been suggested that significantly more massive black-hole binaries could form through a number ofalternative channels. In this dissertation, the question of the existence of black
holes with masses larger than hundreds of solar masses is investigated inchapter 8; this section focuses instead on a search for binary systems withtotal mass between 25 and 100 M . Several simulations over the past years
have indicated that dense stellar environments —such as globular and nuclear
star clusters— could contribute to the expected rates of compact binary coalescences via dynamical formation [185, 191, 190]. Mass-segregation mechanisms
make the most massive black holes sink towards the center of the clusters,
favouring the dynamical formation of massive black-hole binaries. Besides,
simulations of the evolution of the merger remnants after repeated stellar
collisions seem to indicate that runaway mergers at solar metallicity evolve to∼ 100 M Wolf-Rayet stars [133]. The star rapidly loses mass and turns into a∼ 10 M black hole, but there are hints pointing out that lower metallicities
can lead to higher remnant masses.
In light of these results, a search targeting more massive systems thanthose considered in section 6.2 appears promising. Most of these systems
would merge within the most sensitive part of the LIGO band; searchingfor them with complete filters for the full coalescence increases the reachof the detectors and, thus, the chance of detecting gravitational waves fromfaraway sources. Figure 31 illustrates the convenience of extending the inspiralsearches to higher masses and frequencies. The horizon distance for standard
post-Newtonian filters in the stationary-phase approximation is comparedwith the reach obtained with full inspiral-merger-ringdown templates. It isclear that this kind of searches provide a considerably enlarged reach, both insurveyed effective distance and in masses.
Inspiral-Merger-Ringdown Waveforms
The high-mass search intends to go up to binary systems with higher masses,
therefore it requires waveforms that describe the coalescence up to higher
frequencies. Diverse waveform models are used in the search, both to efficiently filter the data for signals and to assess the sensitivity of the instruments
and the data analysis procedures via simulations. In the mass range that thehigh-mass search wants to explore, the use of full inspiral-merger-ringdown
waveforms that model all of the observable signal and naturally decay away

6.3 the high-mass search in s5 ligo data

Figure 31: Expected horizon distance of the LIGO detectors for inspiral-only (red) andfull inspiral-merger-ringdown signals (blue). The merger and ringdown
contribute significantly to the SNR; a search employing full filters is able
to explore more massive systems and to reach further.

during the ringdown phase is highly desirable. As already discussed in chapter 4, numerical relativity is now able to reveal the nature of the merger andringdown stages of the binary black hole coalescence. The most optimal way
to integrate these new results into existing data analysis codes is by means ofanalytical models that capture the features of the coalescence.
So far, this effort has led to two different approaches, the effective-onebody method calibrated to numerical relativity data (EOBNR) [80, 83, 98, 103]
and the phenomenological models obtained by matching post-Newtonianand numerical waveforms [24, 22, 23, 25]. These approaches have alreadybeen described in section 5.1; in particular, the model developed in chapter 5 belongs to the second category. The high mass search uses EOBNR as
search templates and also as injected waveforms to test the detection efficiency.
The non-spinning phenomenological waveforms are used for injections andprovide a check that that search pipeline can detect waveforms which areslightly different than the search templates. Future searches for systems inthis mass range will make use of the spinning equivalents of the EOBNR andphenomenological models, including our model of chapter 5.
The Data Analysis Procedure
The search strategy for the high-mass search follows all the main steps of thepipeline employed for the low-mass search of section 6.2, with only a different
choice for the filter waveforms and minor technical details. Data from thethree detectors under consideration —H1, H2 and L1— is read in and usedas the base to compute the power spectral density. In a first filtering stage,
double- or triple-coincident events above a single-detector SNR threshold ofρ∗ = 5.5 are recorded. A subsequent second filtering stage re-filters the datawith the χ2 veto using p = 10 bins and χdof = 18 in equation 6.10.

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One particularity of the high-mass search has to do with the choice oftemplate bank placement. The templates ought to cover the parameter spaceof total mass between 25 − 100 M , with component masses in the range
1 − 99 M . The bank is laid out using a hexagonal placement algorithm [93],such that the maximum loss in SNR would be 3%. The template spacing isdetermined using the metric calculated for the frequency-domain templates
in the stationary phase approximation [51], as has been used in previous
searches for signals from low-mass systems. Although this metric has notbeen calculated for signals featuring inspiral, merger and ringdown phases, it
has been found that the bank gives the desired minimal match for most of
the parameter space. We do not achieve exactly the desired minimal matchfor the highest masses. To take into account this limitation, the coincidence
requirements chosen are somewhat looser than those of previous searches.
This limitation should be mitigated in future searches when metrics for fullinspiral-merger-ringdown waveforms become available.
After the two filtering stages mentioned above, events are required to becoincident in at least two detectors. The background is estimated via 100 fivesecond time slides between the two LIGO sites. All H1H2 double-coincidence
events are ignored due to correlated noise. H2L1 events found when H1 wasin operation are likewise discarded. These choices agree with those made forthe low-mass search. The coincident candidates are ranked according to theireffective SNR ρeff , and the false alarm rate (FAR) is calculated in the manner
described in section 6.1.5. The three different mass categories in which thetriggers are separated, due to the mass-dependent background, are chosen
as Mlow < 50 6 Mmid < 85 6 Mhigh . After being separated in categories
according to their total mass and type of coincidence, the inverse false alarm
rate is used as an intermediate statistic to finally compute the combined FAR;
candidates are ranked accordingly. Potential candidates for gravitational-wave
detection are identified and followed up. If no plausible candidate is found,
upper limits on the rate of coalescences for systems in the considered mass
range are calculated.
6.3.2 Preliminary Results of the High-Mass Search

The author analyzed
months 1 and 2 of the S5
LIGO data, from
November 4, 2005 to
January 6, 2006

The high-mass search analyzed the complete S5 LIGO data in a similar fashion
to the second year S5 low-mass search of section 6.2. The data was split in 12
two-month long blocks, that were analyzed separately by different analysts. We
report results of the analysis of playground data for months 1 and 2 of the S5
LIGO data. Figure 32 illustrates the reach of the detectors during these months.
A large number of simulated signals are injected into the pipeline and analyzed
in order to test the data analysis finding algorithm. In the high-mass search,
EOBNR and phenomenological waveforms are injected at physical distances
between 1 and 750 Mpc. These injections are considered found if they produce
coincident triggers above the SNR threshold ρ∗ = 5.5. In figure 32 we plot inblue and green the injections that were found in triple and double coincidence
respectively; in black those that were missed. The line that separates theregions of found and missed injections gives an estimation of the reach of
the detectors, which ranges from tens of Mpc at low masses to hundreds ofMpc at the high end of the mass range, confirming the predictions of figure 31.
Rather than reaching a maximum at around total mass M = 30 M anddecreasing afterwards, like we saw in figure 28 for the low-mass search, the6.3 the high-mass search in s5 ligo data

Figure 32: Found and missed injection during months 1 and 2 of the high-mass S5
LIGO search. The plot corresponds to the times when the three LIGO
interferometers were operating. Double- and triple- found coincidences
are compared with missed injections according to their chirp mass M.

horizon distance reaches its maximum at total mass of few hundred M
when inspiral-merger-ringdown templates are used.
As already mentioned, the data is filtered against the elements of the template bank twice; at the second filter stage the χ2 veto is applied. This signalbased veto is very important in order to separate potential real signals frombackground noise. Figure 33 shows how the veto differentiates among them;we plot the value of χ2 versus the SNR ρ for three different kinds of events:
software injections, background triggers estimated via time slides and zerolag triggers from the playground. The figure corresponds to the L1 detector;
plots for H1 and H2 show a similar structure. Firstly, one should notice theefficiency of the veto in separating background and injections. The χ2 test
provides significant separation from noise for a large fraction of simulated
signals in this search. Background triggers associated to glitches might bevery loud and pass the SNR cut, but for the most part they present a large
χ2 value; in general, injections are correlated with lower χ2 values. In principle we could draw a line in the ρ-χ2 plane and discard all triggers above
it. In practice, however, one should be careful when applying this procedure,
since the separation between injections and background is not always perfectly clean, especially at low ρ values. Thus, stringent cuts would put us indanger of discarding real signals with slightly-above-threshold SNR. This isa well-identified issue; future search efforts in this mass range will employ
new signal-based vetoes and multivariate classifiers to hopefully achieve a
better separation of signal from background. In addition, figure 33 shows thatamong the zero-lag triggers for months 1 and 2 of S5 no obvious candidatefor detection is present, for all blue crosses in the plot are consistent with thetime-slided background triggers.

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Figure 33: Plot of SNR versus χ2 in L1 during months 1 and 2 of the high-mass S5
LIGO search. Time-slided background triggers (black) present larger χ2values than the injections, showing how the χ2 veto helps discriminate
between them. Additionally, we observe no obvious real candidate among
the zero-lag playground triggers.

This time is a functionof the instruments thatwere in operation

Once the data has been filtered and the triggers have been subject to thecoincidence test, the next step is the ranking of the candidates in order toestablish their significance. For the high-mass search this ranking is done using
the false alarm statistic. As it happened in the low-mass search, the false alarm
rate depends on the filter —despite the expectation for Gaussian noise in which
it does not— and also on how many detectors were operating and participatedin the event. For the high-mass search, we compute the false alarm rate as adiscrete function of four parameters, which each coincident event possesses.
Each parameter is an index for an event E. The first index, i, describes the
instruments that were functioning during the event and is a member of theset {H1L1, H2L1, H1H2L1}. The second index, j, indexes the instruments thatparticipated and is also a member of the set {H1L1, H2L1, H1H2L1}. The thirdindex, k, denotes a range for the average total mass estimated for the event andis in the set {[25, 50), [50, 85), [85, 100)}. The fourth index, m, is the rank of
the event. This index is determined by assigning the event with the lowest
combined effective SNR defined in 6.13 the value 0 and the the next lowest,
1, etc. until all N events are ranked. We calculate the false alarm rate, ξ for agiven event by summing all background events B ∈ E with a rank larger thanthat event dividing by the background time analyzed Ti .
X
ξijkl =
Bijkm / Ti .
(6.15)
m>lThis is essentially equivalent to equation 6.14 and to the calculation done
for the low-mass search, but new algorithms have been developed in thehigh-mass search in order to parse events from the database.
In order to assess the total FAR of events, independent of the second andthird indices, j, k, we use the inverse FAR ξ−1 as an intermediate ranking

6.3 the high-mass search in s5 ligo data

statistic to replace combined effective SNR as the fourth index. Then the
combined FARX
ξil =Eijkl / Ti
(6.16)
m>l,j,kis only a function of the detectors i that were functioning during the eventand the inverse FAR computed at the previous step. The results of the FAR
calculation for the zero-lag playground triggers occurred during months 1
and 2 of the high-mass S5 LIGO search are shown in figure 34. We observe
values of ρeff compatible with the background; however the FAR plot shows asignificant event above the 3-σ level. Such an event is eventually followed up
with appropriate diagnosis tools developed by the group; the description of thefull follow-up pipeline falls beyond the scope of this dissertation. Nevertheless,the true significance of this event found in the playground can only be statedafter the analysis of the full data set is complete; the playground results areaffected by small statistics associated to the reduced data-set that is analyzed.
Full results of the high-mass search, including in-depth description of theloudest candidates found and —if applicable— computation of upper limits
on the rate of binary coalescences will be presented in in [2].
Limitations and Future Prospects
There are a number of limitations in the current approach, which will be
addressed for future searches. The main limitation is that the template waveforms neglect the effects of spin. As mentioned in previous chapters of thisdissertation, the statistical distribution of the spins of black holes in binaries
is not well known; nevertheless there are examples of black holes in X-ray
binaries which have been observed to have a large spin [181]. For a binary
with spinning components, we have shown that the expected observed gravitational wave signal will differ substantially from the non-spinning case;the observed length can be different, and in the case of non-aligned spinand orbital angular momenta, there would be modulation of the amplitude
and phase of the gravitational waveform due to the precession of the orbital
plane. Neglecting such effects in the search templates will affect the detection
efficiency for binaries with spinning components. Due to the current lackof analytical inspiral-merger-ringdown waveforms for systems with genericspins, we are not yet able to fully quantify how large an effect this is. The
model presented in chapter 5 constitutes a first step towards the incorporationof spins in current GW searches and will be included in the near future.
Another limitation of the search is that, due to the shorter duration andbandwidth of the signals in comparison to searches for lower mass systems,
it is harder to distinguish between genuine signals and background events.
Since the signals themselves resemble short-duration glitches, tests whichhave previously proved very effective in the lower mass searches, such as theχ2 -test described in 6.1.3, and the consistency between measured distances inthe two Hanford detectors, are not as effective here. New approaches to theranking of candidate events are being developed to improve this situation in
the future.
This chapter has presented the current status of searches for binary coalescences with ground-based interferometers. Over the past years, significant
advances, especially in waveform modelling, have opened the door to improvements in the design of the searches. In particular, the possibility of

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Figure 34: Playground zero-lag events observed compared to background duringmonths 1 and 2 of the high-mass S5 LIGO search. The results correspond to
the times when the three LIGO detectors were operating. The upper panel
shows the number events versus ρeff , whereas the lower panel shows thefalse alarm rate. The apparent 3-σ significance of an event in the FAR plotis due to insufficient analyzed data in the playground. In order to establish
its true significance, the event would need to be properly analyzed against
the full data set.

incorporating input from numerical relativity is slowly taking form. The next
chapter presents the results of the first project directly targeted to study the
influence of numerical relativity waveforms into current gravitational-wave
searches.

NUMERICAL INJECTIONS IN GRAVITATIONAL SEARCHES

So far we have stressed the importance of binary systems of compact objects
formed by black holes and/or neutron stars for testing general relativity andstudying its astrophysical implications [253]. We have seen how detection ofgravitational radiation from these objects is very likely with future generations of gravitational-wave interferometers. Two important advances haveoccurred in recent years that approach us to the goal of observing and interpreting signals from coalescing compact objects. The first is the successful
construction and operation of the world-wide network of gravitational-wave
interferometers described in depth in chapter 1. The second has been the
success of numerical relativity in simulating the merger phase of binary blackhole (BBH) coalescence introduced in chapter 4. Since the already-mentioned
breakthroughs occurred in 2005, a number of numerical relativity groups
around the world have successfully evolved various configurations startingfrom the inspiral phase all the way through the merger to the final remnant
black hole. This has led to important new physical insights in BBH mergers.
Since the coalescence of black holes is among the most important targetsof gravitational-wave detectors, detailed information provided by numericalsimulations should be used to increase the reach and to quantify the efficacy
of data analysis pipelines. Indeed the driving motivation of research on numerical simulations of black-hole binaries over the last few decades has been
their use in gravitational-wave observations.
This chapter presents the results of the first project established with thosespecific goals. Below is the description of the scope and goals of the NINJA
project as well as the contributed numerical waveforms and the constructionof the simulated gravitational-wave detector data used in the analyses. The
core of the author’s work for NINJA, corresponding to the implementation ofa search that uses the non-spinning phenomenological template bank of [24]is described in section 7.2. The chapter concludes with the discussion of theresults and future directions for NINJA in sections 7.2.3 and 7.2.4.
7.1

the ninja projectThus far, most searches for gravitational waves from BBH mergers have reliedon post-Newtonian results, which are valid when the black holes are sufficiently far apart, as explained in section 3.1 of chapter 3. Within its range ofvalidity, post-Newtonian theory provides a convenient analytic descriptionof the expected signals produced by binary systems. The numerical relativity
results, on the other hand, have not yet been synthesised into an analytic
model for the merger phase covering a broad range of parameters, i.e., a wide
range of mass ratios, spins and if necessary, eccentricity. As a matter of fact,
one of the core results of this thesis, presented in chapter 5, is the constructionof a phenomenological model for the full coalescence of non-precessing BBH
systems that aims at incorporating the spins of the black holes in a simplified
way.

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7

This chapter
summarizes the work of
the author in thecontext of the NINJA
collaboration, which
resulted in the
publications [49], [87]and [228]

118

NINJA: The Numerical
INJection Analysis
Project
http://www.
ninja-project.org/

numerical injections in gravitational searches

Similarly, despite significant progress, there is not yet a complete detailed
description over the full parameter space of how post-Newtonian and numerical simulations are to be matched with each other. The waveform family ofchapter 5 is a first step in this direction, but more work is necessary to expand
this model to generically spinning systems. On the data analysis side, many
pipelines, especially ones that rely on a detailed model for the signal waveform,
have made a number of choices based on post-Newtonian results, and it isimportant to verify that these choices are sufficiently robust. More generally,
it is necessary to quantify the performance of these data analysis pipelines forboth detection and parameter estimation. This is critical for setting astrophysical upper limits in case that no detection is made, for following up interesting
detection candidates, and of course for interpreting direct detections. To date,available research has primarily used post-Newtonian waveforms. Numerical
relativity now provides an important avenue for extending these studies tothe merger phase.
There are significant challenges to be overcome before numerical relativity
results can be fully exploited in data-analysis pipelines. The NINJA project wasstarted in the spring of 2008 with the aim of addressing these challenges andfostering close collaboration between numerical relativists and data analysts.
The purpose of NINJA is to study the sensitivity of existing gravitational-wave
search algorithms using numerically generated waveforms that are injected
into simulated noise. Only BBH simulations are considered, leaving out resultsfrom supernova simulations or simulations containing neutron stars; thewaveform data comes purely from numerical simulations; the NINJA data set
is constructed using Gaussian noise to model the response of the Initial LIGO
and Virgo detectors —no attempt has been made to include non-Gaussian
noise transients found in real detector data.
7.1.1 Numerical Waveforms
The NINJA project studied BBH coalescence waveforms submitted by ten
individuals and teams. Participation in NINJA was open to anyone and theonly restrictions were that each contribution: (i) was a numerical solutionof the full Einstein equations, (ii) consisted of only two waveforms, or up to
five waveforms if they were part of a one-parameter family. No restrictions
were placed on the accuracy of each waveform. All contributions followed theformat specified in [75]. The contributed waveforms, plotted in Figures 35 and36, cover a variety of physical and numerical parameters. Most simulations
model low-eccentricity inspiral, the mass ratio q = m1 /m2 ranges from 1 to4. The initial angular frequency of the ` = m = 2 mode ranges from 0.033/M
to 0.203/M (where M denotes the sum of the initial black-hole masses). Thisinitial angular frequency marks where contributors consider the waveform
sufficiently clean to represent the physical system (e.g. this will be chosen
after initial unphysical radiation content, often referred to as “junk radiation”
in numerical relativity, is radiated away). The length of the waveforms varies
between a few 100 M to over 4000 M. The contributions naturally differ inaccuracy, both regarding how well they capture the black-hole dynamics andin the extraction of the gravitational-wave signal.
Table 10 lists a few key parameters that distinguish the waveforms, andintroduces the following tags for the different contributions and NR codes:
BAM HHB [77, 156, 146, 145, 78] and BAM FAU [77, 156, 254, 78] are contribu-

7.1 the ninja project

0.3

SpEC q=1

0.3

0
-0.3
0.3

BAM HHB S00

0.3

BAM HHB S50

0.3

Hahndol kickHahndol non

Lean 2PU CP

PU T52W

UIUC cp

UIUC punc-0.3
BAM HHB S75

BAM HHB S85

0.3

0

0

-0.3

-0.3
LazEv

Lean c

0.3

0

0

-0.3

-0.3
MayaKranc e0

MayaKranc e02

0.3

0

0

-0.3

-0.3

-2000

-1000

0 -2000

t/M

0.3

CCATIE s6

0

-0.3

0.3

CCATIE r6-0.3
BAM HHB S25

00.3

CCATIE r4

0

-0.3

0.3

CCATIE r2-0.3
BAM FAU

00.3

CCATIE r0

0

-1000

0

-600

-300

0

-600

t/M

t/M

-300

0

t/M

BAM FAUBAM HHB S00

BAM HHB S25

BAM HHB S50

BAM HHB S75

BAM HHB S85

CCATIE r0

CCATIE r2

CCATIE r4

CCATIE r6

CCATIE s6

Hahndol kickHahndol non

LazEv

Lean 2

Lean cMayaKranc e0

MayaKranc e02

PU CP

PU T52W

SpEC q=1

UIUC cp

UIUC punc

0
-0.3
0.3
0
-0.3
0.3
0
-0.3
0.3
0
-0.3
-100

t/M

0

-100

0

t/M

-100

t/M

0

-100

t/M

0

-100

t/M

0

-100

t/M

0

Figure 35: Summary of all submitted numerical waveforms: r/M Re(h22 ). The xaxis shows time in units of M and the y-axis shows the real part of the(`, m) = (2, 2) component of the dimensionless wave strain rh = rh+ −irh× . The top panels show the complete waveforms: the top-left panel
includes waveforms that last more than about 700M, and the top-rightpanel includes waveforms shorter than about 700M. The bottom panel
shows an enlargement of the merger phase for all waveforms. (Figure credit:
Harald Pfeiffer)

119

120

numerical injections in gravitational searches

1 SpEC q=1

1 CCATIE r0

0.1
0.01

0.01

1 BAM FAU

BAM HHB S00

1 CCATIE r4

0.1

CCATIE r6

0.1

0.01

0.01

1 BAM HHB S25

BAM HHB S50

1 CCATIE s6

0.1

Hahndol kick

0.1

0.01

0.01

1 BAM HHB S75

BAM HHB S85

1 Hahndol non

0.1

Lean 2

0.1

0.01

0.01

1 LazEv

Lean c1 PU CP

0.1

PU T52W

0.1

0.01

0.01

1 MayaKranc e0

MayaKranc e02

1 UIUC cp

0.1

0.1

0.01

0.01

-2000

-1000

0 -2000

t/M

1

CCATIE r2

0.1

-1000

0

-600

UIUC punc

-300

0

-600

t/M

t/M

-300

0

t/M

BAM FAUBAM HHB S00

BAM HHB S25

BAM HHB S50

BAM HHB S75

BAM HHB S85

CCATIE r0

CCATIE r2

CCATIE r4

CCATIE r6

CCATIE s6

Hahndol kickHahndol non

LazEv

Lean 2

Lean cMayaKranc e0

MayaKranc e02

PU CP

PU T52W

SpEC q=1

UIUC cp

UIUC punc

0.1
0.01
1
0.1
0.01
1
0.1
0.01
1
0.1
0.01
-100

t/M

0

-100

t/M

0

-100

t/M

0

-100

t/M

0

-100

t/M

0

-100

t/M

0

Figure 36: Distribution of power into different spherical harmonics. The blue line
1/2
shows Σ`,m |h`m r/M|2. A dashed red line, if present, shows thesame sum, but excluding the (`, m) = (2, ±2) modes. The separation
between the two lines gives the relative importance of non (2, ±2) modes.
If no red line is present for a certain run, then only the (2, ±2) modes were
supplied. The layout is as in figure 35: The top panels show the complete
waveforms, whereas the bottom panel shows an enlargement of the merger
phase. The x-axis shows time in units of M. (Figure credit: Harald Pfeiffer)

7.1 the ninja project

121

tions using the BAM code, CCATIE is the AEI/LSU code [28, 29, 170, 206, 222],
Hahndol is the Goddard Space Flight Center’s code [158, 256], LazEv is the RIT

code [269, 88, 96], Lean is Ulrich Sperhake’s code [244, 246, 245], MayaKrancis the Georgia Tech/Penn State code [255, 153], PU stands for the Princeton
University code [213, 212, 82, 214], SpEC for the Cornell/Caltech collaboration
code [231, 204, 69, 232], and UIUC stands for the University of Illinois at
Urbana-Champaign team [114].
The codes listed above use different formulations of the Einstein equations,gauge conditions, mesh structures, initial data and wave extraction methods;
they follow either of two approaches to solving the Einstein equations: (i) thegeneralized harmonic formulation, which was the basis of Pretorius’ initial
breakthrough simulation of coalescing black holes [212], or (ii) the movingpuncture approach, following [88, 52]. Both approaches result in canonicalchoices for the construction of initial data, the evolution system for the Einstein equations, and the treatment of the singularity inside the black-holehorizons. Full details of each individual code are given in the references above
and a comparative description of their main features is presented in [49].
7.1.2 Creation of NINJA Data
The data provided by the numerical relativity groups follows the format outlined in [75], which is based on the mode decomposition of the gravitationalradiation field at large distances from the source. If we specify a gravitational
waveform hµν in the Transverse-Traceless (TT) gauge, we only need the spatial components hij . We assume that we are sufficiently far away from thesource so that the 1/r piece dominates:
hij = AijM
+ O r−2 ,r(7.1)

where M is the total mass of the system, r is the distance from the source, andAij is a time-dependent TT tensor. In the TT gauge, hij has two independent
polarizations denoted h+ and h× and the complex function h+ − ih× canbe decomposed into modes using spin-weighted spherical harmonics −2 Y lm
of weight -2:∞

h+ − ih× =M X X̀
H`m (t) −2 Y `m (ι, φ) .
r

(7.2)

`=2 m=−`The expansion parameters Hlm are complex functions of the retarded timet − r, however if we fix r to be the radius of the sphere at which we extract
waves then Hlm are functions of t only. The angles ι and φ are respectively
the polar and azimuthal angles in a suitable coordinate system centered on thesource. This decomposition is directly applicable to non-precessing binaries.
Otherwise, a comparison of the waveforms requires a careful treatment ofmode-mixing effects due to rotations of the frame; see for instance [137].
The numerical data contributed to NINJA is given in the form of an ASCII
data file for each mode (`, m), with accompanying meta-data describing thesimulation [75]. Only modes that contribute appreciably to the final waveform
are included, at the discretion of the contributing group. Each data file consistsof three columns: time in units of the total mass, and the real and imaginaryparts of the mode coefficients H`m as a function of time. Note that the total

The list of contributing
groups to NINJA
reflects the success ofthe project, for all
research groups capableof performing NR
simulations of BBH
space-times did
contribute theirwaveforms.

122

numerical injections in gravitational searches

Table 10: Initial conditions for numerical waveforms submitted to the NINJA project.
The columns list, in order from left to right, the name of the contribution orcode, the name of the run where appropriate, the mass ratio q = m1 /m2
where m1 > m2 , the spins of the black holes in vector form (if only one spinis given, both spins are equal), the initial frequency of the (`, m) = (2, 2)
mode (rounded to three digits) and the initial coordinate separation ofeither the black-hole punctures or the excision surfaces. All binaries startout in the xy-plane with initial momenta tangent to the xy-plane. See text
for the identification of each contribution. The dimensionless spins of the
BAM FAU run are (−0.634, −0.223, 0.333) and (−0.517, −0.542, 0.034).
Code [Ref.]Runq~Si /m2i

ω22 M

D/M

BAM FAU[254]

1

see caption

0.06

9.58 ŷS00 [146]

1

0

0.045

12 ŷ

[77, 156]
BAM HHB[77, 156]

S25 [145]10.25 ẑ

0.045

12 ŷ

S50 [145]10.50 ẑ

0.052

11 ŷ

S75 [145]10.75 ẑ

0.06

10 ŷ

S85 [145]10.85 ẑ

0.06

10 ŷ

CCATIEr0 [206]

1

0.6 ẑ, −0.6 ẑ

0.079

8 x̂

[28, 29, 170, 206]

r2 [206]

1

0.6 ẑ, −0.3 ẑ

0.078

8 x̂

r4 [206]

1

0.6 ẑ, 0

0.076

8 x̂

r6 [206]

1

0.6 ẑ, 0.3 ẑ

0.075

8 x̂

s6 [222]1

0.6 ẑ

0.074

8 x̂

Hahndolkick

3

0.2 x̂, 0.022 x̂

0.078

8.007 ŷ[158, 256]

non

4

0

0.070

8.470 ŷLazEv

MH [96]1

0.92 ẑ

0.07

8.16 x̂[269, 88]
Lean

c

4

0

0.05

10.93 x̂

[244]

2

1

0.926 ẑ

0.11

6.02 x̂MayaKranc

e0 [153]

1

0

0.05

12 x̂

[255]

e02 [153]

1

0

0.05

15.26 x̂PU

CP [82]

1

0.063 ẑ

0.07

9.5 x̂

[213, 212]

T 52W [214]

1

0

0.07

SpEC

q = 1 [69, 232]

1

0

0.033

UIUC

cp [114]

1

0

0.194

4.790 x̂[114]

punc [114]

1

0

0.203

4.369 ŷ

15 x̂

[231]

7.1 the ninja project

Table 11: Characteristic duration, mass and frequencies of the waveforms summarizedin table 10. The columns ∆T100 and fi,100 give the duration and initialfrequency of the waveform when scaled to total mass M = 100 M . M30Hz
is the total mass of the waveform when it is scaled so that the initial frequency
is 30Hz (this sets the lowest mass at which each waveform can be injected
into the NINJA data).
q∆T100 (s)fi,100 (Hz)

M30Hz (M )

1

0.54

19

65

S00

1

1.03

15

48

S25

1

1.15

15

48

S50

1

1.03

17

56

S75

1

0.81

19

65

S85

1

0.87

19

65

r0

1

0.34

26

85

r2

1

0.37

25

84

r4

1

0.40

25

82

r6

1

0.45

24

81

s6

1

0.59

24

80

kick

3

0.25

25

84

non

4

0.32

23

75

LazEv

MH

1

0.43

23

75

Lean

c

4

0.92

16

54

2

1

0.20

36

118

e0

1

1.23

16

54

e02

1

0.74

16

54

Code

Run

BAM FAUBAM HHBCCATIE

Hahndol

MayaKranc

CP

1

0.29

23

75

T 52W

1

0.16

23

75

SpEC

q=1

1

1.96

11

36

UIUC

cp

1

0.10

63

209

punc

1

0.10

66

219

PU

123

124

numerical injections in gravitational searches−20

Detector noise (Sn(f))1/2 [strain Hz−1/2]

10

H1
H2
L1
V1
LIGO 4km design
LIGO 2km design
Virgo Design

−21

10

−22

10

−23

10

1

10

2

10
Frequency [Hz]3

10

Figure 37: Design spectra of the first generation LIGO and Virgo detectors and the
NINJA noise curves generated via coloured Gaussian data to mimic the
response of the real detectors.

This is not true in thecase of simulations
which include matterfields, but we do notconsider such
waveforms here

See, for example, [253]mass M scales both the time and the amplitude; thus the BBH waveforms foreach simulation can be scaled to an arbitrary value of the mass. To model thesignal seen by a gravitational-wave detector, we need to calculate the detector
strain h(t) from the above mode decomposition. To do this, we must choose
particular values of the total mass, orientation and distance from the detector.
Given the H`m , the total mass, the distance to the source, and the angles(ι, φ), we calculate h+,× using equation 7.2, and use the detector response
functions F+,× to calculate the observed strainh(t) = h+ (t)F+ (α, δ, ψ) + h× (t)F× (α, δ, ψ) .

(7.3)

Here (α, δ) are sky-angles in the detector frame, ψ is the polarization angle
and the time t is measured in seconds. In this analysis, we wish to simulatesignals that might be observed by the Initial LIGO and Virgo detectors. Since
the location and alignment of the three observatories differ, we must use
the appropriate detector response and arrival time to compute the strain
waveform h(t) seen at each observatory. This ensures that the waveforms arecoherent between the detectors and simulate a true signal.
To model the detector noise, we generated independent Gaussian noisetime series n(t), sampled at 4096 Hz, for each detector. This sample rate
was chosen to mimic that used in LSC-Virgo searches and assures a tolerable
loss in signal-to-noise ratio due to the discrete time steps. Stationary white
noise time series are generated and coloured by a number of time-domain
filters designed to mimic the design response of each of the LIGO andp Virgodetectors. Figure 37 shows the one-sided amplitude spectral density Sn (f)of each time detector’s time series, where Sn (f) is defined by equation 5.4.
We see from figure 37 that the noise power spectrum of the NINJA data set
closely approximates the Initial LIGO design sensitivity in the frequency range
of interest (30 − 103 Hz). There is a slight discrepancy with the Virgo design7.1 the ninja project

Figure 38: Total mass and distance of the 126 NINJA injections, with the colour code
encoding the modulus of the dimensionless total spin |S~1 /m21 + S~2 /m22 |of the black holes. The total mass of the injected signals lies within therange 36 M 6 M 6 460 M and they are located at a distance between
52 and 480Mpc. The vertical line delimits the mass parameter space with
M < 160 M that the phenomenological template bank for the search
described in section 7.2 covers. The full NINJA data set spans a duration ofa little over 30 hours and contains a total of 126 signals injected in simulated
noise, 67 of which overlap with the parameter space of our template bank.

curve at low frequencies (between approximately 20 and 150 Hz), which is anartefact of the Virgo noise generation procedure. Narrow-band features suchas the violin and mirror modes were removed from the detector response used
to compute the NINJA data, but were included in the calculation of the Virgo
design curve. The 1/f tails of these narrow-band features are responsible forthe small discrepancy.
Having produced the simulated detector data, we then generated a population of simulated signals using the numerical relativity data. This population
was constructed to cover a broad range of masses and signal amplitudes. We
required that the starting frequency of the dominant ` = m = 2 mode of
the signal was not more than 30 Hz, an appropriate threshold given the sensitivity curve of the Initial LIGO and Virgo detectors. This sets a minimum
mass at which each waveform can be injected, which is given in Table 11. The
minimum possible injection mass is therefore 36 M . The maximum mass
was chosen as 350 M . To get a good sample of long injected waveforms, we
systematically chose a lower range of masses for the longer waveforms. No
restrictions were placed on the other simulation parameters, i.e., the spins,
mass-ratios and eccentricities. We ensured that waveforms from all the participating groups were equitably represented by generating approximately 12
signals from the waveforms supplied by each group. The time interval between
adjacent injected signals was chosen to be a random number in the range
700 ± 100 s.

125

126

The software forcarrying out thisprocedure is freely
available as part of the
LSC Algorithm Library
http://www.
lsc-group.phys.uwm.
edu/lal [178].

numerical injections in gravitational searches

Given these constraints, we generated the parameters of the signal population. The logarithm of the distance to the binary was drawn from a uniform
distribution ranging from 50 Mpc to 500 Mpc, and the source locations andorientations were drawn from an isotropic distribution of angles. We thencomputed waveforms corresponding to this population and at the appropriate
sampling rate. We required that the optimal matched filter signal-to-noise
ratio of any injection be greater than five in at least one of the four simulated
detectors. Any waveform that did not satisfy this constraint was discarded
from the population. Subject to this condition, the distances of injected signals
varied from 52 Mpc to 480 Mpc (median at 145 Mpc), the injected total mass
range was 36 M 6 M 6 346 M (median at 155 M ), with individualcomponent masses in the range 11 M 6 mi 6 193 M .
Finally, the waveforms h(t) were added to the simulated detector noisen(t) to generate the NINJA data set s(t) = n(t) + h(t). As described above,care was taken to ensure that signals were coherently injected in the data
streams from the four detectors. The data set used in this analysis consistedof a total of 126 signals injected in a total of 106 contiguous segments of noise
each 1024 s long, thus spanning a duration of a little over 30 hours. Figure 38
shows the mass, spin and distance of the waveforms contained in the NINJA
data set, as well as the part of the parameter space explored by the search
described in section 7.2
7.2 search with a phenomenological template bankAnalysis of the NINJA data was open and nine groups submitted contributions
using a variety of analysis techniques. Participating groups were provided with
the NINJA data set containing signals embedded in noise and the parametersof the injected signals. Analysts were not given access to the raw numericalrelativity waveforms or noiseless injection data. Methods used to analyze the
NINJA data include: matched-filter based searches, unmodeled waveform
searches using excess-power techniques, and Bayesian model-selection andparameter-estimation techniques. For a full description of all analyses andresults we refer the reader to the main NINJA paper [49]. In this section we
present the results of our analysis employing a matched-filter search using
phenomenological waveforms.
7.2.1 The Non-Spinning Phenomenological Waveform Model
Most standard searches for gravitational waves from BBHs use the PN approximation of general relativity to construct banks of templates that account for
the inspiral stage of the coalescence process, and the final ringdown can also
be computed via perturbative techniques. However, the full calculation of the
waveform in the merger stage requires numerical methods. These numericalsimulations are in general rather expensive, and it is at present not feasible tomodel a coalescing binary over hundreds of orbits with sufficient accuracy.
It is in fact also unnecessary to do so, because PN theory provides a valid
description of the system when the black holes are sufficiently separated andthe gravitational field is weak. Thus, a promising approach for constructing
long waveform models covering the inspiral, merger and ringdown regimes
is to stitch together the results of PN and NR calculations.

7.2 search with a phenomenological template bank

127

One procedure for constructing such hybrid waveforms is presented in[24, 23, 22], where PN and NR waveforms are matched in an appropriate
regime (−750 6 t/M 6 −550) prior to the merger (M is the total massof the binary system in solar masses). Restricted 3.5PN waveforms at massquadrupole order are used for the inspiral phase, as given by equation (3.1)
of [23]. For the numerical part, the model is based on long unequal-mass
waveforms from simulations run by the Jena group using the BAM code [77,146, 105]. These simulations span a range of mass ratios corresponding to
0.16 6 η 6 0.25, where η = (m1 m2 )/M2 is the symmetric mass ratio of
the binary system. The matching of PN and NR data is performed over anoverlapping region, under the assumption that both approaches to the true
BBH waveform are approximately correct at the late inspiral stage. Once thehybrid waveforms are constructed, they are fit to a phenomenological model
determined entirely by the physical parameters of the binary system. This fit
to an analytical expression is performed in the Fourier domain, assuming afunctional dependence of the formu(f) = Aeff (f)Ψeff (f) ,(7.4)

with amplitude and phase given by the following equations:

−7/6

if f < fmerg
 f/fmerg −2/3
Aeff (f) ≡ C
f/fmerg
if fmerg 6 f < fring
w L(f, fring , σ) if fring 6 f < fcut
Ψeff (f) = 2πft0 + ϕ0 +7Xψk f(k−5)/3 ,

(7.5)

(7.6)

k=0

Each waveform is parameterized by the physical parameters of the system,
which in the non-spinning case are solely the masses m1 and m2 of the black
holes. As a result of the matching and fitting procedures described above, a twodimensional template family of waveforms that attempt to model the entire
coalescence of non-spinning binary black hole systems has been obtained.
7.2.2

The Search Strategy

The LSC inspiral pipeline infrastructure developed by the CBC group anddescribed in chapter 6 has been employed to analyze the data released for
the NINJA project. The pipeline, without major conceptual modifications,has been used in LSC searches for compact binaries from the third LIGO
science run onward [10]. The same pipeline has been modified for the analysis of the NINJA data with the non-spinning phenomenological templatefamily [24, 23, 22] described in section 7.2.1. Since the signals present in thesimulated noise are known to be numerical simulations of BBH coalescences,
the search method consists of a matched-filter technique [31] using an IMR
waveform model based on hybrid NR-PN waveforms. The findings presented
here concentrate solely on the simulated LIGO detectors H1, H2 and L1, although the NINJA data was generated for the simulated Virgo interferometer
V1 as well.
The tuning of the pipeline was realized according to the usual choices madeby the group in the analysis of real LIGO data. For our NINJA analysis, a SNR
threshold ρ∗ = 5.5 was employed, in agreement with the value currently used

The phenomenologicalcoefficients
fmerg , fring , fcut , σ andψk are functions of thebinary symmetric mass
ratio η and total mass
only and can becomputed from Tables I
and II of [23].

128

numerical injections in gravitational searches

Original
Whitened
ISCO
Light ring

H20 + 20L M
127.80127.85127.90Time HsL

127.95128.00127.90Time HsL

127.95128.00

Original
Whitened
ISCO
Light ringH80 + 80L M
127.80127.85Figure 39: Two time-domain phenomenological waveforms from the template bankused in this search, corresponding to equal-mass binaries in the cornersof our parameter space, namely (20 + 20) M and (80 + 80) M BBH
systems. The original and “whitened” [99] waveforms are shown, with theiramplitudes arbitrarily resized. The dotted and dashed vertical lines markthe points where the ISCO and light ring frequencies are reached. The LRD
frequency is not shown, since it basically extends up to the full waveform.
A matched-filter search that starts at 30 Hz and ends at the ISCO will not
be able to pick the most massive binaries, since the inspiral phase of the
coalescence falls below the LIGO interferometers’ detection band. It is
expected that the light ring and LRD frequencies, which extend up to the
BH merger and ringdown respectively, will show improved performanceat recovering high-mass signals.

in recent LSC searches for binary coalescences [15]. Whenever triggers arefound with comparable coalescence time and parameters —in this case, component masses—, they are stored as coincident [224]. The detection statistic
is the combined SNR of the single detector triggers.
Once the initial matched filter has produced a list of triggers that pass thefirst coincidence step, a second stage follows where data is again filtered, but
only through the templates that previously matched a trigger. Additionally the7.2 search with a phenomenological template bank129

χ2 [30] and r2 [225] signal-based vetoes, designed to separate true inspiralsignals from fluctuations in non-stationary noise, are applied. At this point
an effective SNR ρeff,i (ρi , χ2i ) is calculated combining the standard SNR withthe χ2 value characterizing the mismatch between the spectral content of thetemplate and the data. After further coincidence tests, the surviving triggers
are listed as true gravitational wave candidates and constitute the output of
the search. The significance of the triggers is based on the combined effective
SNR, namelyvuNuXρeff = t (ρeff,i )2 .
(7.7)
i

A direct comparison between the list of injections performed on the NINJA
noise and the triggers found by the pipeline allows for conclusions about
the sensitivity of the analysis and the relative performance of the different
template banks.
The phenomenological template bank has been included in the LSC inspiralpipeline routinely used by the CBC group as a new waveform for filtering in
the time domain. A search on the NINJA data has been performed, within amass range of 20 M 6 m1 , m2 6 80 M for the component masses, with
40 M 6 M 6 160 M for the total mass of the binary. The template bankis constructed using the standard second order post-Newtonian metric, anduses a hexagonal placement algorithm in mass space with a minimal match of
0.99 [93]. The number of signals that are recovered by the pipeline depends
strongly on the choice for the upper frequency cutoff used in the matchedfilter integral, as we have observed in our investigations with the integration
stopping at the ISCO (Innermost Stable Circular Orbit, r = 6M), light ring(the unstable circular orbit for photons orbiting a Schwarzschild black hole,r = 3M) and Lorentzian ringdown (LRD) frequencies.
In Figure 39 we show two waveforms from our phenomenological templatebank, which correspond to equal-mass binaries in the corners of our parameter space, namely total mass M = 40 M and 160 M . Displayed are boththe original time domain waveform and its “whitened” form [99], as the initial
LIGO detector perceives it, and the relative amplitudes have been arbitrarily
resized. The whitened waveform is computed as the p
inverse Fourier transformof the original signal multiplied by the function 1/ Sh (f) in the frequency
domain, where Sh (f) is the one-sided noise power spectral density of the simulated LIGO detectors. In each plot the vertical lines correspond to the ISCO
and light ring frequencies. In our searches we have started filtering against
the phenomenological templates at either 30 or 40 Hz and we have stopped
the integration at the three frequencies discussed in the above paragraph. It isevident that whereas a cut at the ISCO frequency still retains a good portionof the inspiral signal for low-mass binaries, it is insufficient for higher masses.
The light ring and LRD frequencies, on the other hand, extend roughly up to
the BH merger and to the Lorentzian tail (from the decay of the quasi-normal
modes of the ringdown), respectively, and are therefore expected to produce
better results for a matched-filter search for high-mass signals.

The Lorentzian
ringdown frequency isdefined as 1.2 times thefundamental ringdown
frequency of Berti,
Cardoso and Will, Phys.
Rev. D 73 064030(2006).

130

numerical injections in gravitational searches

Table 12: Results of the search for NINJA signals using the non-spinning phenomenological template bank. There were 126 injections performed into the analyzed
data for H1, H2 and L1, 67 of which fell within the mass range of our phenomenological template bank (M < 160 M ). We explicitly show thata much better efficiency in trigger recovery is achieved when the cutoff
frequency is pushed beyond the ISCO frequency, up to the light ring andLorentzian ringdown frequencies. Likewise we observe improved efficiency
in finding the signals that lie within the mass range of our template bank.
In both cases the signal-based vetoes have little influence in the rejection of
triggers, confirming their efficiency in separating inspiral-like signals fromother kind of glitches.
Frequency Cutoff

ISCO

LightRing

LRD

LRD

Filter Start Frequency

30 Hz

30 Hz

30 Hz

40 Hz

78, 54, 69

94, 66, 90

92, 61, 87

93, 60, 86

Found Coincidence

59

78

81

80

Found Second Coincidence

59

78

80

79

Complete set of 126 NINJA Injections
Found Single (H1, H2, L1)

Reduced set of 67 NINJA Injections with M < 160 M
40, 17, 32

55, 41, 50

55, 41, 50

56, 40, 50

Found Coincidence

30

47

48

47

Found Second Coincidence

30

47

48

47

Found Single (H1, H2, L1)

7.2.3 Efficiency for Detection
The main results of our search for numerical relativity signals injected insimulated LIGO noise employing a phenomenological template bank arepresented in Table 12. We show here a summary of the found triggers at
different stages of the pipeline for several runs, with the starting frequency
for the matched-filter integral being either 30 or 40Hz and the integration
stopping at three different frequencies —ISCO, light ring and LRD— displayedin ascending order. We have separated our results in two sections, according
to performance in recovering the full set of 126 NINJA injections and the
reduced set of 67 injections whose total mass falls below 160 M . This choice
is motivated by the construction of the phenomenological bank.
A time window of 120 ms has been used in order to cluster the triggers
found by the pipeline in a single detector. Similarly the coincidence has been
determined within a 80 ms injection window. Given these choices for the parameters used in clustering the triggers, we report recovery of 80/126 triggers
in double or triple coincidence for the full injection set and 48/67 triggers forthe reduced set with M < 160 M . These are triggers that survive the second
coincidence stage (including the signal-based vetoes) for our best run, whichcorresponds to the matched-filter integral starting at 30Hz and ending at the
LRD frequency. The number of recovered triggers in the full mass range iscompatible with the results quoted by other participants in the NINJA project
employing searches with higher-order corrections PN templates extended up
to larger frequencies. The efficiency of the search improves, however, when7.2 search with a phenomenological template bank131

Figure 40: Found and missed injections in the mass region 32 M 6 M 6 160 M
as a function of their total mass and distance for our best search, startingat 30 Hz and stopping at the LRD frequency. The circles represent triggers
that were recorded as either double or triple coincidences after the second
stage (including the signal-based vetoes), whereas the triangles represent
missed injections. The colour code displayed in the vertical scale represents
the modulus of the dimensionless total spin |S~1 /m21 + S~2 /m22 | of the black
holes. The red border of the triangles serves solely as visual aid to facilitate
their quick identification as missed injections.

we restrict ourselves to signals with masses overlapping those of our template bank. It is worthwhile noting that among the triggers recovered by thepipeline we find not only non-spinning simulations but also signals with nonprecessing spins, such as the CCATIE and BAM_BBH waveforms (except in thecase of equal S/m2 = 0.25 spins aligned in the z-direction, which we discusslater). This supports existing evidence for the fact that non-spinning templates
should be able to detect non-precessing spinning signals with moderate individual spins. The capability of non-spinning templates for recovering signals
with precession and large spins could however be compromised, as we discuss
below. Due to the low statistics of the present analysis, these statements should
be taken with the appropriate reservations.
Figure 40 provides an overview of the found and missed injections corresponding to total mass below 160 M . The colour code encodes the modulusof the dimensionless total spin |S~1 /m21 + S~2 /m22 | of the black holes, and gives
an indication of the injections that significantly deviate from the non-spinningcase modelled by the phenomenological waveforms. We observe how signals
located at distances above 350Mpc are systematically lost, giving us an indication of the distance reach of the pipeline; nevertheless, several nearby
injections are missed as well. In order to track down the missed injectionsin the mass region below 160 M , a compilation of their relevant physicalparameters and associated information is given in Table 13. A similar analysis of the missed and found injections has been recently performed by theHere S is the modulusof the angularmomentum ~S (which inthis particular BAM
simulation is the samefor both black holes).

132

numerical injections in gravitational searches

Table 13: Overview of the 19 missed injections with total mass below 160 M for thebest run reported. The ID column stores an index that identifies each of theinjections of the NINJA set. The last column displays the modulus of the sumof the black holes individual spins. Among the missed signals we stress thepresence of waveforms with eccentricity, large spins and precession and also
those injected at distances further than 350 Mpc. Note that the IDs enclosed
in asterisks correspond to signals also reported as missed in [50], wherea Bayesian inference search on the NINJA data using a Nested Samplingalgorithm is presented.

ID

NR

Tot. mass

Distance

Eff. dist.

Simulation

(M )

(Mpc)

H / L (Mpc)ηTot. SpinP| S~i /m2i |

136

LazEv

94.6

444.5

15831.9 / 2941.3

0.25

1.84

*141*

LazEv

75.6

355.8

1047.2 / 746.6

0.25

1.84

*142*

LazEv

129.7

442.5

2221.8 / 1537.7

0.25

1.84

*59*

CC

69.7

87.5

469.2 / 1573.0

0.25

0

*41*

BAM_HHB_S25

112.0

212.1

1150.8 / 802.2

0.25

0.5

47

BAM_HHB_S00

150.0

131.0

648.5 / 908.0

0.25

0

27

BAM_FAU

77.8

121.3

764.1 / 741.1

0.25

1.43

*29*

BAM_FAU

72.3

185.1

1325.7 / 885.4

0.25

1.43

*30*

BAM_FAU

73.9

351.1

896.9 / 771.4

0.25

1.43

114

PU_T52W

82.6

133.1

364.0 / 320.2

0.25

0

116

PU_T52W

88.2

164.2

654.7 / 533.7

0.25

0

118

PU_T52W

90.0

86.0

1055.4 / 452.3

0.25

0

120

PU_T52W

108.6

106.0

202.0 / 205.9

0.25

0

*125*

PU_T52W

96.8

258.8

463.8 / 464.7

0.25

0

*126*

PU_T52W

105.8

398.5

1175.2 / 1797.1

0.25

0

*64*

GSFC_X4

134.7

149.6

1320.4 / 856.0

0.16

0

*68*

GSFC_X3

160.0

220.8

819.3 / 1123.2

0.1875

0.222

*76*

GSFC_X4

158.0

145.4

722.2 / 558.5

0.16

0

*95*

Lean_c138

68.6

246.5

333.9 / 407.7

0.16

0

Total number of missed injections: 19

Birmingham group in [50], applying Bayesian inference on the NINJA datausing a Nested Sampling algorithm. The work of Aylott explores how different
waveform families affect the confidence of detection of NR waveforms. Their
Bayes factor B is a metric for assessing the level of confidence that a signal
has been detected, and their defined thresholds for log10 B allow for classification of the signals as found or missed. The IDs displayed in bold type in our
Table 13 correspond to signals that are reported as missed by the Birmingham
group in Table 2 of [50]; 12 of our 19 missed injections are also lost by them,a correlation that seems worth following up. Future versions of the NINJA
project will certainly benefit from combined searches and cross-checks of thiskind between different DA methods.
Signals with large eccentricity, such as the Princeton e & 0.5 PU_T52W
run are invariably lost by our pipeline. Likewise, the phenomenological templates are not able to pick up signals with considerable spin, such as theequal-mass, spinning waveforms from LazEv with individual spins of equal7.2 search with a phenomenological template bankvalue S/m2 = 0.92 aligned along the z-axis and BAM_FAU with precessing
spins S/m2 = 0.75 outside the xy-plane and misaligned with respect tothe z-direction. Our template bank was developed to search for signals inwhich spin is unimportant and no precession is present, so these results areunderstandable. Further work targeted to incorporating spins within the phenomenological model is desirable and will be undertaken in the future. The
missed GSFC signals with mass ratio 1 : 3 and 1 : 4 correspond to runs with few
orbits before merger and moreover they are injected at total masses borderingon the edge of our template bank, which could explain them being lost. Morebewildering is however the fact that the pipeline misses a couple of long equalmass non-spinning simulations injected at rather close distances, such as SpEC,
which is also missed by the search reported in [50], and BAM_HHB_spp00. A
look at the columns of Table 13 that list the effective distance in H and L (thedistance to an equivalent source with optimal location and orientation) indicates that it might be the poor orientation of these injections that prevents thepipeline from finding them. Aside from these individual cases, which would
need a careful follow-up that is below the scope of this paper, we can justify therest of missed injections as those either placed at large distances, presenting
large spin values and/or precession and containing few orbits before merger.
Among the obvious improvements that a search with phenomenologicaltemplates could benefit from we can mention the following. Firstly, and once
the technical issue with the generation of high-mass templates is resolved, thesearch would clearly improve with the use of a template bank that fully covers
the parameter space of the signals searched for. Additionally, the inclusionof the fourth interferometer V1 in our pipeline shall provide a larger numberof recovered triggers, in the manner reported by the search using EOBNR
templates that is described in Section 4.1.3 of [49]. Both improvements will
be most likely incorporated to searches with the phenomenological templatebank in future realizations of the NINJA project.
7.2.4 Accuracy for Parameter Estimation
The number of found versus missed triggers is not the only relevant metricfor assessing the performance of the standard GW searches. If astrophysically
relevant statements are to be made from GW observations, the ability ofaccurately estimate the physical parameters of the measured signals is crucial.
The inspiral pipeline returns estimated values for the individual and total
masses of the detected system, effective distance, coalescence time and event
duration, among others. Figure 41 shows two parameter estimation plots
for the phenomenological search on the NINJA injections. The left panel
displays the fractional difference for recovery of the chirp mass of the system.
The vertical colour bar encodes the total mass in solar masses. We report
substantial improvement in parameter estimation with respect to LIGO/Virgostandard searches that make use of PN templates, which we can see in Figure
8 of the NINJA paper [49]. While standard PN searches recover most of theinjections with a fractional difference in chirp mass of 0.5 or above, the left
panel of our Figure 41 shows an overall better accuracy, with the exceptionof one outlier. Using the phenomenological template bank, the chirp mass isrecovered within a 20% accuracy for values of M below 40 M and ∼ 40% forsignals with larger chirp mass. In any case, it should not be forgotten that the
PN searches reported in [49] make use of banks with masses up to 90 M

133

134

numerical injections in gravitational searches

Figure 41: Accuracy in the recovery of chirp mass and end time (left and rightpanel respectively) of the reduced set of NINJA injections with total
mass below 160 M using the phenomenological template bank. In bothplots, the colour scale is given by the total mass of the system. The
chirp mass, a common quantity in data analysis which is defined as
M = (m1 m2 )3/5 (M)−1/5 , is typically recovered within a 20%–40%
accuracy, depending on the chirp mass and total mass of the system. The
results for parameter estimation with our IMR bank constitute a significant improvement over current LIGO/Virgo searches with standard PN
templates.

only. The outlier that can be spotted at M ∼ 30 M corresponds to a SpEC
waveform injected at 132Mpc with total mass 56.6 M . For this particular
injection the accuracy in parameter recovery is rather poor, and further work
to understand this behaviour will be undertaken in the future.
The panel in the right shows the accuracy in end time recovery of thefound signals, with the colour code again displaying the total mass of thesystem. The injection time of the numerical waveforms corresponds to the
maximum of their amplitude, which happens roughly at the merger of the two
black holes. Current PN templates stop before that point while IMR templates
extend beyond the merger into the ringdown. The sign convention for ∆tend
corresponds to the injected minus the recovered parameters, so that a trigger
that presents a positive value ∆tend > 0 indicates that the signal was reallyinjected at a later time than the value recorded by our pipeline. Most of thesignals displayed in the left panel of Figure 41 are recovered at a time within a
few hundredths of second from the injected end time value, with the outliers
corresponding partially to signals with larger total mass. These results are
consistent with the other IMR search reported in the NINJA paper. Again
we expect that the use of a template bank overlapping the mass region of theinjections would lead to a reduction of the outliers, but certain improvementwith respect to the left panel in Figure 8 of [49] can still be acknowledged.
Even though the number of total recovered triggers for the phenomenological search on the NINJA data is similar to the results quoted by the standard
PN searches, there is a reasonable indication that the use of a full IMR template bank helps the estimation of the physical signal parameters. In view ofthese and other coincident results quoted in [49, 50, 115], we conclude thatsearches that attempt to recover and estimate the physical parameters of BBH
signals in the mass range 102 − 103 M would profit from using an IMR
template bank that fully models the inspiral, merger and ringdown of thebinary system. This is of crucial important for future LIGO/Virgo searches7.3 looking ahead: the ninja 2 project

135

that aim at targeting coalescences of compact objects in the above-mentioned
range, for which a full template bank adapted to arbitrarily high masses (andideally also to non-zero spin values) needs to be developed. Attempts in thisdirection are already underway within the LIGO/Virgo collaboration.
7.3

looking ahead: the ninja 2 project

The second NINJA project, aimed at testing the gravitational wave analysis
pipelines’ sensitivity to numerical waveforms will build upon the success ofthe first NINJA project while extending it to be a more systematic test, using
real GW detector data and will proceed in a similar manner as for NINJA 1,
by building a collaboration of all interested researchers to tackle this problem.
Other NINJA projects, including one focused on matter, are envisioned tooccur in parallel, although not using real data.
The ultimate goal of this sort of projects is to develop an optimal searchand parameter estimation approach for the full BBH coalescence, including
rapidly spinning black holes. NINJA 2, the second phase of the NINJA project,
expected to start in summer 2010, is widely regarded as the key means bywhich these investigations will be conducted by the international community.
The scope of NINJA 2 will be much broader than NINJA 1: longer and more
numerous spinning and non-spinning signals —including the model developed by the author and introduced in chapter 5— will be injected in old
LIGO data containing glitches and non-Gaussian features, which make theanalyses more realistic.
The analysis of the NINJA 2 data will be a challenging task so far never
explored by the current search pipelines: extension to spins and injection offull IMR waveforms. The presence of spins increases the dimensionality ofthe problem from 9 to 15. The study of how this enlarged parameter space
impacts the searches poses a challenge, which will have to be undertaken byparts, first understanding the effect of including non-precessing spin templates
and later addressing the general, misaligned spin case, as the corresponding
NR simulations become available. Furthermore, accurate estimation of thephysical parameters of the binary is crucial in order to maximize the science
exploitation of the data. Extensive parameter estimation studies are neededin order to establish the reliability of our search pipelines for recovering thecorrect physical information of the binary systems.
The final goal is to identify and conceptually develop a full analysis approachfor BBH detection and parameter estimation that can be ported into the LSCVirgo analysis. The techniques and analyses needed to bring the full NINJA 2
project to completion fall outside the scope of this thesis, but some of themethods developed in this work will be of direct relevance for forecoming
research in this direction.

https://www.
ninja-project.org/
doku.php?id=ninja2:
home

8

ASTROPHYSICS OF INTERMEDIATE-MASS BLACK HOLES

Experimental evidence exists to support the existence of both solar-mass andsupermassive black holes with masses ranging between ∼ 1.4 − 20 M and∼ 105−9 M respectively. Gravitational-wave searches targeted to observing
the former have been discussed in chapter 6. Detection of the latter should beenabled once the space antenna LISA flies. Quite to the contrary, the existenceof intermediate-mass black holes (IMBHs) with masses of roughly 102−4 M
has not yet been corroborated observationally, despite the high interest thatthese objects have for astrophysics. Our understanding of formation and evolution of supermassive black holes, as well as galaxy evolution modeling andcosmography would dramatically change if an IMBH was observed. Froma point of view of traditional photon-based astronomy, which relies on themonitoring of innermost stellar kinematics, the direct detection of an IMBH
does not seem plausible until the next-generation telescopes start operating
within the next decade. However, the prospect of detection and characterization of an IMBH has good chances in lower-frequency gravitational-wave
astrophysics with ground-based detectors such as LIGO, Virgo and the future
Einstein Telescope. The prospects for IMBH detection and characterization
with ground-based gravitational-wave observatories would not only provide
us with a robust test of general relativity, but would also corroborate the existence of these systems. Such detections should allow astrophysicists to probe
the stellar environments of IMBHs and their formation processes.
This chapter discusses the mechanisms that might give rise to one or more
IMBHs inside globular stellar clusters, and provides an estimation of the
expected event rates for binaries of IMBHs that future gravitational-wave
observatories might observe; the waveform model developed in chapter 5 isemployed to estimate the sensitivity of the detector to these systems; finally,
we explore the implications that a detection would have.

This chapter
summarizes the work
presented in [39]8.1 formation of an intermediate-mass black hole
By following the stellar dynamics at the center of our Galaxy, we have now
the most well-established evidence for the existence of a SMBH. The close
examination of the Keplerian orbits of the S-stars has revealed the nature of
the central dark object located at the Galactic Center. By following S2 (SO2),
the mass of SgrA∗ was estimated to be about 3.7 × 106 M within a volume
with radius no larger than 6.25 light-hours [234, 128]. More recent data basedon 16 years of observations set the mass of the central SMBH to ∼ 4×106 M[111, 129, 130, 131].
Massive black holes in a lower range of masses may exist in smaller stellar
systems such as globular clusters. These are called intermediate-mass black
holes because their masses range between M ∼ 102−4 M , if we assume
that they follow the observed correlations between SMBHs and their hoststellar environments. Nevertheless, the existence of IMBHs has never been
confirmed, though we have some evidences that could favor them.

137

These also called
SO-stars, where theletter “S” stands simply
for source

See [184, 183] andreferences therein

138

astrophysics of intermediate-mass black holes

10-19

Initial LIGO
Initial Virgo
Adv LIGO base
Adv LIGO BHBH
Adv Virgo
ET baseET Xylophone

æ
æ

à

10

æ
á

à
à
á

10-21
á
á
10-22

10-23
200M 

300M

10-24

600M

1000M
Sn HfL and 2ÈhHfLÈ f H1 Hz L

à
æ

-20

10-25
1

10

100

1000

f HHzL
Figure 42: Amplitude of equal-mass, non-spinning BBH systems scaled to various
total masses of the IMBH binary compared with the sensitivity curvesof various detectors. The sources are optimally oriented and placed at
100 Mpc of the detectors. The symbols on top of the waveforms markvarious stages of the BBH evolution: solid circles represent fISCO , squares
the fLR and open squares fLRD (see section ). The sensitivity curves ofcurrent and future ground-based detectors are shown as well.

This is, by following thestellar dynamics
around it

If we wanted to apply the same detection technique to detect IMBHs in
globular clusters as we do with SMBHs in galactic centers, ultra-precise astronomy would be required, since the sphere of influence of an IMBH is ∼ few arc
seconds. The number of stars enclosed in that volume is only a few. Currently,
with adaptive optics, one can aspire to have a couple of measurements ofvelocities if the target is about ∼ 5 kpc away in the time basis of 10 yrs. The
measures depend on a number of factors, such as the required availability ofa bright reference star, in order to have a good astrometric reference system.
Also, the sensitivity limits correspond to a K-band magnitude of ∼ 15 — BMS stars at 8 kpc, like e.g. S2 in our Galactic Center.
This means that, in order to detect an IMBH or, at least, a massive dark
object in a globular cluster center with traditional astronomy, one has to resort
to the Very Large Telescope interferometer and to one of the next-generation
instruments, the VSI or GRAVITY [132, 112]. In this case we can hope to
improve the astrometric accuracy by a factor of ∼ 10. Only in that scenario wewould be in the position of following closely the kinematics around a potential
IMBH, so as to determine its mass.
Incidentally, gravitational-wave astronomy could contribute to IMBH detection. Current and future ground-based interferometers are sensitive to thefrequencies associated to the late inspiral, merger and ringdown of coalescing
binaries of IMBHs. Figure 42 illustrates this fact. Systems with total mass

8.2 life of a massive binary

139

above 600 M fall almost completely below the 40 Hz “seismic wall” of theinitial LIGO detectors; however they will become very interesting sources
for the second generation of GW interferometers and the proposed Einstein
Telescope. Indeed, they will also be seen by the future space-borne LISA.
Next section 8.2 expands the astrophysical context to this problem andgives a description of the different efforts made to address the evolution of a
black-hole binary in a stellar cluster, from its birth, to the final coalescence. In
section 8.3 we compute the reach of the detectors for systems of IMBH binaries,
which is needed for the calculation of the expected event rates presented in
section 8.4.
8.2 life of a massive binary
The aim of this section is not to give a detailed explanation of the processes offormation of IMBHs and binaries of IMBHs (BBHs), but a description of the
global picture so as to introduce the two different scenarios that play a role in
the formation of BBHs.
8.2.1

Birth

Up to now, the IMBH formation process which has drawn more attention
is that of a young cluster in which the most massive stars sink down to thecenter due to mass segregation. There, a high-density stellar region builds
and stars start to physically collide. One of them gains more and more massand forms a runaway star whose mass is much larger than that of any otherstar in the system. Later, that runaway star may collapse and form an IMBH
[210, 123, 209, 121].
In particular, Freitag et al. [122, 121] described the requirements from thepoint of view of the host cluster to form an IMBH in the center of the system.
By starting with a cluster of main-sequence stars with a determined initialmass function, the authors find that, after the cluster reaches core-collapse due
to mass segregation in the system, if there are not too hard binaries, the time
to reach core collapse is shorter than 3 Myrs and the environmental velocity
dispersion is not much larger than ∼ 500 km s−1 , the runaway formation ofa very massive star (VMS) is possible. Not yet well understood are the later
evolution of the VMS and the conditions to impose upon it, so that it does
not evolve into a super-massive star (SMS) in this particular scenario. Also
not completely clear are the factors that could limit the mass of such an object
so that it could collapse and turn into an IMBH. The process depends on anumber of factors and assumptions, such as e.g. the role of metallicity,stellar
winds and the collisions on to the runaway star from a certain mass upwards.
On the other hand, Suzuki et al. [248] investigated the process of growing up
of a runaway particle by coupling direct N−body simulations with smooth
particle hydrodynamics (SPH) to analyze the evolution of the star and foundthat stellar winds would not inhibit the formation of a very massive star.
More recently, the effects of the stellar evolution on the runaway collision
product have been considered by analyzing the succession of collisions from a
dynamical evolution[133]. It is found that for low-metallicity models, the final
remnant of the merger tree is expected to explode as a supernova, and in theirhigh-metallicity models the possibility of forming an IMBH is negligible andend up with a mass of 10–14 M at the onset of carbon burning. Nevertheless,A very massive star(VMS) is a star of masslarger than  100 M
See forinstance [37, 38, 33] andreferences therein
See e.g. [56], though it
is rather unclear how to
extrapolate their results(which are limited tostars with masses ofmaximum 150 M ) to
the masses found in the
runaway scenario,
which are typically atleast one order ofmagnitude larger

140

See [35] for a detailed
explanation of theprocess

astrophysics of intermediate-mass black holes

these develop an extended envelope, so that the probability of further collisions
is higher. The authors did not change the masses accordingly in the dynamical
simulation. In any case, self-consistent direct-summation N-body simulations
with evolution of the runaway process are called in to investigate the final
outcome. Hence, assuming that they form, we can theoretically explain theformation of a binary of two IMBHs (BBH) in a cluster in two different ways.
(i) The double-cluster channel:
In this scenario, two clusters born in a cluster of clusters, such as thosefound in the Antennæ galaxy, are gravitationally bound and doomed to collide.
When this happens, the IMBHs sink down to the center of the resulting mergedstellar system due to dynamical friction. They form a BBH whose semi-major
axis continues to shrink due to slingshot ejections of stars coming from thestellar system. In each of the processes, a star removes a small fraction of theenergy and angular momentum of the BBH, which becomes harder. At laterstages in the evolution of the BBH, GW radiation takes over efficiently andstarts to circularize, though one can expect these systems to have a residual
eccentricity when entering the LISA band [35]. This detector will typically beable to see systems of binaries of IMBHs out to a few Gpc. For this channel
and volume, the authors estimated an event rate of 4 − 5 yr−1 .
(ii) The single-cluster channel:
Gürkan et al. [138] added a fraction of primordial binaries to the initialconfiguration in the scenario of formation of a runaway star in a stellar cluster.
In their simulations they find that not one, but two very massive stars form inrich clusters with a binary fraction of 10%. Fregeau et al. [120] investigatedthe possibility of emission of GWs by such a BBH and estimated that LISA
and Advanced LIGO can detect tens of them depending on the distribution ofcluster masses and their densities. More recently, Gair et al. [124] addressed
the event rate that the proposed Einstein Telescope could see and quoted
a few to a few thousand events of comparable-mass IMBH mergers of thesingle-cluster channel.
8.2.2 Growing Up (Shrinking Down): The role of triaxiality on Centrophilic
OrbitsIn the case of the double-cluster channel, the cluster, which is in rotation,
results from the merger of the two initial clusters and may develop a triaxiality
sufficient to produce enough centrophilic orbits. These boxy orbits are typicalof systems that do not possess a symmetry around any of their axes [57].
On the contrary to loop orbits, a characteristic of spherically symmetric or
axisymmetric systems, boxy orbits bring stars arbitrarily close to the centerof the system, since it oscillates independently along the three different axes.
Therefore, such stars, due to the fact of being potential sling-shots, can feed
the process of shrinkage of the BBH semi-major axes by removing energy
and angular momentum out of it after a strong interaction. In the strong
triaxial systems of [57], the rotation caused in the process of merger creates an
unstable structure in the form of a bar. Within the bar the angular momentum
will not be conserved and thus the BBH loss-cone is full due to the stars on
centrophilic orbits, independently of the number of stars N? . In the modelsof [35], the initial conditions are a realistic parabolic merger of two stellar
clusters. The resulting merged cluster does not show the strong axisymmetryof [57]. In the simulations we address for the results of this work, the BBH of

8.2 life of a massive binary

141

1.00
0.95

ba, ca

0.90
0.85
0.80
0.75
0.70
0.65

Mass fraction:
200

300

400

0.1

500

0.3

0.5

600

0.8
700

Time @N-bodyD
1.0

ba

ba, ca, Τ

0.9

Τ=Hb-cLHa-cL

0.8

ca
0.7
0.6

Mass fraction:

0.5

0.5
200

300

400

500

600

700

Time @N-bodyD

Figure 43: Triaxiality of the resulting merged cluster for different mass fractions(upper panel) and the mass fraction 0.5. We calculate the semi-major axesof the ellipsoid of inertia a, b and c (where a > b > c) according to four
different mass fractions which, in turn, are distributed on the basis of theamount of gravitational energy. The shorter the distance to the center ofthe resulting cluster, the lower the mass fraction. Displayed are b/a (solidlines) and c/a (dashed lines). The lower panel shows the shape indicators
for the mass fraction 0.5, together with the evolution of the parameter τ,
an indicator for the triaxiality of the system, which tends to one as time
elapses; i.e. the system tends to be oblate. The evolution of τ is similar forthe rest of mass fractions

IMBHs is not stalling, in spite of the reduced number of centrophilic orbits
due to the architecture of the stellar system.
In Figure 43 we show the role of the cluster symmetry explicitly by depicting
the evolution of the triaxiality of the cluster formed as a result of the mergerof the two clusters for our fiducial model in the case of the double-clusterchannel. After a merger which is the result of a parabolic orbit, the final
system is oblate rather than prolate; i.e. a ∼ b > c, where a, b and c are
the cluster axes. At the outskirts the resulting merged cluster is flatter and at
the center the binary of IMBHs makes it rather spherical. Amaro-Seoane etal. [36] addressed the single-cluster channel scenario after the formation of
the IMBHs and used additional simulations to further evolve the BBH. They
used scattering experiments of three bodies including relativistic precession to
1st post-Newtonian order, as well as radiation reaction caused by GW, so thatthey did not have to integrate every single star in the cluster to understand theposterior evolution of the BBH. In their work, between the strong encounters,a and e of the BBH were evolved by resorting to the quadrupole formulæ
of [203].

This is the reference
model of [35]

142

astrophysics of intermediate-mass black holes

The BBH will have completely circularized when it reaches the frequencies
probed by Advanced LIGO and the ET, because the emission of GWs takes
over the dynamics of the system.
8.2.3 Death
While the emission of GWs is present all the time from the very first momentin which the BBH is formed, the amplitude and frequency of the waves is
initially so low that no present or planned detector would be able to register
any information from the system. Only when the semi-major axis shrinks
sufficiently, the frequency increases enough so as to enter the LISA band, which
we assume starts at 10−4 Hz. The BBH then crosses the entire detector window
during its inspiral phase, as we can see in Figure 44. We depict the evolutionof a BBH of mass 439.2 + 439.2 M . The reason for this particular choiceof masses is to give the reader a point of reference to understand the whole
picture. Recently, Amaro-Seoane et al. [34] included the effect of rotation of
the host cluster and addressed the dynamical evolution of the global system.
The authors have shown that LISA will see the system of Figure 44 with a
median SNR of few tens. The fact that the system merges outside its band
prevents LISA from observing the loudest part of the BBH coalescence. In
order to follow the system at this early stage of its evolution in the LISA band,a simple post-Newtonian approach suffices for modeling the GW radiation.
We are far enough from the highly relativistic regime and only the inspiralphase of the BBH coalescence is visible to the space antenna.
As the binary system depicted in Figure 44 leaves the LISA band andenters the strong field regime, higher order post-Newtonian corrections andeventually input from numerical relativity simulations need to be considered
in order to model the GW waveform. Three reference frequencies in theevolution of a compact BBH that approaches its merger are the innermost
stable circular orbit (fISCO ) defined in equation 2.7 of a test particle orbitinga Schwarzschild black hole, the light-ring frequency (fLR ) corresponding to
the smallest unstable orbit of a photon orbiting a Kerr black hole and thefundamental ringdown frequency (fFRD ) of the decay of the quasi-normal
modes computed by [59].
For the binary system shown in Figure 44, the values of these three frequencies are fISCO |878.4M ' 5 Hz, fLR |878.4M ' 14.2 Hz and fFRD |878.4M '21.4 Hz. Should such a binary exist at a distance of 100 Mpc, and if it was to be
detected with Advanced LIGO, it would produce a sky-averaged SNR of ∼ 450,
assuming a low frequency cut-off of 10 Hz. To that total SNR, the contributionof parts of the inspiral happening before the system reaches the characteristicfrequencies fISCO , fLR and fFRD would be 0%, 37% and 95% respectively. It is
immediately noticed that, for the binaries of IMBHs of interest in this study,
most of the SNR that these binaries will produce in Advanced LIGO comesfrom the last stages of the the BBH coalescence.
We can estimate the time that the binary system takes to evolve fromf = 0.01 Hz, a frequency where the BBH can be seen by LISA, to the lower
cut-off frequency of 10 Hz of Advanced LIGO or of 1 Hz of the ET. A lower
order approximation based on the Newtonian quadrupole formula [203] leads

8.2 life of a massive binary143Sn HfL and 2ÈhHfLÈ f H1 Hz L

10-17
10-18
ì
10-19
0.2

10-20

yrs

ì
-21

10

10-22
10-23
10-24
10-25
10-4

LISA
ET baseAdv LIGO base
H439.2 + 439.2L M  1Gpc

10-3

10-2

10-1

1

10

100

1000

f HHzL
Figure 44: Amplitude of the GW emitted by a system of two equal-mass IMBHs
of total mass 878.4 M placed at 100 Mpc as seen by different√ GW observatories. Note that we have multiplied |h̃(f)| by a factor 2 f, with f
the frequency of the system. From left to right we depict the sensitivity
windows of the future space-borne LISA (dashed, grey curve), the Einstein Telescope (dotted, grey curve) and Advanced LIGO (solid, grey line
starting sharply at 10 Hz). The strain of the BBH of IMBHs spends mostof its inspiral in the LISA band, whilst the ringdown and merger occur at
higher frequencies, only observable by ground-based detectors. Notably,
the ET captures an important extent of the inspiral as well as the whole
ringdown and merger. The averaged SNR produced by this system wouldbe SNRLISA = 854, SNRET = 7044 and SNRAdvLIGO = 450. The BBH
system spends approximately 0.2 yrs to go from f = 0.01 Hz (well into the
LISA band) up to the lower cut-off frequency of Advanced LIGO, 10 Hz.
These two points are pinpointed on the plot

to the following expression for the evolution of the frequency in terms of the
chirp mass M and frequency of the systemdf96= π8/3 M5/3 f11/3 .
dt
5

(8.1)

We find a delay of only 0.2 yrs (80 days) for a BBH with total mass M =878.4 M to go from 0.01 Hz to the beginning of the ET band —taken to
be at 1 Hz— and almost similar numbers to the beginning of the Advanced
LIGO band —at 10 Hz.
In view of these figures, LISA could be used as an alarm to prepare groundbased detectors to register in detail the final coalescence, the death of the
BBH as such, by adjusting their “sweet spots” to the particular BBH. The highaccuracy of which LISA is capable for parameter estimation during the inspiralphase could be combined with the information obtained from the large-SNR
triggers that the BBH merger and ringdown will produce in Advanced LIGO
or ET to achieve a more complete characterization of the system.

The evolution of thesystem is
extraordinarily quick in
the late inspiral phase,
which explains the fastevolution from 1 to
10 Hz

astrophysics of intermediate-mass black holes
200

17.00

100

9.40

50

ET

5.20

20

2.40

10

1.40

5

0.79

AdvLIGOΗ = 0.25, Χ=0
Η = 0.25, Χ=0.75
Η = 0.19, Χ=0
Η = 0.25, Χ=0
Η = 0.25, Χ=0.75
Η = 0.19, Χ=0

2

101

102

103

Redshift z

Luminosity Distance HGpcL

144

0.37

104

Mz M
Figure 45: Sky-averaged horizon distance versus redshifted mass for three binary
configurations obtained with the design sensitivity curves of Advanced
LIGO and the Einstein Telescope. The dashed line corresponds to thehang-up configuration with χ1,2 = 0.75.

8.3 horizon distance and signal-to-noise ratio

Averaging for the whole
sky corresponds todividing the optimalvalue by 2.26
This is the “hang-up”
configuration

This result is confirmedin [220], although they
do not consider thevariation of M with z

See Gair et al. [125]A commonly-used quantity to estimate the reach of a detector is the horizon
distance. It is defined as the distance at which a detector measures an SNR =8 for an optimally-oriented and optimally-located binary, i.e. an overhead,face-on orbit. Non-optimally located and oriented sources are detected with
SNR = 8 at closer distances. At the large distances that Advanced LIGO andthe ET are expected to survey, the calculation of the horizon distance needs
to take into account the redshift and, thus, a cosmological model needs to be
assumed. For the results shown here, we adopt the standard ΛCDM universe
with parameters given by the first five years of the WMAP sky survey [154].
These are ΩΛ = 0.73, Ωb = 0.046, Ωc = 0.23, H0 = 70.5 km s−1 Mpc−1
and t0 = 13.72 Gyr.
Using the full inspiral-merger-ringdown waveforms described in chapter 5
and the corresponding redshift function z(d) for the ΛCDM model, we
compute the sky-averaged horizon distance for non-spinning systems withsymmetric mass ratio η = 0.25, 0.1875 and for an equal-mass system with
spins χ1,2 = 0.75 aligned in the direction of the angular moment. The results
can be seen in figure. 45 for the Advanced LIGO and the ET. We plot thehorizon distance for IMBH binaries in Gpc and z versus the redshifted —observed— total mass Mz ≡ M(1 + z). Firstly, we observe how the detectors’
reach depends on the spins of the binary. “Loud” configurations, such as thehang-up case, can increase the observed distance by a factor of ∼ 2–3. In
the mass range of interest regarding IMBH binaries, i.e. total mass between
200 and 2 × 104 M , we can see that Advanced LIGO will be able to survey
cosmological distances up to z ≈ 2. The Einstein Telescope will be able toprobe much larger distances, possibly up to z > 10. This opens the possibilityof studying seed black hole formation that will provide information about theformation of structure in the early Universe.

8.3 horizon distance and signal-to-noise ratio

145

100000SNR HaverageL

10000

1000

100
Initial LIGO
Initial Virgo
Adv LIGO base
Adv LIGO BHBH
Adv Virgo
ET baseET Xylophone
LISA

10

1

0.1
100

200

500

1000

2000

5000

10 000 20 000

Total mass HML
Figure 46: Signal-to-noise ratio as a function of the total mass of the BBH for thepresent and future generations of GW detectors and LISA. The sources
are placed at a distance of 100 Mpc and the SNRs are angle-averaged

In Figure 46 we show the angle-averaged SNR expected for these sourcesin current and future gravitational-wave detectors as a function of the total
mass of the system. The sources are placed at a distance of 100 Mpc simply
because this is an easily scalable number. The redshift z = 0.023 in this case
can therefore be neglected. For more distant sources however, the total mass
Mz = (1 + z) M would need to be considered, as we do in figure 45.
Unsurprisingly, the SNRs calculated for the third generation of groundbased detectors beat the expectations for initial and Advanced Virgo and
LIGO at all masses. SNRs of the order of 10 are expected for current LIGO
and Virgo interferometers for binaries with total mass up to a few hundredsof solar masses at 100 Mpc. The first-generation detectors are most sensitive
to neutron star binaries and stellar-mass back holes, hence they miss most of
the inspiral part of an IMBH binary coalescence and can only see a fractionof its merger and ringdown phases. Advanced LIGO and Virgo will be able tomeasure averaged SNRs of the order of 102−3 at 100 Mpc, with a maximal
response to BBH systems with total mass in the range of 400 to 1000 M . For
the Einstein Telescope the SNR values are expected to lie within the 103−5
range, and it is expected to be sensitive to binaries with total masses of theorder of 104 M , a significantly larger range than that surveyed by Advanced
LIGO and Virgo. It is noticeable how the ET xylophone configuration increases
the detectability of binaries with masses above 600 M with respect to thebroadband ET configuration. This is due to its improved sensitivity precisely
at frequencies in the range of 1 − 30 Hz, which is where systems of mass
above hundreds of solar masses accumulate most of their SNR. As for LISA,IMBH binaries with masses of hundreds of solar masses will be seen by the
space antenna with a moderate SNR — it is only at masses above tens of
thousands of solar masses that LISA will start taking over the ground-based
observatories, as can be seen in Figure 46. Although the space antenna will

The xylophone
configuration is
described in [151]

See Figure 42

146

Some studies also
include mild
eccentricities, see [34]astrophysics of intermediate-mass black holes

be most sensitive to BBH binaries with masses in the range of 106−7 M ,the possibility that it can act as a complementary observatory to the Einstein
Telescope for IMBH binaries is a very promising one. Parameter accuracy
studies for IMBHs in LISA are already available using the inspiral part ofthe coalescence . There are indications that masses and sky positions will berecovered with a high accuracy level. In order to complete the characterizationof IMBHs with the information given by the second and third generations ofground-based detectors, a comprehensive study of parameter recovery thattakes the BBH coalescence into account is very much desirable.
8.4 event rates

These are typicallystellar black holes
merging with IMBHs

This is equation 2
in [120]

Miller [182] estimated for the first time the event rate for intermediate-massmergers of IMBHs in clusters by calculating the luminosity distance for theinspiral, merger and ringdown [118] out to which these three stages can bedetected with a SNR ratio larger than 10. With no cosmological corrections,the maximum distance for the detector was found to be 3 Gpc (z ∼ 0.53). The
event rate was calculated as
Z
4π
D(M)3 ν(M) nng f(M) dM
(8.2)
R=
3
In this equation nng is the number density of globular clusters, which wastaken to be nng ∼ 8h3 /Mpc3 , as in the work of [210]. The rate of coalescence ofstellar-mass compact objects with the IMBH is ν(M) and f(M) = dN/dM
is
R the mass distribution of massive enough black holes in clusters. Obviously,f(M) dM = ftot < 1. Therefore, everything boils down to the calculationof the maximum distance to which the detector, in this specific case LISA,
can observe these sources; i.e. the observable volume of the detector. The estimation of [118] is used in [182] to find that a few per year should be detectable
during the last phase of their inspiral. Two years later, Will [264] revisited the
problem using matched filtering for the parameter estimation, an updated
curve for the sensitivity of the detector and restricted post-Newtonian waveforms to calculate an analytical expression for the luminosity distance DL . He
finds that the detection rate for binaries in a mass range of 10 M –100 M
is of about 1 per Myr.
A more detailed and updated analysis is given by Fregeau et al. [120]. They
calculated the number of events that LISA and Advanced LIGO could see
from the single-cluster channel. In their estimation, they assume that thevery massive stars formed in the runaway scenario do not merge into one,but evolve separately; each of them eventually form an individual IMBH,
following the numerical results of the Monte Carlo experiments of [138]. Theyfind a generalized form for the event rate observed at z = 0
dNeventR==
dt0

Z zmax0d2 MSFdte dVc
gcl g
dVc dtedt0 dzZ Mcl, max
dN2cl
×dMcl dz.
Mcl, min dMSF, cl dMcl

(8.3)

In this expression, d2 MSF /dVc dte is the star formation rate (SFR) per unitof comoving volume per unit of local time; gcl is the fraction of mass that
goes into the massive clusters of interest; g is the fraction of massive clusters

8.4 event rates

RSF Hh65 M yr-1 Mpc-3 L

0.12

SFR1
SFR2
SFR3

0.10
0.08
0.06
0.04
0.02
0.00
0

1

2

3

4

Redshift z
Figure 47: Three parameterizations of the stellar formation rate per unit comoving
volume compared in the calculation of the event rates. We assume a Λdominated cosmology with H0 = 70.5 km s−1 Mpc−1 = 65 h65 , ΩΛ =0.73 and Ω = 0.27. The three stellar formation rate functions increase
with the redshift until ∼ z = 2; for larger distances they exhibit different
behaviours: SFR1 decreases, SFR2 keeps approximately constant and SFR3
increase. Nevertheless, the rates integrand is not too strongly dependent
on the particular functional form for redshifts z > 2.

which form IMBHs; dte /dt0 = (1 + z)−1 is the relation between local andobserved time; dVc /dz is the change of comoving volume with redshift; anddN2cl /dMSF cl dMcl is the distribution function of clusters over individual
cluster mass Mcl and total star-forming mass in clusters MSF cl .
We calculate the event rate using the values for DL (Mz ) that we obtained
in section 8.3 for Advanced LIGO and the ET, which are based on the waveform model of chapter 5 and depend on the mass ratio and spins of thebinary. We consider three different cases, corresponding to the three different
configurations shown in figure 45. In addition, we compare three different
parameterizations of the stellar formation rate per unit comoving volume,
RSF1,2,3 (z) as given by equations 4, 5 and 6 of [208]. The three models are
depicted in figure 47; they are similar for close distances until ∼ z = 2, differing from there on. For the distribution of cluster masses, the factor can be
expressed as
dN2clf(Mcl )
=R.
dMSF, cl dMcl
Mcl f(Mcl ) dMcl

(8.4)

We take dNcl /dMcl ∝ 1/M2cl , following the power-law form observed foryoung star clusters in the Antenna. The validity of assuming the same law for
the larger volume of the Universe surveyed by Advanced LIGO or the ET is,
however, a generalization not based on direct observations. Unfortunately, amore precise distribution function based on measured data is at present not
known. Thus, we should take this premise with care. By assuming an efficiency
factor of fGC ∼ 2 × 10−3 , based in the results of [138], we can set the values
for the limits based on the masses of the IMBHs and the observable volume

147

148

astrophysics of intermediate-mass black holes

5.0

Redshift z

2.0

ET

1.0
0.5

AdvLIGO
0.2

200

500

1000

2000

5000

1 ´ 1042 ´ 104

M HML
Figure 48: Sky-averaged reach the Advanced LIGO and the ET detectors up to z = 5
versus intrinsic total mass of the IMBH binary. Red lines are for Advanced
LIGO and blue are for the ET. The solid, dotted, dashed curves represent
the same physical configurations displayed in figure 45.

—i.e. the maximum redshift zmax — of the detector of interest. The integral can
now be expressed as
Z MIMBHB, max
fGC g gcl
dMIMBHBR=
ln(Mcl, max )/ ln(Mcl, min ) MIMBHB, min M2IMBHB
Z zmax
dVc
1
×
dz,
(8.5)
SFRi (z) F(z)
(1
+
z)
dz
0
Following the definitionof IMBH, we take
MIMBHB from 200 to2 × 104 M

where MIMBHB, max (min) is the range of total mass of the IMBH binary that
we are considering, SFRi (z), i = 1, 2, 3 is any of the three considered stellar
formation rates of [208] displayed in figure 47 andpΩM (1 + z)3 + Ωk (1 + z)2 + ΩΛF(z) =(8.6)(1 + z)3/2
is the factor that relates the stellar formation rate function in different cosmologies with respect to the Einstein-de Sitter Universe. The maximum value
zmax in the integral on z is a function of the mass of the IMBH system, itsconfiguration —mass ratio, spin— and the particular waveform model used in
the calculation of the horizon distance. The results employing the waveformmodel of chapter 5 can be seen in figure 45.
The maximum values for DL (Mz ) obtained with the ET range betweenz ∼ 5.5 and z ∼ 17. This implies that the ET will be able to probe the different
proposed scenarios to produce the first generation of black hole seeds, as
pointed out by [237]. We note that the values for z that we obtain are significantly larger than assumed in [237]. However, at these very large cosmologicaldistances the stellar formation rate is unknown. We therefore set a maximum
cut-off value of zmax = 5 in the computation of the integral when DL (Mz )exceeds this value. The value of zmax (MIMBH ) that we have used in the computation of the rates integral is show in figure 48 for our three particular physicalconfigurations.

8.4 event rates

149

Table 14: Event rates formed in the single-cluster channel for IMBH binaries potentially observable by Advanced LIGO and the ET per year. We take g = 0.1,
gcl = 0.1 as standard values.

Rh g   g 
clDetector

Configuration

Advanced LIGO

η = 0.25, χ = 048

η = 0.25, χ = 0.75283

η = 0.19, χ = 0

16

η = 0.25, χ = 0868

η = 0.25, χ = 0.75

940

η = 0.19, χ = 0

741

Einstein Telescope

0.1

0.1

yr−1

i

We have evaluated the integral 8.5 for the three stellar formation rates of
figure 47 and the three configurations of figure 45 for Advanced LIGO andthe ET. In agreement with [120], we find that the event rate does not depend
strongly on the assumed stellar formation rate; the differences are negligible
for Advanced LIGO and of a ∼ 15% for the ET. We therefore quote the results
found for SFR2 (z) only.
The event rates do, however, depend on the spins and mass ratio of thebinary. As expected, “loud” configurations like the hang-up case increase
the event rate by a factor of ∼ 6 in the case of Advanced LIGO. Smaller
mass ratios decrease the rate, the difference between η = 0.25 and η =0.1875 being of a factor of 3. The differences are not so extreme in the caseof the ET, due to the fact that we are cutting off zmax at a value of 5 and,thus, neglecting contributions at higher redshift; we lack observational data
to make statements about the stellar formation rate at larger values of z.
Assuming that IMBH binary systems exhibit comparable masses, as it was
found in [138], and random spin values, we can estimate the single-channel
event rate roughly as ∼ 100 yr−1 for Advanced LIGO and ∼ 850 yr−1 for the
ET. The exact computation of the rates would imply further integration on
the mass ratios and spins of the binary system. At present it is not clear what
those distribution might be, therefore we simply summarize our results for all
configurations under consideration in table 14. These rates assume formationof IMBH binaries in the single-cluster channel.
So far, we have concentrated on the single-cluster channel scenario. AmaroSeoane et al. [35] gave a prescription to calculate an estimate of the event rates
for the double-cluster channel by resorting to the detailed calculation of [120].
This was based in the fact that the only difference between both astrophysicalscenarios in terms of the event calculation involves (i) the fact that in the
double-cluster channel one has one single IMBH in one cluster and (ii) thesetwo clusters have to collide so that the IMBHs form a BBH when they sink tothe center due to dynamical friction.
As explained in section 4 of [35], the connection between the event rateestimation of the two channels isΓ doub = Pmerg g Γ sing ,(8.7)

This is the function thatkeeps constant after
z=2

150

astrophysics of intermediate-mass black holes

where Γ doub is the event rate of the double-cluster channel, Γ sing of the singlechannel and Pmerg is the probability for two clusters to collide in the scenarioof [35]. They find Pmerg ∈ [0.1, 1].
The two different works assumed that the probability that a cluster gets into
the runaway phase is g. Fregeau et al. [120] took this value as a parameterbecause of the large uncertainties and set it to 0.1 as an example. Nevertheless,as proven in the simulations of [122], it could be as large as 0.5. We therefore
can define the (absolute) optimistic upper limit and pessimistic lower limit of
the event rates for Advanced LIGO and the ET by assigning all parameters
their maximum and minimum values and summing the contributions of thetwo channels:
total
2
3
−1
ΓAdv.
LIGO ∈ [(0) 1.1 × 10 , 3 × 10 ] yr
total
ΓET
∈ [(0) 9.35 × 102 , 2.55 × 104 ] yr−1

(8.8)
(8.9)

Even though the optimistic upper limit is to be taken carefully, these event
rates are obviously more than encouraging to address the problem of detection
and characterization of systems of IMBH binaries with of GWs. On the other
hand, one should bear in mind that the existence of IMBHs altogether has
not yet been corroborated, so that the pessimistic estimate is still somewhat
optimistic. This is why we have added a (0) in the previous rates as the absolute
lower limit.
8.5 summary and conclusions
The existence of IMBHs is a subject of particular interest in theoretical astrophysics. Even though we do not have any evidence of these objects sofar, a number of theoretical works have addressed their formation in dense
stellar clusters. If we were to follow the same techniques that have led us todiscover the SMBH in our Galaxy, we would need the Very Large Telescopeinterferometer and next-generation instruments, such as the VSI or GRAVITY, which should be operative in the next ∼ 10 yrs. An alternative, or evencomplementary way of discovering IMBHs is via their emission of GWs when
they are in a BBH system.
The identification and characterization of these systems relies on accurate
waveform modeling of their GW emission, which has been made possible dueto the success of numerical relativity in simulating the last orbits of the BBH
coalescence and the coupling of these results to analytical post-Newtonian
calculations of the inspiral phase, as we discuss in this dissertation.
We have computed the sky-averaged horizon distance corresponding to
Advanced LIGO and the proposed ET and the space-based LISA. The results
show that binaries with total mass around to 500 M will be seen by Advanced
LIGO at redshifts as high as z = 1 in the non-spinning case and z = 2 if thespin is significant. The ET will see IMBH binaries up to tens of thousands ofsolar masses up to cosmological distances above z = 5 and as high as z > 15
for particularly loud configurations. This could open the door to explorationof seed black hole formation and other discoveries in the early Universe.
We have revisited the event rate of BBHs for Advanced LIGO and the ET
and found hundreds to tens of hundreds per year for the former and rates oneorder of magnitude larger for the latter. The first of these predictions should be
able to be tested within five years after Advanced LIGO comes into operation.

8.5 summary and conclusions

The observations of Advanced LIGO and the ET in the intermediate-massrage could be complementary to those of LISA, which is expected to detect
these systems with moderate SNRs and to be more sensitive to SMBH binaries
instead. More remarkably, in principle if LISA and the ET are operative at
the same time, they could complement each other and be used to track aparticular event.
Current LIGO and Virgo matched-filter searches for BBH coalescences
are solely targeted to stellar-mass black holes, for those are one of the typesof systems that first-generation ground-based detectors are most sensitive
to. The elevated rates of IMBHs events that we predict for Advanced LIGO
and Virgo and the ET based on our improved waveforms should bring thesemore massive systems to the attention of the GW data analysis community.
Future matched filter searches specifically targeted towards detections of
IMBH binaries with ground-based detectors should be able to shed lightinto the question of their existence and corroborate or invalidate the current
theoretical estimations on their event rate.
Advanced ground-based detectors are designed to be able to operate indifferent modes so that their sensitivity can be tuned to various kinds ofastrophysical objects. Considering the importance of an eventual detection of
an IMBH binary, the design of an optimized Advanced LIGO configuration
for systems with M ∼ 102−4 M would be desirable in order to increase thepossibility of observing such a system. In case an IMBH binary coalescence
was detected, the recovery and study of the physical parameters of the system
could serve to test general relativity and prove or reject other alternative
theories, such as scalar-tensor type or massive graviton theories.
The direct identification of an IMBH with GWs will be a revolutionary
event not only due to the uncertainty that surrounds their existence andtheir potential role to test general relativity. The information encoded in thedetection will provide us with a detailed description of the environment of
the BBH/IMBH.
The information which we will recover from the data analysis of thesesystems, once they have been detected with GWs, will provide us with restrictions on the models which will constrain the various unknowns. Also, by
combining this information with that from forthcoming instruments such as
the Very Large Telescope interferometer and next-generation observatories,
as e.g. VSI or GRAVITY, we will have a more accurate description of the stellar
environment surrounding the IMBH. Thanks to an accurate identificationof the system, we will be in position to “reverse-engineer” the astrophysicalhistory of the stellar cluster, since this will leave a fingerprint in the detected
IMBHs.

151

9

EPILOGUE

This dissertation has provided a global picture of the problem of modellingand detecting the gravitational-wave signature associated to binaries of black
holes. Black-hole binary systems can be approached as a purely theoreticalchallenge, in the sense that they constitute one of the simplest non-trivial
problems to be solved in General Relativity. From that point of view, analyticaland numerical methods are needed to arrive to a full solution for the metric of
the space-time, a solution that even nowadays is available only for a reduced
subset of the most general parameter space. We have presented a full modelfor the coalescence of black hole binaries that covers a particular subset of thefull pararameter space, as a first step towards a further generalization. Our
phenomenological waveform model for non-precessing, spinning binaries
belongs precisely in the category of work targeted towards a proper theoreticalunderstanding of gravitational-wave sources.
But black-hole binaries are also among the most fascinating objects thatastrophysicists expect to observe in our Universe. Direct measurement of thegravitational-wave emission of such objects would turn into a revolution in
the field of astronomy. With that goal, an international scientific community
is gathered around the LIGO and Virgo gravitational-wave detectors, developing the necessary tools to properly analyze and understand their data. We
have presented the results of the latest search for low-mass binary black-holecoalescence on LIGO data and have introduced preliminary results of an
ongoing search for more massive systems with total mass up to hundreds ofsolar masses. But this is by no means the end; the Enhanced and Advanced
configurations of the detectors hold great promise for detection of a binary
system, perhaps of a pair of black holes. The incorporation of up-to-date
results in waveform models for searches of black-hole binary coalescences is,thus, another major focus of this dissertation.
Upon entering the era of gravitational-wave astronomy, the detectors willturn into observatories. Their data will not only confirm or confound our
expectations about the Universe around us, but they also should allow us to test
current hypotheses about expected and conjectured astrophysical populations.
In this sense, predictions such as the existence of intermediate-mass blackhole binaries, the astrophysical background of which has also been discussed
in this dissertation, could eventually be ascertained. We have revisited the
expected astrophysical rates for these kind of objects and stated the prospect
for their detection with gravitational-wave astronomy.
We currently find ourselves on the verge of finishing the analyses of theenhanced interferometers’ output and entering a time of preparation for thedata from the advanced detectors. Collaborations will be formed to devise
and design the most promising data analysis strategies and techniques; input
from scientists working in electromagnetic counterparts will be sought; faster,
automatized searches will be run. Some of the results obtained in this dissertation will be employed in this challege. The final reward will be sensational.

153

BIBLIOGRAPHY[1] Abadie, J. et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), “Predictions for the Rates of Compact Binary Coalescences Observable by
Ground-based Gravitational-wave Detectors”, (2010). (Cited on page 34.)
[2] Abadie, J. et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), “Search for gravitational waves from black hole binary inspiral, merger andringdown”, (2010). In preparation. (Cited on pages 110 and 115.)[3] Abadie, J. et al. (LIGO Scientific Collaboration and Virgo Collaboration),“Sensitivity to Gravitational Waves from Compact Binary Coalescences Achieved
during LIGO’s Fifth and Virgo’s First Science Run”, arXiv e-print, (mar 2010).
[arXiv:1003.2481]. (Cited on page 79.)
[4] Abbott, B. et al. (LIGO Scientific Collaboration), “Search for gravitational waves
from galactic and extra-galactic binary neutron stars”, Phys. Rev. D, 72, 082001, (2005).
(Cited on pages 95 and 100.)[5] Abbott, B. P. et al. (LIGO Scientific Collaboration), “Analysis of LIGO data forgravitational waves from binary neutron stars”, Phys. Rev. D, 69, 122001, (2004). (Cited
on pages 95 and 100.)[6] Abbott, B. P. et al. (LIGO Scientific Collaboration), “Search for gravitational
waves from primordial black hole binary coalescences in the galactic halo”, Phys. Rev. D,
72, 082002, (2005). (Cited on pages 95 and 100.)[7] Abbott, B. P. et al. (LIGO Scientific Collaboration), “Search for gravitational
waves from binary black hole inspirals in LIGO data”, Phys. Rev. D, 73, 062001, (2006).
(Cited on pages 95 and 100.)[8] Abbott, B. P. et al. (The LIGO Scientific Collaboration), Tuning matched filter
searches for compact binary coalescence, (2007). URL (cited on April 13, 2010):
http://www.ligo.caltech.edu/docs/T/T070109-01.pdf. (Cited on page 103.)[9] Abbott, B. P. et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), “Beating the Spin-Down Limit on Gravitational Wave Emission fromthe Crab Pulsar”, Astrophys. J. Lett., 683(1), L45–L49, (2008). (Cited on page 28.)
[10] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “Search for gravitational
waves from binary inspirals in S3 and S4 LIGO data”, Phys. Rev. D, 77, 062002, (2008).
(Cited on pages 65, 95, 100, 102, and 127.)[11] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “Search of S3 LIGO
data for gravitational wave signals from spinning black hole and neutron star binary
inspirals”, Phys. Rev. D, 78, 042002, (2008). (Cited on page 65.)[12] Abbott, B. P. et al. (The LIGO Scientific Collaboration and the Virgo Collaboration), “An Upper Limit on the Stochastic Gravitational-Wave Background of
Cosmological Origin”, Nature, 460, 990, (2009). (Cited on page 28.)[13] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “LIGO: The Laser
Interferometer Gravitational-Wave Observatory”, Rep. Prog. Phys., 72, 076901, (2009).
(Cited on pages 69 and 95.)
[14] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “Search for gravitational
wave ringdowns from perturbed black holes in LIGO S4 data”, Phys. Rev. D, 80, 062001,(2009). (Cited on page 50.)
[15] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “Search for Gravitational
Waves from Low Mass Binary Coalescences in the First Year of LIGO’s S5 Data”, Phys.
Rev. D, 79, 122001, (2009). (Cited on pages 65, 102, 103, 105, 107, 109, and 128.)[16] Abbott, B. P. et al. (The LIGO Scientific Collaboration), “Search for Gravitational
Waves from Low Mass Compact Binary Coalescence in 186 Days of LIGO’s fifth Science
Run”, Phys. Rev. D, 80, 047101, (2009). (Cited on pages 65 and 101.)[17] Abrahams, A. M. and Evans, C. R., “Reading off gravitational radiation waveforms
in numerical relativity calculations: Matching to linearized gravity”, Phys. Rev. D, 37,
318–332, (1988). (Cited on page 66.)
[18] Abrahams, A. M. and Evans, C. R., “Gauge invariant treatment of gravitational radiation near the source: Analysis and numerical simulations”, Phys. Rev. D, 42, 2585–2594,(1990). (Cited on page 66.)

155

156

bibliography[19] Acernese, F. et al., “Virgo status”, Class. Quantum Grav., 25, 184001, (2008). (Cited
on page 69.)
[20] “Advanced LIGO anticipated sensitivity curves”. URL (cited on April 13, 2010):(Cited on
https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=2974.
page 79.)[21] Aguiar, O. D., Andrade, L. A., Barroso, J. J., Bortoli, F., Carneiro, L. A., Castro,
P. J., Costa, C. A., Costa, K. M. F., de Araujo, J. C. N., de Lucena, A. U., de Paula,W., de Rey Neto, E. C., de Souza, S. T., Fauth, A. C., Frajuca, C., Frossati, G.,
Furtado, S. R., Magalhaes, N. S., Marinho, R. M., Melo, J. L., Miranda, O. D.,
Oliveira, N. F., Ribeiro, K. L., Stellati, C., Velloso, W. F. and Weber, J., “TheBrazilian gravitational wave detector Mario Schenberg: status report”, Class. Quantum
Grav., 23(8), S239–S244, (2006). (Cited on page 16.)[22] Ajith, P., “Gravitational-wave data analysis using binary black-hole waveforms”, Class.
Quantum Grav., 25, 114033, (2008). (Cited on pages 66, 71, 91, 111, and 127.)[23] Ajith, P., Babak, S., Chen, Y., Hewitson, M., Krishnan, B., Sintes, A. M., Whelan,
J. T., Brügmann, B., Diener, P., Dorband, N., Gonzalez, J., Hannam, M., Husa,
S., Pollney, D., Rezzolla, L., Santamaría, L., Sperhake, U. and Thornburg, J.,“A template bank for gravitational waveforms from coalescing binary black holes: I.
non-spinning binaries”, Phys. Rev. D, 77, 104017, (2008). (Cited on pages 66, 71, 85, 91,
111, and 127.)[24] Ajith, P., Babak, S., Chen, Y., Hewitson, M., Krishnan, B., Whelan, J. T., Brügmann, B., Diener, P., González, J., Hannam, M., Husa, S., Koppitz, M., Pollney,D., Rezzolla, L., Santamaría, L., Sintes, A. M., Sperhake, U. and Thornburg,J., “Phenomenological template family for black-hole coalescence waveforms”, Class.
Quantum Grav., 24, S689–S699, (2007). (Cited on pages 71, 91, 111, 117, and 127.)[25] Ajith, P., Hannam, M., Husa, S., Chen, Y., Brügmann, B., Dorband, N., Müller,D., Ohme, F., Pollney, D., Reisswig, C., Santamaría, L. and Seiler, J., ““Complete”
gravitational waveforms for black-hole binaries with non-precessing spins”, (2009).
(Cited on pages 66, 71, 87, 91, and 111.)[26] Alcubierre, M., Introduction to 3+1 Numerical Relativity, (Oxford University Press,
2008). (Cited on pages 60 and 62.)[27] Alcubierre, M. and Brügmann, B., “Simple excision of a black hole in 3+1 numerical
relativity”, Phys. Rev. D, 63(10), 104006, (2001). (Cited on page 60.)
[28] Alcubierre, M. et al., “Towards a stable numerical evolution of strongly gravitating
systems in general relativity: The conformal treatments”, Phys. Rev. D, 62, 044034, (2000).
(Cited on pages 121 and 122.)[29] Alcubierre, M. et al., “Gauge conditions for long-term numerical black hole evolutions without excision”, Phys. Rev. D, 67, 084023, (2003). (Cited on pages 121 and 122.)[30] Allen, B., “A χ2 time-frequency discriminator for gravitational wave detection”, Phys.
Rev. D, 71, 062001, (2005). (Cited on pages 98, 99, 103, and 129.)[31] Allen, B., Anderson, W. G., Brady, P. R., Brown, D. A. and Creighton, J. D. E.,“FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries”, (2005). (Cited on pages 103 and 127.)[32] Alvi, K., “Energy and angular momentum flow into a black hole in a binary”, Phys. Rev.
D, 64, 104020, (2001). (Cited on page 42.)[33] Amaro-Seoane, P., “Dynamics of dense gas-star systems. Black holes and their precursors”, (2004). (Cited on page 139.)[34] Amaro-Seoane, P., Eichhorn, C., Porter, E. and Spurzem, R., “Binaries of massive
black holes in rotating clusters: Dynamics, gravitational waves, detection and the roleof eccentricity”, Mon. Not. R. Astron. Soc., 401, 2268–2284, (2009). (Cited on pages 142
and 146.)[35] Amaro-Seoane, P. and Freitag, M., “Intermediate-Mass Black Holes in Colliding
Clusters: Implications for Lower Frequency Gravitational-Wave Astronomy”, Astrophys.
J. Lett., 653, L53–L56, (2006). (Cited on pages 140, 141, 149, and 150.)[36] Amaro-Seoane, P., Miller, M. C. and Freitag, M., “Gravitational Waves from Eccentric Intermediate-Mass Black Hole Binaries”, Astrophys. J. Lett., 692, L50–L53, (2009).
(Cited on page 141.)[37] Amaro-Seoane, P. and Spurzem, R., “The loss-cone problem in dense nuclei”, Mon.
Not. R. Astron. Soc., 327, 995–1003, (2001). (Cited on page 139.)

bibliography[38] Amaro-Seoane, P., Spurzem, R. and Just, A., “Super-Massive Stars: Dense Star-Gas
Systems”, in Lighthouses of the Universe: The Most Luminous Celestial Objects and Their
Use for Cosmology, Proceedings of the MPA/ESO/MPE/USM Joint Astronomy Conference
Held in Garching, Germany, 6–10 August 2001, p. 376. Springer, (2002). (Cited on
page 139.)[39] Amaro-Seoane, P. and Santamaría, L., “Detection of IMBHs with ground-based
gravitational wave observatories: A biography of a binary of black holes, from birth todeath”, (2009). (Cited on page 137.)[40] Apostolatos, T. A., Cutler, C., Sussman, G. J. and Thorne, K. S., “Spin induced
orbital precession and its modulation of the gravitational wave forms from merging
binaries”, Phys. Rev. D, 49, 6274–6297, (1994). (Cited on pages 40, 41, and 108.)[41] Arnowitt, R., Deser, S. and Misner, C. W., “The Dynamics of General Relativity”, in
Witten, L., ed., Gravitation: An Introduction to Current Research, pp. 227–265, (Wiley,
New York, 1962). (Cited on page 54.)[42] Arun, K. G., Buonanno, A., Faye, G. and Ochsner, E., “Higher-order spin effects inthe amplitude and phase of gravitational waveforms emitted by inspiraling compact
binaries: Ready-to-use gravitational waveforms”, Phys. Rev. D, 79, 104023, (2009). (Cited
on pages 41, 45, and 46.)[43] Arun, K. G., Iyer, B. R., Sathyaprakash, B. S. and Sundararajan, P. A., “Parameter
estimation of inspiralling compact binaries using 3.5 post-Newtonian gravitational wave
phasing: The non-spinning case”, Phys. Rev. D, 71, 084008, (2005). (Cited on pages 41,
44, and 45.)[44] Ashby, N., “Relativity in the Global Positioning System”, Living Rev. Relativity, 6(1),(2003). URL (cited on April 13, 2010):
http://www.livingreviews.org/lrr-2003-1. (Cited on page 3.)[45] Ashtekar, A. and Krishnan, B., “Isolated and dynamical horizons and their applications”, Living Rev. Relativity, 7, 10, (2004). URL (cited on April 13, 2010):
http://www.livingreviews.org/lrr-2004-10. (Cited on page 75.)[46] Astone, P., Bassan, M., Bonifazi, P., Carelli, P., Castellano, M. G., Cavallari,G., Coccia, E., Cosmelli, C., Fafone, V., Frasca, S., Majorana, E., Modena, I.,
Pallottino, G. V., Pizzella, G., Rapagnani, P., Ricci, F. and Visco, M., “Long-term
operation of the Rome “Explorer” cryogenic gravitational wave detector”, Phys. Rev. D,
47(2), 362–375, (1993). (Cited on page 15.)[47] Astone, P., Bassan, M., Bonifazi, P., Carelli, P., Coccia, E., Cosmelli, C., Fafone,
V., Frasca, S., Marini, A., Mazzitelli, G., Minenkov, Y., Modena, I., Modestino,G., Moleti, A., Pallottino, G. V., Papa, M. A., Pizzella, G., Rapagnani, P., Ricci,
F., Ronga, F., Terenzi, R., Visco, M. and Votano, L., “The gravitational wave detector
NAUTILUS operating at T = 0.1 K”, Astropart. Phys., 7(3), 231–243, (1997). (Cited on
page 15.)[48] “Australian International Gravitational Observatory”. URL (cited on April 13, 2010):
http://www.gravity.uwa.edu.au/. (Cited on page 19.)[49] Aylott, B. et al., “Testing gravitational-wave searches with numerical relativity waveforms: Results from the first Numerical INJection Analysis (NINJA) project”, Class.
Quantum Grav., 26, 165008, (2009). (Cited on pages 70, 117, 121, 126, 133, and 134.)[50] Aylott, B., Veitch, J. and Vecchio, A., “Bayesian inference on the Numerical INJection Analysis (NINJA) data set using a nested sampling algorithm”, Class. Quantum
Grav., 26, 114011, (2009). (Cited on pages 132, 133, and 134.)[51] Babak, S., Balasubramanian, R., Churches, D., Cokelaer, T. and Sathyaprakash,
B., “A template bank to search for gravitational waves from inspiralling compact binaries
I: physical models”, Class. Quantum Grav., 23, 5477–5504, (2006). (Cited on page 112.)[52] Baker, J. G., Centrella, J., Choi, D.-I., Koppitz, M. and van Meter, J., “Gravitational
wave extraction from an inspiraling configuration of merging black holes”, Phys. Rev.
Lett., 96, 111102, (2006). (Cited on pages 33, 53, 63, 65, 69, and 121.)[53] Baker, J. G., van Meter, J. R., McWilliams, S. T., Centrella, J. and Kelly, B. J.,“Consistency of post-Newtonian waveforms with numerical relativity”, (2006). (Cited
on page 70.)[54] Baumgarte, T. W. and Shapiro, S. L., “On the Numerical Integration of Einstein’s
Field Equations”, Phys. Rev. D, 59, 024007, (1999). (Cited on page 57.)
[55] Beetle, C., “Approximate Killing Fields as an Eigenvalue Problem”, arXiv e-print,(2008). [arXiv:0808.1745 [gr-qc]]. (Cited on pages 74 and 75.)

157

158

bibliography[56] Belkus, H., Van Bever, J. and Vanbeveren, D., “The Evolution of Very Massive Stars”,
Astrophys. J., 659, 1576–1581, (2007). (Cited on page 139.)[57] Berczik, P., Merritt, D., Spurzem, R. and Bischof, H.-P., “Efficient Merger of BinarySupermassive Black Holes in Non-Axisymmetric Galaxies”, Astron. J., 642, L21, (2006).
(Cited on page 140.)[58] Berti, E., Cardoso, V., Gonzalez, J. A., Sperhake, U. and Brügmann, B., “Multipolar
analysis of spinning binaries”, Class. Quantum Grav., 25, 114035, (2008). (Cited on
page 45.)[59] Berti, E., Buonanno, A. and Will, C. M., “Testing general relativity and probing themerger history of massive black holes with LISA”, Class. Quant. Grav., 22, S943–S954,(2005). (Cited on page 142.)[60] Berti, E. et al., “Inspiral, merger and ringdown of unequal mass black hole binaries:
A multipolar analysis”, Phys. Rev. D, 76, 064034, (2007). (Cited on page 67.)[61] Bildsten, L., “Gravitational radiation and rotation of accreting neutron stars”, Astrophys.
J., 501, L89, (1998). (Cited on page 28.)[62] Biswas, R., Brady, P. R., Creighton, J. D. E. and Fairhurst, S., “The Loudest Event
Statistic: General Formulation, Properties and Applications”, Class. Quantum Grav., 26,
175009, (2009). (Cited on page 105.)[63] Blair, D. G., Ivanov, E. N., Tobar, M. E., Turner, P. J., van Kann, F. and Heng, I. S.,“High Sensitivity Gravitational Wave Antenna with Parametric Transducer Readout”,
Phys. Rev. Lett., 74(11), 1908–1911, (1995). (Cited on page 15.)[64] Blanchet, L., “Gravitational Radiation from Post-Newtonian Sources and Inspiralling
Compact Binaries”, Living Rev. Relativity, 9(4), (2006). URL (cited on April 13, 2010):
http://www.livingreviews.org/lrr-2006-4. (Cited on pages 31, 40, and 41.)[65] Blanchet, L., Buonanno, A. and Faye, G., “Higher-order spin effects in the dynamics
of compact binaries II. Radiation field”, Phys. Rev. D, 74, 104034, (2006). (Cited on
pages 41 and 42.)[66] Blanchet, L., Damour, T., Esposito-Farèse, G. and Iyer, B. R., “Gravitational radiation from inspiralling compact binaries completed at the third post-Newtonian order”,
Phys. Rev. Lett., 93, 091101, (2004). (Cited on page 41.)[67] Blanchet, L., Faye, G., Iyer, B. R. and Sinha, S., “The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspirallingcompact binaries in quasi-circular orbits”, Class. Quantum Grav., 25, 165003, (2008).
(Cited on pages 41, 44, and 45.)[68] Bona, C., Masso, J., Seidel, E. and Stela, J., “A New formalism for numerical relativity”, Phys. Rev. Lett., 75, 600–603, (1995). (Cited on page 60.)
[69] Boyle, M. et al., “High-accuracy comparison of numerical relativity simulations with
post-Newtonian expansions”, Phys. Rev. D, 76, 124038, (2007). (Cited on pages 42, 63,
65, 70, 121, and 122.)[70] Boyle, M. et al., “High-accuracy numerical simulation of black-hole binaries: Computation of the gravitational-wave energy flux and comparisons with post-Newtonian
approximants”, Phys. Rev. D, 78, 104020, (2008). (Cited on page 70.)[71] Brady, P. R., Creighton, J. D. E. and Wiseman, A. G., “Upper limits on gravitationalwave signals based on loudest events”, Class. Quantum Grav., 21, S1775–S1781, (2004).
(Cited on page 105.)[72] Brady, P. R. and Fairhurst, S., “Interpreting the results of searches for gravitational
waves from coalescing binaries”, Class. Quantum Grav., 25(10), 1050002, (2008). (Cited
on pages 105 and 106.)[73] Brandt, S. and Brügmann, B., “A Simple Construction of Initial Data for Multiple
Black Holes”, Phys. Rev. Lett., 78(19), 3606–3609, (1997). (Cited on page 61.)[74] Brill, D. R. and Lindquist, R. W., “Interaction Energy in Geometrostatics”, Phys. Rev.,
131(1), 471–476, (1963). (Cited on page 61.)[75] Brown, D. A., Fairhurst, S., Krishnan, B., Mercer, R. A., Kopparapu, R. K., Santamaría, L. and Whelan, J. T., “Data formats for numerical relativity waves”, arXiv
e-print, (2007). [arXiv:0709.0093 [gr-qc]]. (Cited on pages 118 and 121.)[76] Brügmann, B., Gonzalez, J. A., Hannam, M., Husa, S. and Sperhake, U., “Exploring
black hole superkicks”, Phys. Rev. D, 77, 124047, (2008). (Cited on page 66.)[77] Brügmann, B., González, J. A., Hannam, M., Husa, S., Sperhake, U. and Tichy, W.,“Calibration of Moving Puncture Simulations”, Phys. Rev. D, 77, 024027, (2008). (Citedbibliographyon pages 63, 64, 66, 118, 122, and 127.)[78] Brügmann, B., Tichy, W. and Jansen, N., “Numerical simulation of orbiting black
holes”, Phys. Rev. Lett., 92, 211101, (2004). (Cited on page 118.)[79] Bulik, T. and Belczyński, K., “Constraints on the Binary Evolution from Chirp Mass
Measurements”, Astrophys. J. Lett., 589, L37–L40, (2003). (Cited on page 110.)[80] Buonanno, A. and Damour, T., “Effective one-body approach to general relativistic
two- body dynamics”, Phys. Rev. D, 59, 084006, (1999). (Cited on pages 70 and 111.)[81] Buonanno, A., Chen, Y. and Damour, T., “Transition from inspiral to plunge inprecessing binaries of spinning black holes”, Phys. Rev. D, 74, 104005, (2006). (Cited on
page 41.)[82] Buonanno, A., Cook, G. B. and Pretorius, F., “Inspiral, merger and ring-down ofequal-mass black-hole binaries”, Phys. Rev. D, 75, 124018, (2007). (Cited on pages 70,
121, and 122.)[83] Buonanno, A. and Damour, T., “Transition from inspiral to plunge in binary blackhole coalescences”, Phys. Rev. D, 62, 064015, (2000). (Cited on pages 70 and 111.)[84] Buonanno, A. et al., “Toward faithful templates for non-spinning binary black holes
using the effective-one-body approach”, Phys. Rev. D, 76, 104049, (2007). (Cited on
page 70.)[85] Buonanno, A. et al., “Effective-one-body waveforms calibrated to numerical relativity
simulations: coalescence of non-spinning, equal- mass black holes”, Phys. Rev. D, 79,
124028, (2009). (Cited on page 70.)[86] Buonanno, A., Iyer, B., Ochsner, E., Pan, Y. and Sathyaprakash, B. S., “Comparisonof post-Newtonian templates for compact binary inspiral signals in gravitational-wave
detectors”, (2009). (Cited on pages 42 and 81.)[87] Cadonati, L. et al., “Status of NINJA: the Numerical INJection Analysis project”,
Class. Quantum Grav., 26(11), 114008, (2009). (Cited on pages 70 and 117.)[88] Campanelli, M., Lousto, C. O., Marronetti, P. and Zlochower, Y., “Accurate
evolutions of orbiting black-hole binaries without excision”, Phys. Rev. Lett., 96, 111101,(2006). (Cited on pages 33, 53, 63, 65, 69, 121, and 122.)[89] Campanelli, M., Lousto, C. O., Nakano, H. and Zlochower, Y., “Comparison of
Numerical and Post-Newtonian Waveforms for Generic Precessing Black-Hole Binaries”,
Phys. Rev. D, 79, 084010, (2009). (Cited on page 70.)[90] Campanelli, M., Lousto, C. O., Zlochower, Y., Krishnan, B. and Merritt, D.,“Spin Flips and Precession in Black-Hole-Binary Mergers”, Phys. Rev. D, 75, 064030,(2007). (Cited on page 75.)[91] Cao, Z.-j., Yo, H.-J. and Yu, J.-P., “A Reinvestigation of Moving Punctured Black Holes
with a New Code”, Phys. Rev., D78, 124011, (2008). (Cited on page 63.)[92] Chu, T., Pfeiffer, H. P. and Scheel, M. A., “High accuracy simulations of black holebinaries:spins anti-aligned with the orbital angular momentum”, Phys. Rev. D, 80, 124051,(2009). (Cited on page 65.)[93] Cokelaer, T., “Gravitational waves from inspiralling compact binaries: hexagonal
template placement and its efficiency in detecting physical signals”, Phys. Rev. D, 76,
102004, (2007). (Cited on pages 97, 103, 112, and 129.)[94] Collins, H., Gravity’s shadow: the search for gravitational waves, (University of Chicago
Press, 2004). (Cited on page 9.)[95] Corinaldesi, E. and Papapetrou, A., “Spinning test particles in general relativity. 2”,
Proc. R. Soc. London, Ser. A, 64, 259–268, (1952). (Cited on page 74.)[96] Dain, S., Lousto, C. O. and Zlochower, Y., “Extra-Large Remnant Recoil Velocities
and Spins from Near- Extremal-Bowen-York-Spin Black-Hole Binaries”, Phys. Rev. D,
78, 024039, (2008). (Cited on pages 121 and 122.)[97] Damour, T., “Coalescence of two spinning black holes: An effective one- body approach”,
Phys. Rev. D, 64, 124013, (2001). (Cited on page 41.)[98] Damour, T., Iyer, B. R. and Sathyaprakash, B. S., “Improved filters for gravitational
waves from inspiralling compact binaries”, Phys. Rev. D, 57, 885–907, (1998). (Cited on
pages 70, 73, and 111.)[99] Damour, T., Iyer, B. R. and Sathyaprakash, B. S., “Frequency-domain P-approximant
filters for time-truncated inspiral gravitational wave signals from compact binaries”,
Phys. Rev. D, 62, 084036, (2000). (Cited on pages 128 and 129.)

159

160

bibliography[100] Damour, T., Iyer, B. R. and Sathyaprakash, B. S., “A comparison of search templates
for gravitational waves from binary inspiral”, Phys. Rev. D, 63, 044023, (2001). (Cited
on pages 41 and 44.)[101] Damour, T., Iyer, B. R. and Sathyaprakash, B. S., “A comparison of search templates
for gravitational waves from binary inspiral: 3.5-PN update”, Phys. Rev. D, 66, 027502,(2002). (Cited on pages 41 and 44.)[102] Damour, T., Iyer, B. R. and Sathyaprakash, B. S., “Erratum: Comparison of searchtemplates for gravitational waves from binary inspiral: 3.5PN update [Phys. Rev. D 66,
027502 (2002)]”, Phys. Rev. D, 72(2), 029901, (2005). (Cited on pages 41 and 44.)[103] Damour, T., Jaranowski, P. and Schäfer, G., “On the determination of the last stableorbit for circular general relativistic binaries at the third post-Newtonian approximation”,
Phys. Rev. D, 62, 084011, (2000). (Cited on pages 70 and 111.)[104] Damour, T., Jaranowski, P. and Schäfer, G., “Dimensional regularization of thegravitational interaction of point masses”, Phys. Lett. B, 513, 147–155, (2001). (Cited on
page 41.)[105] Damour, T., Nagar, A., Hannam, M., Husa, S. and Brügmann, B., “Accurate EffectiveOne-Body waveforms of inspiralling and coalescing black-hole binaries”, Phys. Rev. D,
78, 044039, (2008). (Cited on page 127.)[106] de Waard, A., Benzaim, Y., Frossati, G., Gottardi, L., van der Mark, H., Flokstra,J., Podt, M., Bassan, M., Minenkov, Y., Moleti, A., Rocchi, A., Fafone, V. andPallottino, G. V., “MiniGRAIL progress report 2004”, Class. Quantum Grav., 22(10),S215–S219, (2005). (Cited on page 16.)[107] Dreyer, O., Krishnan, B., Shoemaker, D. and Schnetter, E., “Introduction to
Isolated Horizons in Numerical Relativity”, Phys. Rev. D, 67, 024018, (2003). (Cited on
pages 74 and 75.)[108] Droz, S., Knapp, D. J., Poisson, E. and Owen, B. J., “Gravitational waves from inspiraling compact binaries: Validity of the stationary-phase approximation to the Fouriertransform”, Phys. Rev. D, 59, 124016, (1999). (Cited on page 103.)[109] Echeverria, F., “Gravitational-wave measurements of the mass and angular momentum of a black hole”, Phys. Rev. D, 40(10), 3194–3203, (1989). (Cited on page 50.)[110] Einstein, A., “Zur allgemeinen Relativitätstheorie”, Sitzungsber. K. Preuss. Akad. Wiss.,
778–786, (1915). (Cited on page 9.)[111] Eisenhauer, F. et al., “SINFONI in the Galactic Center: Young Stars and Infrared
Flares in the Central Light-Month”, Astrophys. J., 628, 246–259, (2005). (Cited on
page 137.)[112] Eisenhauer, F. et al., “GRAVITY: getting to the event horizon of Sgr A*”, in Opticaland Infrared Interferometry, Proc. SPIE, vol. 7013, (2008). (Cited on page 138.)[113] Estabrook, F. B. and Wahlquist, H. D., “Response of Doppler spacecraft tracking togravitational radiation”, General Relativity and Gravitation, 6, 439–447, (1975). (Cited
on page 14.)[114] Etienne, Z. B., Faber, J. A., Liu, Y. T., Shapiro, S. L. and Baumgarte, T. W., “Fillingthe holes: Evolving excised binary black hole initial data with puncture techniques”,
Phys. Rev. D, 76, 101503, (2007). (Cited on pages 63, 121, and 122.)[115] Farr, B., Fairhurst, S. and Sathyaprakash, B. S., “Searching for binary coalescenceswith inspiral templates: Detection and parameter estimation”, Class. Quantum Grav.,
26, 114009, (2009). (Cited on page 134.)[116] Faye, G., Blanchet, L. and Buonanno, A., “Higher-order spin effects in the dynamics
of compact binaries. I: Equations of motion”, Phys. Rev. D, 74, 104033, (2006). (Cited on
page 42.)[117] Finn, L. S., “The response of interferometric gravitational wave detectors”, Phys. Rev.,
D79, 022002, (2009). (Cited on page 14.)[118] Flanagan, É. É. and Hughes, S. A., “Measuring gravitational waves from binary blackhole coalescences. I. Signal to noise for inspiral, merger, and ringdown”, Phys. Rev. D,
57, 4535–4565, (1998). (Cited on page 146.)[119] Forward, R. L., “Multidirectional, multipolarization antennas for scalar and tensor
gravitational radiation.”, Gen. Relativ. Gravit., 2, 149–159, (1971). (Cited on page 16.)[120] Fregeau, J. M., Larson, S. L., Miller, M. C., O’Shaughnessy, R. and Rasio, F. A.,“Observing IMBH-IMBH Binary Coalescences via Gravitational Radiation”, Astrophys.
J. Lett., 646, L135–L138, (2006). (Cited on pages 140, 146, 149, and 150.)

bibliography[121] Freitag, M., Gürkan, M. A. and Rasio, F. A., “Runaway collisions in young starclusters - II. Numerical results”, Mon. Not. R. Astron. Soc., 368, 141–161, (2006). (Cited
on page 139.)[122] Freitag, M., Rasio, F. A. and Baumgardt, H., “Runaway collisions in young starclusters - I. Methods and tests”, Mon. Not. R. Astron. Soc., 368, 121–140, (2006). (Cited
on pages 139 and 150.)[123] Gürkan, M. A., Freitag, M. and Rasio, F. A., “Formation of Massive Black Holes in
Dense Star Clusters. I. Mass Segregation and Core Collapse”, Astrophys. J., 604, 632–652,(2004). (Cited on page 139.)[124] Gair, J. R., Mandel, I., Miller, M. C. and Volonteri, M., “Exploring intermediate
and massive black-hole binaries with the Einstein Telescope”, arXiv e-print, (jul 2009).
[arXiv:0907.5450]. (Cited on page 140.)[125] Gair, J. R., Mandel, I., Sesana, A. and Vecchio, A., “Probing seed black holes using
future gravitational-wave detectors”, Class. Quant. Grav., 26, 204009, (2009). (Cited on
page 144.)[126] Galaviz, P., Bruegmann, B. and Cao, Z., “Numerical evolution of multiple black
holes with accurate initial data”, (2010). (Cited on page 63.)[127] “GEO600 detector”. URL (cited on April 13, 2010):
http://www.geo600.org/. (Cited on pages 18 and 19.)[128] Ghez, A. M. et al., “The First Measurement of Spectral Lines in a Short-Period Star
Bound to the Galaxy’s Central Black Hole: A Paradox of Youth”, Astrophys. J., 586,
L127–L131, (2003). (Cited on page 137.)[129] Ghez, A. M., Salim, S., Hornstein, S. D., Tanner, A., Lu, J. R., Morris, M., Becklin, E. E. and Duchêne, G., “Stellar Orbits around the Galactic Center Black Hole”,
Astrophys. J., 620, 744–757, (2005). (Cited on page 137.)[130] Ghez, A. M., Salim, S., Weinberg, N. N., Lu, J. R., Do, T., Dunn, J. K., Matthews,
K., Morris, M. R., Yelda, S., Becklin, E. E., Kremenek, T., Milosavljevic, M. andNaiman, J., “Measuring Distance and Properties of the Milky Way’s Central Supermassive Black Hole with Stellar Orbits”, Astrophys. J., 689, 1044–1062, (2008). (Cited on
page 137.)[131] Gillessen, S., Eisenhauer, F., Trippe, S., Alexander, T., Genzel, R., Martins, F.
and Ott, T., “Monitoring Stellar Orbits Around the Massive Black Hole in the Galactic
Center”, Astrophys. J., 692, 1075–1109, (2009). (Cited on page 137.)[132] Gillessen, S., Perrin, G., Brandner, W., Straubmeier, C., Eisenhauer, F., Rabien, S., Eckart, A., Lena, P., Genzel, R., Paumard, T. and Hippler, S., “GRAVITY:
the adaptive-optics-assisted two-object beam combiner instrument for the VLTI”, inAdvances in Stellar Interferometry, Proc. SPIE, vol. 6268, (2006). (Cited on page 138.)[133] Glebbeek, E., Gaburov, E., de Mink, S. E., Pols, O. R. and Portegies Zwart, S. F.,“The evolution of runaway stellar collision products”, Astron. Astrophys., 497, 255–264,(2009). (Cited on pages 110 and 139.)[134] Goldberger, W. D. and Rothstein, I. Z., “An effective field theory of gravity forextended objects”, Phys. Rev. D, 73, 104029, (2006). (Cited on page 74.)[135] Gopakumar, A., Hannam, M., Husa, S. and Brügmann, B., “Comparison between
numerical relativity and a new class of post-Newtonian gravitational-wave phase evolutions: the non-spinning equal-mass case”, Phys. Rev. D, 78, 064026, (2008). (Cited on
page 70.)[136] Grote, H. (LIGO Scientific), “The status of GEO 600”, Class. Quantum Grav., 25,114043, (2008). (Cited on page 69.)[137] Gualtieri, L., Berti, E., Cardoso, V. and Sperhake, U., “Transformation of themultipolar components of gravitational radiation under rotations and boosts”, Phys.
Rev. D, 78, 044024, (2008). (Cited on page 121.)[138] Gürkan, M. A., Fregeau, J. M. and Rasio, F. A., “Massive Black Hole Binaries from
Collisional Runaways”, Astrophys. J. Lett., 640, L39–L42, (2006). (Cited on pages 140,
146, 147, and 149.)[139] Hahn, S. G. and Lindquist, R. W., “The two-body problem in geometrodynamics”,
Ann. Phys., 29, 304–331, (1964). (Cited on pages 33 and 53.)[140] Hannam, M., Husa, S., Brügmann, B., González, J. A., Sperhake, U. and Murchadha, N. Ó., “Where do moving punctures go?”, J. Phys.: Conf. Ser., 66(1), 012047,(2007). (Cited on page 62.)

161

162

bibliography[141] Hannam, M., Husa, S., Pollney, D., Brügmann, B. and Murchadha, N. Ó., “Geometry and Regularity of Moving Punctures”, Phys. Rev. Lett., 99(24), 241102, (2007).
(Cited on page 62.)[142] Hannam, M., “Status of black-hole-binary simulations for gravitational- wave detection”,
Class. Quantum Grav., 26, 114001, (2009). (Cited on page 70.)
[143] Hannam, M. et al., “The Samurai Project: verifying the consistency of black- holebinary waveforms for gravitational-wave detection”, Phys. Rev. D, 79, 084025, (2009).
(Cited on pages 65 and 70.)
[144] Hannam, M. et al., “Gravitational waveforms from equal-mass black-hole binaries
with non-precessing spins”, (2010). In preparation. (Cited on pages 65 and 66.)[145] Hannam, M., Husa, S., Brügmann, B. and Gopakumar, A., “Comparison between
numerical-relativity and post-Newtonian waveforms from spinning binaries: the orbital
hang-up case”, Phys. Rev. D, 78, 104007, (2008). (Cited on pages 65, 66, 70, 118, and 122.)[146] Hannam, M., Husa, S., Sperhake, U., Brügmann, B. and González, J. A., “Where
post-Newtonian and numerical-relativity waveforms meet”, Phys. Rev. D, 77, 044020,(2008). (Cited on pages 66, 70, 118, 122, and 127.)[147] Harte, A. I., “Approximate spacetime symmetries and conservation laws”, Class. Quantum Grav., 25, 205008, (2008). (Cited on pages 74 and 75.)[148] Helmstrom, C. W., Statistical Theory of Signal Detection, (Pergamon Press, London,
1968), 2 edition. (Cited on page 95.)
[149] Helstrom, C. W., Elements of signal detection and estimation, (Prentice-Hall, Inc.,
Upper Saddle River, NJ, USA, 1995). (Cited on page 72.)[150] Herrmann, F., Silberholz, J., Bellone, M., Guerberoff, G. and Tiglio, M., “Integrating Post-Newtonian Equations on Graphics Processing Units”, Class. Quantum
Grav., 27, 032001, (2010). (Cited on page 40.)[151] Hild, S., Chelkowski, S., Freise, A., Franc, J., Morgado, N., Flaminio, R. and
DeSalvo, R., “A Xylophone Configuration for a third Generation Gravitational Wave
Detector”, Class. Quantum Grav., 27, 015003, (2010). (Cited on pages 25 and 145.)[152] Hinder, I., “The Current Status of Binary Black Hole Simulations in Numerical Relativity”, (2010). (Cited on page 70.)
[153] Hinder, I., Vaishnav, B., Herrmann, F., Shoemaker, D. and Laguna, P., “Universality
and Final Spin in Eccentric Binary Black Hole Inspirals”, Phys. Rev. D, 77, 081502, (2008).
(Cited on pages 63, 121, and 122.)[154] Hinshaw, G. et al. (WMAP), “Five-Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Data Processing, Sky Maps, & Basic Results”, Astrophys. J.
Suppl., 180, 225–245, (2009). (Cited on page 144.)[155] Hulse, R. A. and Taylor, J. H., “Discovery of a pulsar in a binary system”, Astrophys. J.
Lett., 195, L51–L53, (1975). (Cited on page 31.)[156] Husa, S., Gonzalez, J. A., Hannam, M., Brügmann, B. and Sperhake, U., “Reducing
phase error in long numerical binary black hole evolutions with sixth order finite
differencing”, Class. Quantum Grav., 25, 105006, (2008). (Cited on pages 74, 118, and 122.)[157] Husa, S., Hannam, M., Gonzalez, J. A., Sperhake, U. and Brügmann, B., “Reducing eccentricity in black-hole binary evolutions with initial parameters from postNewtonian inspiral”, Phys. Rev. D, 77, 044037, (2008). (Cited on pages 64 and 66.)[158] Imbiriba, B. et al., “Evolving a puncture black hole with fixed mesh refinement”, Phys.
Rev. D, 70, 124025, (2004). (Cited on pages 63, 121, and 122.)[159] Jasiulek, M., “A new method to compute quasi-local spin and other invariants on
marginally trapped surfaces”, Class. Quantum Grav., 26, 245008, (2009). (Cited on
pages 74 and 75.)[160] Kalogera, V., Belczynski, K., Kim, C., O’Shaughnessy, R. and Willems, B., “Formation of double compact objects”, Phys. Rep., 442, 75–108, (2007). (Cited on page 35.)[161] Kalogera, V. et al., “Erratum: “The Cosmic Coalescence Rates for Double NeutronStar Binaries (ApJ, 601, L179 [2004])”, Astrophys. J. Lett., 614, L137–L138, (2004). (Cited
on page 109.)[162] Kalogera, V., Kim, C., Lorimer, D. R., Burgay, M., D’Amico, N., Possenti, A.,
Manchester, R. N., Lyne, A. G., Joshi, B. C., McLaughlin, M. A., Kramer, M.,
Sarkissian, J. M. and Camilo, F., “The Cosmic Coalescence Rates for Double NeutronStar Binaries”, Astrophys. J., 601, L179–L182, (2004). Erratum-ibid. 614 (2004) L137. (Cited
on pages 35 and 109.)

bibliography[163] Kalogera, V., Narayan, R., Spergel, D. N. and Taylor, J. H., “The Coalescence Rateof Double Neutron Star Systems”, Astrophys. J., 556, 340, (2001). (Cited on page 105.)[164] Kerr, R. P., “Gravitational Field of a Spinning Mass as an Example of AlgebraicallySpecial Metrics”, Phys. Rev. Lett., 11, 237–238, (1963). (Cited on page 34.)
[165] Kidder, L. E., “Coalescing binary systems of compact objects to postNewtonian 5/2
order. 5. Spin effects”, Phys. Rev. D, 52, 821–847, (1995). (Cited on page 41.)[166] Kim, C., Kalogera, V. and Lorimer, D. R., “Effect of PSR J0737-3039 on the DNS
Merger Rate and Implications for GW Detection”, ArXiv e-prints, (2006). (Cited on
page 35.)[167] Kiuchi, K., Sekiguchi, Y., Shibata, M. and Taniguchi, K., “Longterm general relativistic simulation of binary neutron stars collapsing to a black hole”, Phys. Rev. D, 80,
064037, (2009). (Cited on page 103.)[168] Kokkotas, K. D. and Schmidt, B., “Quasi-Normal Modes of Stars and Black Holes”,
Living Rev. Relativity, 2(2), (1999). URL (cited on April 13, 2010):
http://www.livingreviews.org/lrr-1999-2. (Cited on page 48.)[169] Kopparapu, R. K., Hanna, C., Kalogera, V., O’Shaughnessy, R., Gonzalez, G.,
Brady, P. R. and Fairhurst, S., “Host Galaxies Catalog Used in LIGO Searches for
Compact Binary Coalescence Events”, Astrophys. J., 675(2), 1459–1467, (2008). (Cited
on pages 105, 106, and 109.)[170] Koppitz, M. et al., “Getting a kick from equal-mass binary black hole mergers”, Phys.
Rev. Lett., 99, 041102, (2007). (Cited on pages 121 and 122.)[171] Kramer, M. et al., “Tests of general relativity from timing the double pulsar”, Science,
314, 97–102, (2006). (Cited on page 31.)[172] Kyrian, K. and Semerák, O., “Spinning test particles in a Kerr field - II”, Mon. Not. R.
Astron. Soc., 382, 1922–1932, (2007). (Cited on page 75.)[173] “Laser Interferometer Gravitational-wave Observatory”. URL (cited on April 13, 2010):
http://www.ligo.caltech.edu/. (Cited on pages 18 and 19.)[174] Leaver, E. W., “An Analytic representation for the quasi normal modes of Kerr black
holes”, Proc. R. Soc. London, Ser. A, 402, 285–298, (1985). (Cited on pages 48 and 49.)
[175] Lindblom, L., Owen, B. J. and Brown, D. A., “Model Waveform Accuracy Standards
for Gravitational Wave Data Analysis”, Phys. Rev. D, 78, 124020, (2008). (Cited on
pages 72 and 73.)[176] Lousto, C. O., Nakano, H., Zlochower, Y. and Campanelli, M., “Statistical studiesof Spinning Black-Hole Binaries”, (2009). (Cited on page 40.)
[177] Lovelace, G., Owen, R., Pfeiffer, H. P. and Chu, T., “Binary-black-hole initial datawith nearly-extremal spins”, Phys. Rev. D, 78, 084017, (2008). (Cited on pages 74 and 75.)
[178] “LSC Algorithm Library”. URL (cited on April 13, 2010):
http://www.lsc-group.phys.uwm.edu/lal. (Cited on page 126.)[179] Manchester, R. N., Hobbs, G. B., Teoh, A. and Hobbs, M., “The Australia Telescope
National Facility Pulsar Catalogue”, Astron. J., 129(4), 1993–2006, (2005). (Cited on
page 108.)[180] Mauceli, E., Geng, Z. K., Hamilton, W. O., Johnson, W. W., Merkowitz, S., Morse,
A., Price, B. and Solomonson, N., “The Allegro gravitational wave detector: Dataacquisition and analysis”, Phys. Rev. D, 54(2), 1264–1275, (1996). (Cited on page 15.)[181] McClintock, J. E., Shafee, R., Narayan, R., Remillard, R. A., Davis, S. W. and Li,
L.-X., “The Spin of the Near-Extreme Kerr Black Hole GRS 1915+105”, Astrophys. J., 652,
518, (2006). (Cited on page 115.)[182] Miller, M. C., “Gravitational Radiation from Intermediate-Mass Black Holes”, Astrophys. J., 581, 438–450, (2002). (Cited on page 146.)[183] Miller, M. C., “Intermediate-mass black holes as LISA sources”, Class. Quantum Grav.,
26(9), 094031, (2009). (Cited on page 137.)[184] Miller, M. C. and Colbert, E. J. M., “Intermediate-Mass Black Holes”, Int. J. Mod.
Phys. D, 13, 1–64, (2004). (Cited on page 137.)[185] Miller, M. C. and Lauburg, V. M., “Mergers of Stellar-Mass Black Holes in Nuclear
Star Clusters”, Astrophys. J., 692, 917–923, (2009). (Cited on page 110.)[186] Nagy, G., Ortiz, O. E. and Reula, O. A., “Strongly hyperbolic second order Einstein’s
evolution equations”, Phys. Rev., D70, 044012, (2004). (Cited on page 57.)
[187] Nakamura, T., Oohara, K. and Kojima, Y., “General Relativistic Collapse to Black
Holes and Gravitational Waves from Black Holes”, Prog. Theor. Phys. Suppl., 90, 1–218,

163

164

bibliography(1987). (Cited on page 57.)
[188] Nielsen, A., Jasiulek, M., Schnetter, E. and Krishnan, B., (2010). In preparation.
(Cited on page 74.)[189] “Numerial INJection Analysis project”. URL (cited on April 13, 2010):
http://www.ninja-project.org. (Cited on page 70.)[190] O’Leary, R. M., Kocsis, B. and Loeb, A., “Gravitational waves from scattering ofstellar-mass black holes in galactic nuclei”, Mon. Not. R. Astron. Soc., 395, 2127–2146,(2009). (Cited on page 110.)[191] O’Leary, R. M., O’Shaughnessy, R. and Rasio, F. A., “Dynamical interactions andthe black-hole merger rate of the Universe”, Phys. Rev. D, 76(6), 061504, (2007). (Cited
on page 110.)[192] Oliynyk, T. A., “Cosmological post-Newtonian expansions to arbitrary order”, (2009).
(Cited on page 74.)[193] Orosz, J. A., McClintock, J. E., Narayan, R., Bailyn, C. D., Hartman, J. D., Macri,
L., Liu, J., Pietsch, W., Remillard, R. A., Shporer, A. and Mazeh, T., “A 15.65-solarmass black hole in an eclipsing binary in the nearby spiral galaxy M 33”, Nature, 449,
872–875, (2007). (Cited on page 110.)[194] O’Shaughnessy, R., Kalogera, V. and Belczynski, K., “Binary compact object coalescence rates: The role of elliptical galaxies”, arXiv e-print, (aug 2009). [arXiv:0908.3635].
(Cited on page 110.)[195] O’Shaughnessy, R., Kim, C., Fragos, T., Kalogera, V. and Belczynski, K., “Constraining Population Synthesis Models via the Binary Neutron Star Population”, Astrophys. J., 633, 1076–1084, (2005). (Cited on page 109.)[196] O’Shaughnessy, R., Kim, C., Kalogera, V. and Belczynski, K., “Constraining Population Synthesis Models via Empirical Binary Compact Object Merger and Supernova
Rates”, Astrophys. J., 672, 479–488, (2008). (Cited on pages 35 and 109.)[197] Owen, B. J., “Search templates for gravitational waves from inspiraling binaries: Choiceof template spacing”, Phys. Rev. D, 53, 6749–6761, (1996). (Cited on page 97.)[198] Owen, B. J. and Sathyaprakash, B. S., “Matched filtering of gravitational waves frominspiraling compact binaries: Computational cost and template placement”, Phys. Rev.
D, 60, 022002, (1999). (Cited on page 103.)[199] Pan, Y. et al., “Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-precessing, spinning, equal-mass black holes”, (2009). (Cited
on page 70.)[200] Papapetrou, A., “Spinning test particles in general relativity”, Proc. R. Soc. London, Ser.
A, 209, 248–258, (1951). (Cited on page 74.)[201] Papoulis, A. and Unnikrishna Pillai, S., Probability, Random Variables and Stochastic
Processes, (McGraw Hill Higher Education, 2002), 4 edition. (Cited on page 72.)[202] Penrose, R. and Rindler, W., Spinors and Space-time: Two-spinor calculus and relativistic fields, (Press Syndicate of the University of Cambridge, 1986). (Cited on page 66.)[203] Peters, P. C., “Gravitational Radiation and the Motion of Two Point Masses”, Phys.
Rev., 136, 1224–1232, (1964). (Cited on pages 141 and 142.)[204] Pfeiffer, H. P. et al., “Reducing orbital eccentricity in binary black hole simulations”,
Class. Quantum Grav., 24, S59–S82, (2007). (Cited on page 121.)[205] Poisson, E., “Gravitational waves from inspiraling compact binaries: The quadrupolemoment term”, Phys. Rev. D, 57, 5287–5290, (1998). (Cited on page 41.)[206] Pollney, D. et al., “Recoil velocities from equal-mass binary black-hole mergers: asystematic investigation of spin-orbit aligned configurations”, Phys. Rev. D, 76, 124002,(2007). (Cited on pages 63, 65, 66, 121, and 122.)[207] Pollney, D., Reisswig, C., Schnetter, E., Dorband, N. and Diener, P., “Highaccuracy binary black hole simulations with an extended wave zone”, (2009). (Cited on
pages 63, 66, and 67.)[208] Porciani, C. and Madau, P., “On the Association of Gamma-Ray Bursts with Massive
Stars: Implications for Number Counts and Lensing Statistics”, Astrophys. J., 548, 522–531,(2001). (Cited on pages 147 and 148.)[209] Portegies Zwart, S. F., Baumgardt, H., Hut, P., Makino, J. and McMillan, S. L. W.,“Formation of massive black holes through runaway collisions in dense young starclusters”, Nature, 428, 724–726, (2004). (Cited on page 139.)

bibliography[210] Portegies Zwart, S. F. and McMillan, S. L. W., “Black Hole Mergers in the Universe”,
Astrophys. J. Lett., 528, L17–L20, (2000). (Cited on pages 139 and 146.)[211] Porto, R. A. and Rothstein, I. Z., “Spin(1)Spin(2) Effects in the Motion of Inspiralling
Compact Binaries at Third Order in the Post-Newtonian Expansion”, Phys. Rev. D, 78,
044012, (2008). (Cited on page 74.)[212] Pretorius, F., “Evolution of binary black hole spacetimes”, Phys. Rev. Lett., 95, 121101,(2005). (Cited on pages 33, 53, 61, 63, 65, 69, 121, and 122.)[213] Pretorius, F., “Numerical Relativity Using a Generalized Harmonic Decomposition”,
Class. Quantum Grav., 22, 425–452, (2005). (Cited on pages 63, 65, 121, and 122.)[214] Pretorius, F. and Khurana, D., “Black hole mergers and unstable circular orbits”,
Class. Quantum Grav., 24, S83–S108, (2007). (Cited on pages 121 and 122.)[215] Prodi, G. A. et al., “Initial operation of the gravitational wave detector AURIGA”.
Prepared for 2nd Edoardo Amaldi Conference on Gravitational Waves, Geneva, Switzerland, 1-4 Jul 1997. (Cited on page 15.)[216] Regge, T. and Wheeler, J., “Stability of a Schwarzschild Singularity”, Phys. Rev., 108,
1063–1069, (1957). (Cited on page 48.)[217] Regge, T. and Wheeler, J. A., “Stability of a Schwarzschild singularity”, Phys. Rev., 108,
1063–1069, (1957). (Cited on page 66.)[218] Reisswig, C., Bishop, N. T., Pollney, D. and Szilagyi, B., “Unambiguous determination of gravitational waveforms from binary black hole mergers”, Phys. Rev. Lett., 103,
221101, (2009). (Cited on page 67.)[219] Reisswig, C., Bishop, N. T., Pollney, D. and Szilagyi, B., “Characteristic extraction
in numerical relativity: binary black hole merger waveforms at null infinity”, Class.
Quantum Grav., 27, 075014, (2010). (Cited on page 67.)[220] Reisswig, C. et al., “Gravitational-wave detectability of equal-mass black-hole binaries
with aligned spins”, Phys. Rev. D, 80, 124026, (2009). (Cited on pages 51, 66, and 144.)[221] Rezzolla, L., Dorband, E. N., Reisswig, C., Diener, P., Pollney, D., Schnetter,
E. and Szilágy, B., “Spin Diagrams for equal-mass black-hole binaries with aligned
spins”, (2007). (Cited on page 65.)[222] Rezzolla, L. et al., “Spin Diagrams for Equal-Mass Black-Hole Binaries with Aligned
Spins”, Astrophys. J., 679, 1422–1426, (2008). (Cited on pages 121 and 122.)[223] Rezzolla, L. et al., “The final spin from the coalescence of aligned-spin black- holebinaries”, Astrophys. J., 674, L29–L32, (2008). (Cited on pages 65 and 86.)[224] Robinson, C. A. K., Sathyaprakash, B. S. and Sengupta, A. S., “A geometric algorithm for efficient coincident detection of gravitational waves”, Phys. Rev. D, 78, 062002,(2008). (Cited on pages 103 and 128.)[225] Rodríguez, A., “Reducing false alarms in searches for gravitational waves from coalescing binary systems”, (2007). (Cited on pages 103 and 129.)[226] Rubbo, L. J., Cornish, N. J. and Poujade, O., “Forward modeling of space-borne
gravitational wave detectors”, Phys. Rev. D, 69, 082003, (2004). (Cited on page 14.)[227] Santamaría, L., Ohme, F., Ajith, P., Brügmann, B., Dorband, N., Hannam, M.,
Husa, S., Mösta, P., Müller, D., Pollney, D., Reisswig, C., Seiler, J. and Krishnan,B., “Matching post-Newtonian and numerical relativity waveforms: systematic errors
and a new phenomenological model for non-precessing black hole binaries”, (2010). To
be submitted. (Cited on page 69.)
[228] Santamaría, L., Krishnan, B. and Whelan, J. T., “Searching for numericallysimulated signals of black hole binaries with a phenomenological template family”,
Class. Quantum Grav., 26, 114010, (2009). (Cited on page 117.)[229] Sathyaprakash, B. S. and Dhurandhar, S. V., “Choice of filters for the detection ofgravitational waves from coalescing binaries”, Phys. Rev. D, 44, 3819–3834, (1991). (Cited
on page 103.)[230] Sathyaprakash, B. and Schutz, B. F., “Physics, Astrophysics and Cosmology with
Gravitational Waves”, Living Rev. Relativity, 12(2), (2009). URL (cited on April 13, 2010):
http://www.livingreviews.org/lrr-2009-2. (Cited on page 30.)[231] Scheel, M. A. et al., “Solving Einstein’s equations with dual coordinate frames”, Phys.
Rev. D, 74, 104006, (2006). (Cited on pages 66, 121, and 122.)[232] Scheel, M. A. et al., “High-accuracy waveforms for binary black hole inspiral, merger,
and ringdown”, Phys. Rev. D, 79, 024003, (2009). (Cited on pages 63, 67, 70, 121, and 122.)

165

166

bibliography[233] Schnittman, J. D., “Spin-orbit resonance and the evolution of compact binary systems”,
Phys. Rev. D, 70(12), 124020, (2004). (Cited on page 40.)[234] Schodel, R. et al., “Stellar dynamics in the central arcsecond of our galaxy”, Astrophys.
J., 596, 1015–1034, (2003). (Cited on page 137.)[235] Schutz, B., A First Course in General Relativity, (Cambridge University Press, 2009),
2nd edition. (Cited on page 11.)[236] Searle, A. C., Scott, S. M., McClelland, D. E. and Finn, L. S., “Optimal locationof a new interferometric gravitational wave observatory”, Phys. Rev. D, 73(12), 124014,(2006). (Cited on page 22.)[237] Sesana, A., Gair, J., Mandel, I. and Vecchio, A., “Observing Gravitational Waves
from the First Generation of Black Holes”, Astrophys. J. Lett., 698, L129–L132, (2009).
(Cited on page 148.)[238] Shaddock, D. A., “An overview of the Laser Interferometer Space Antenna”, Publ.
Astron. Soc. Australia, 26, 128–132, (2009). (Cited on page 69.)[239] Shibata, M. and Nakamura, T., “Evolution of three-dimensional gravitational waves:
Harmonic slicing case”, Phys. Rev. D, 52(10), 5428–5444, (1995). (Cited on page 57.)[240] Shibata, M., Kyutoku, K., Yamamoto, T. and Taniguchi, K., “Gravitational waves
from black hole-neutron star binaries I: Classification of waveforms”, Phys. Rev. D, 79,044030, (2009). (Cited on page 103.)[241] Smarr, L. L., “Gauge Conditions, Raidation Formulae, and the Two Black Hole Collision”, in Smarr, L. L., ed., Sources of Gravitational Radiation, pp. 245–274, (Cambridge
University Press, Cambridge, UK, 1979). (Cited on page 53.)[242] Smarr, L. L. and York, Jr., J. W., “Kinematical Conditions in the Construction ofSpacetime”, Phys. Rev. D, 17(10), 2529–2551, (1978). (Cited on page 53.)[243] Smarr, L. L. and York, Jr., J. W., “Radiation Gauge in General Relativity”, Phys. Rev. D,
17(8), 1945–1956, (1978). (Cited on page 53.)[244] Sperhake, U., “Binary black-hole evolutions of excision and puncture data”, Phys. Rev.
D, 76, 104015, (2007). (Cited on pages 63, 121, and 122.)[245] Sperhake, U., Cardoso, V., Pretorius, F., Berti, E. and Gonzalez, J. A., “The highenergy collision of two black holes”, Phys. Rev. Lett., 101, 161101, (2008). (Cited on
page 121.)[246] Sperhake, U. et al., “Eccentric binary black-hole mergers: The transition from inspiralto plunge in general relativity”, Phys. Rev. D, 78, 064069, (2008). (Cited on page 121.)[247] Stuart, A., Keith Ord, K. and Arnold, S., Kendall’s Advanced Theory of Statistics,
Volume 2A: Classical Inference and and the Linear Model, (Wiley, 2009), 6 edition. (Cited
on page 72.)[248] Suzuki, T. K., Nakasato, N., Baumgardt, H., Ibukiyama, A., Makino, J. andEbisuzaki, T., “Evolution of Collisionally Merged Massive Stars”, Astrophys. J., 668,
435–448, (2007). (Cited on page 139.)[249] “TAMA300 detector”. URL (cited on April 13, 2010):
http://tamago.mtk.nao.ac.jp/. (Cited on page 19.)[250] Taylor, J. H. and Weisberg, J. M., “A new test of general relativity: Gravitationalradiation and the binary pulsar PS R 1913+16”, Astrophys. J., 253, 908–920, (1982). (Cited
on pages 3 and 31.)[251] Teukolsky, S. A., “Rotating black holes: Separable wave equations for gravitationaland electromagnetic perturbations”, Phys. Rev. Lett., 29, 1114–1118, (1972). (Cited on
page 49.)[252] Teukolsky, S. A., “Perturbations of a rotating black hole. 1. Fundamental equations
for gravitational electromagnetic and neutrino field perturbations”, Astrophys. J., 185,
635–647, (1973). (Cited on page 86.)[253] Thorne, K. S., “Gravitational radiation”, in Hawking, S. W. and Israel, W., eds., Three
hundred years of gravitation, 9, pp. 330–458, (Cambridge University Press, Cambridge,1987). (Cited on pages 29, 103, 117, and 124.)[254] Tichy, W. and Marronetti, P., “The final mass and spin of black hole mergers”, Phys.
Rev. D, 78, 081501, (2008). (Cited on pages 118 and 122.)[255] Vaishnav, B., Hinder, I., Herrmann, F. and Shoemaker, D., “Matched Filteringof Numerical Relativity Templates of Spinning Binary Black Holes”, Phys. Rev. D, 76,
084020, (2007). (Cited on pages 121 and 122.)bibliography[256] van Meter, J. R., Baker, J. G., Koppitz, M. and Choi, D.-I., “How to move a black hole
without excision: gauge conditions for the numerical evolution of a moving puncture”,
Phys. Rev. D, 73, 124011, (2006). (Cited on pages 63, 121, and 122.)[257] “Virgo detector”. URL (cited on April 13, 2010):
http://www.virgo.infn.it/. (Cited on pages 18 and 19.)[258] Vishveshwara, C., “Scattering of gravitational radiation by a Schwarzschild black-hole”,
Nature, 227, 936–938, (1970). (Cited on page 48.)[259] Wald, R. M., General Relativity, (University Of Chicago Press, 1984). (Cited on pages 10
and 12.)[260] Walther, B., Brügmann, B. and Müller, D., “Numerical black hole initial data withlow eccentricity based on post-Newtonian orbital parameters”, Phys. Rev. D, 79, 124040,(2009). (Cited on page 74.)[261] Weber, J., “Observation of the Thermal Fluctuations of a Gravitational-Wave Detector”,
Phys. Rev. Lett., 17(24), 1228–1230, (1966). (Cited on page 15.)[262] Weber, J., “Evidence for Discovery of Gravitational Radiation”, Phys. Rev. Lett., 22(24),1320–1324, (1969). (Cited on page 15.)[263] Weber, J., “Anisotropy and Polarization in the Gravitational-Radiation Experiments”,
Phys. Rev. Lett., 25(3), 180–184, (1970). (Cited on page 15.)[264] Will, C. M., “On the Rate of Detectability of Intermediate-Mass Black Hole Binaries
Using LISA”, Astrophys. J., 611, 1080–1083, (2004). (Cited on page 146.)[265] Willke, B. et al., “The GEO-HF project”, Class. Quantum Grav., 23(8), S207–S214,(2006). (Cited on page 21.)[266] York, Jr., J. W., “Kinematics and Dynamics of General Relativity”, in Smarr, L. L., ed.,Sources of Gravitational Radiation, pp. 83–126, (Cambridge University Press, Cambridge,
UK, 1979). (Cited on page 54.)[267] Zerilli, F., “Gravitational Field of a Particle Falling in a Schwarzschild Geometry
Analyzed in Tensor Harmonics”, Phys. Rev. D, 2, 2141– 2160, (1970). (Cited on page 48.)[268] Zerilli, F. J., “Effective potential for even parity Regge-Wheeler gravitational perturbation equations”, Phys. Rev. Lett., 24, 737–738, (1970). (Cited on page 66.)[269] Zlochower, Y., Baker, J. G., Campanelli, M. and Lousto, C. O., “Accurate blackhole evolutions by fourth-order numerical relativity”, Phys. Rev. D, 72, 024021, (2005).
(Cited on pages 63, 121, and 122.)

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CURRICULUM VITÆpersonal
Lucía Santamaría Lara
Born on July 23th, 1982 in Burgos, Spain
Nationality: Spanish

education
PhD in Physics (2005-2010)
Albert-Einstein-Institut, Potsdam, Germany
Adviser: Prof. Bernard F. Schutz
University affiliation: Mathematisch-Naturwissenschaftliche Fakultät,
Universität Potsdam
Licenciado en Física (MSc in Physics) (2000-2005)
University of Salamanca, Spain
High School with Honors (1998-2000)
Secondary School “Conde Diego Porcelos”, Burgos, Spain

research fellowships, grants and awards
DAAD (German Academic Exchange Service)
Scholarship for graduate studies in Germany (2006/2009)
DAAD (German Academic Exchange Service)
Scholarship for intensive language courses, Goethe Institute
Munich (Autumn 2006)
CERN (European Organization for Nuclear Research)
Research summership, LHCb experiment (Summer 2005)
Adviser: Prof. Clara Matteuzzi
DESY (German Electron Synchrotron)
Research summership, HERMES experiment (Summer 2004)
Adviser: Dr. Ingrid Gregor
Spanish Ministery of Education
Grant for undergraduate research. Department of Optics, University of Salamanca (2004/2005)
Adviser: Prof. Luis Roso
Spanish Ministery of Education
Scholarship for language courses, London (Summer 2003)
Spanish Ministery of Education
Extraordinary Award for High School Studies (June 2000)

seminars and contributed talks
Black-hole binary coalescence: phenomenological models for nonprecessing systems and astrophysics of intermediate-mass binaries

Department Seminar, Caltech, CA, USA (02/2010)
Department Seminar, UWM, WI, USA (10/2009)
Department Seminar, RIT, NY, USA (10/2009)
A new phenomenological waveform family for modeling non-precessing
spinning black-hole binaries
NRDA09, Potsdam, Germany (07/2009)
8th Edoardo Amaldi Conference on Gravitational Waves, NYC,
USA (06/2009)
The NINJA project: preliminary results
Spanish Relativity Meeting ERE08, Salamanca, Spain (09/2008)
Numerical Modeling of Astrophysical Sources of Gravitational
Radiation, Valencia, Spain (09/2008)
The NINJA project: an Overview
Department Seminar, University of Columbia, NYC, USA (08/2008)
Searching for numerically-simulated signals with a phenomenological template family
NRDA08, Syracuse, USA (08/2008)
Incorporating Numerical Relativity Waveforms into Gravitational
Wave Data Analysis
Pacific Coast Gravity Meeting, UCSB, Santa Barbara, USA (03/2008)
GWDAW12, MIT, Cambrigde, USA (12/2007)
Interplay between Numerical Relativity and Gravitational Wave
Data Analysis
Workshop, KAVLI Institute for Theoretical Physics, Santa Barbara, USA (01/2008)
Numerical Relativity meets Gravitational Wave Data Analysis
Spanish Relativity Meeting ERE07, Tenerife, Spain (09/2007)
Numerical evolutions of Brillwaves
Spanish Relativity Meeting ERE06, Palma, Spain (09/2006)

PUBLICATIONS

1. L. Santamaría, F. Ohme, P. Ajith, B. Brügmann, N. Dorband, M.
Hannam, S. Husa, P. Mösta, D. Pollney, C. Reisswig, J. Seiler
and B. Krishnan, “Matching post-Newtonian and numerical relativity
waveforms: systematic errors and a new phenomenological model for
non-precessing black hole binaries,” to be submitted to Phys. Rev. D,2010.
2. P. Amaro-Seoane and L. Santamaría, “Detection of IMBHs withground-based gravitational wave observatories: A biography of a binaryof black holes, from birth to death,” astro-ph.CO/0910.0254, 2009.
3. P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Bruegmann, N. Dorband,
D. Mueller, F. Ohme, D. Pollney, C. Reisswig, L. Santamaría and
J. Seiler, “”Complete” gravitational waveforms for black-hole binaries
with non-precessing spins,” gr-qc/0909.2867, 2009.
4. L. Santamaría, B. Krishnan and J. T. Whelan, “Searching for numerically-simulated signals of black hole binaries with a phenomenologicaltemplate family,” Class. Quant. Grav., vol. 26, p. 114010, 2009.
5. L. Cadonati et al., “Status of NINJA: the Numerical INJection Analysis project,” Class. Quant. Grav., vol. 26, no. 11, p. 114008 (13pp), 2009.
6. B. Aylott et al., “Testing gravitational-wave searches with numerical relativity waveforms: Results from the first Numerical INJection
Analysis (NINJA) project,” Class. Quant. Grav., vol. 26, p. 165008, 2009.
7. P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes,
J. T. Whelan, B. Brügmann, P. Diener, N. Dorband, J. Gonzalez,
M. Hannam, S. Husa, D. Pollney, L. Rezzolla, L. Santamaría,
U. Sperhake and J. Thornburg, “A template bank for gravitational
waveforms from coalescing binary black holes: I. non-spinning binaries,”
Phys. Rev. D, vol. 77, p. 104017, 2008.
8. D. A. Brown, S. Fairhurst, B. Krishnan, R. A. Mercer, R. K. Kopparapu, L. Santamaría and J. T. Whelan, “Data formats for numerical
relativity waves.” 2007.
9. P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, J. T. Whelan,
B. Brügmann, P. Diener, J. González, M. Hannam, S. Husa, M.
Koppitz, D. Pollney, L. Rezzolla, L. Santamaría, A. M. Sintes,
U. Sperhake and J. Thornburg, “Phenomenological template family
for black-hole coalescence waveforms,” Class. Quantum Grav., vol. 24,
pp. S689–S699, 2007.
Publications with the LIGO and Virgo Scientific Collaborations
10. J. Abadie and others (The LIGO Scientific Collaboration andthe Virgo Collaboration), “Predictions for the rates of compact
binary coalescences observable by ground-based gravitational-wave
detectors,” 2010.

171

172

publications

11. J. Abadie and others (The LIGO Scientific Collaboration andthe Virgo Collaboration), “Search for gravitational waves fromblack hole binary inspiral, merger and ringdown,” 2010. In preparation.
12. J. Abadie et al. (The LIGO Scientific Collaboration), “Search forgravitational-wave inspiral signals associated with short Gamma-Ray
Bursts during LIGO’s fifth and Virgo’s first science run,” 2010.
13. B. P. Abbott et al. (The LIGO Scientific Collaboration andthe Virgo Collaboration), “Searches for gravitational waves fromknown pulsars with S5 LIGO data,” Astrophys. J., vol. 713, pp. 671–685,
2010.
14. B. P. Abbott et al. (The LIGO Scientific Collaboration andthe Virgo Collaboration), “Search for gravitational-wave bursts
associated with gamma-ray bursts using data from LIGO Science Run
5 and Virgo Science Run 1,” 2009.
15. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Search
for Gravitational Waves from Low Mass Compact Binary Coalescencein 186 Days of LIGO’s fifth Science Run,” Phys. Rev., vol. D80, p. 047101,
2009.
16. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Einstein@Home search for periodic gravitational waves in early S5 LIGO
data,” Phys. Rev., vol. D80, p. 042003, 2009.
17. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Searchfor gravitational wave ringdowns from perturbed black holes in LIGO
S4 data,” Phys. Rev., vol. D80, p. 062001, 2009.
18. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Searchfor gravitational-wave bursts in the first year of the fifth LIGO sciencerun,” Phys. Rev., vol. D80, p. 102001, 2009.
19. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Stacked
Search for Gravitational Waves from the 2006 SGR 1900+14 Storm,”
Astrophys. J., vol. 701, pp. L68–L74, 2009.
20. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Searchfor High Frequency Gravitational Wave Bursts in the First Calendar
Year of LIGO’s Fifth Science Run,” Phys. Rev., vol. D80, p. 102002, 2009.
21. B. P. Abbott et al. (The LIGO Scientific Collaboration), “First
LIGO search for gravitational wave bursts from cosmic (super)strings,”
Phys. Rev., vol. D80, p. 062002, 2009.
22. B. P. Abbott et al. (The LIGO Scientific Collaboration), “Search
for Gravitational Waves from Low Mass Binary Coalescences in theFirst Year of LIGO’s S5 Data,” Phys. Rev., vol. D79, p. 122001, 2009.
23. B. P. Abbott et al. (The LIGO Scientific Collaboration and theVirgo Collaboration), “An Upper Limit on the Stochastic Gravitational-Wave Background of Cosmological Origin,” Nature, vol. 460,
p. 990, 2009.

colophon
This thesis was typeset with LATEX 2ε using Robert Slimbach’s Minion andHermann Zapf ’s Euler type faces. The figures were produced with the pythonbased matplotlib plotting package and the Mathematica software.
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