A Search For Gravitational Waves from Perturbed
Black Hole Ringdowns in LIGO Data

Thesis by

Lisa Maria Goggin
In Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy

California Institute of Technology
Pasadena, California

2008(Defended May 13th, 2008)

ii

c

2008

Lisa Maria Goggin
All Rights Reserved

iii

To Mum and Dad

iv

Acknowledgements
The LIGO project has long been a source of wonder and amazement to me and I feel
privileged to have had the opportunity to contribute to it. My involvement with the
LSC has given me access to a brilliant and dedicated group of scientists and engineersand for that I am very grateful. I believe I have learned something from every member
I have encountered, whether through direct interaction or simply by your example of
how good science is done.
There are a number of people who have made a significant contribution to mywork that I would like to give credit to:First and foremost I would like to thank my adviser Alan Weinstein for giving me
my first opportunity to work on LIGO as a SURF student at the 40m, for supervising
my senior thesis and for hiring me once more as a grad student. Thank you for yourguidance, patience and encouragement over the years and allowing me the freedom tofind how I work best. Your breadth of knowledge in and enthusiasm for the project
and field in general has been a great inspiration.
Thank you to Yanbei Chen, Andrew Lange and Frank Porter for serving on mydefense committee and to Albert Lazzarini for being part of my candidacy committee.
Thank you to LIGO lab for funding me, enabling me to attend many conferences
and for the nicest office view I am ever likely to have!
Thanks to Duncan Brown for introducing me to the world of LIGO data analysis
and for your guidance over the last few years. Your expertise was invaluable in getting
the search up and running.
Thanks to my office mates and friends Diego Fazi, Drew Keppel and Pinkesh Patelfor your encouragement, the many interesting discussions and for your K9 friends.

v
Thanks to the members of the CBC group for your help with the analysis. In
particular thanks to co-chairs Patrick Brady and Gabriele González for your guidance and encouragement; to Jolien Creighton for writing the ringdown filtering codeand patiently answering numerous questions; to Stephen Fairhurst for developing the
IMR code and your helpful advice throughout the analysis; and to Sarah Caudill,
Shourov Chatterji, Thomas Cokelaer, Alex Dietz, Nick Fotopolous, Romain Gouaty,
Chad Hanna, Siong Heng, Vicky Kalogera, Eirini Messaritaki, Brian O’Reilly, Richard
O’Shaughnessy, Bangalore Sathyaprakash, Peter Shawhan, Xavier Siemens, Patrick
Sutton, and John Whelan for your countless useful suggestions.
Thanks to the ringdown review committee, Laura Cadonati, Kipp Cannon andStefan Ballmer who patiently and diligently reviewed the code and results.
Thanks to Stuart Anderson, Phil Ehrens, Dan Kozac and Paul Armor for neverbeing too busy to help me out with computer crises.
Thanks to Dennis Ugolini, Steve Vass, and Osamu Miyakawa for many fun andinteresting days at the 40m.
To the friends who have kept me (somewhat) sane during my time in Pasadena:
Guillaume Bres, Alan Hampton, Hannes Helgason, Vala Hjorleifsdottir, Gerdur Jonsdottir, Helen Gardarsdottir, Donal O’Connell, Tosin Otitoju, Jonathan Pritchard,Angel Ruiz, David Seal, Paul Skerritt, Allison Townsend, and Setthivoine You; thanks
for all the good times.
Finally, to my wonderful family, without whose constant love and support thiswould not have been possible; Julie, Amy and Denise, thanks for making 6000 miles
feel so much shorter; Mum and Dad, thank you for always believing in me andencouraging me to follow my dreams.

vi

Abstract
According to General Relativity a perturbed black hole will return to a stable configuration by the emission of gravitational radiation in a superposition of quasi-normal
modes. Such a perturbation will occur due to the coalescence of a black hole binary,following their inspiral and subsequent merger. At late times the waveform, which werefer to as a ringdown, is expected to be dominated by a single mode. As the waveform is well-known the method of matched filtering can be implemented to search forthis signal using LIGO data. LIGO is sensitive to the dominant mode of perturbed
black holes with masses between 10 and 500 M , the regime of intermediate-mass
black holes, to a distance of up to 300 Mpc. We present a search for gravitational
waves from black hole ringdowns using data from the fourth LIGO science run. We
implement a blind analysis of the data. We use Monte Carlo simulations of the expected waveform, and an estimation of the background from timeslides to tune thesearch. We present an analysis of the waveform parameter estimation and estimate
the efficiency of the search. As there were no gravitational wave candidates found,
we place an upper limit on the rate of black hole ringdowns in the local universe.

vii

Preface
The work presented in this thesis was carried out within the LIGO Scientific Collaboration (LSC). The methods and results presented here are under review and arepotentially subject to change. The opinions expressed here are those of the author
and not necessarily those of the LSC.
The author gratefully acknowledges the support of the United States National
Science Foundation for the construction and operation of the LIGO Laboratory, which
provided support for this work.
This thesis has LIGO document number LIGO-P080058-00-Z.

viii

Contents
Acknowledgementsiv

Abstractvi

Prefacevii

List of Figures

xv

List of Tables

xxvii

1 Introduction

1

2 Gravitational Waves

3

2.1

Gravitational Waves in General Relativity . . . . . . . . . . . . . . .

3

2.2

Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . .

6

2.2.1

Effects of Gravitational Waves on Test Masses . . . . . . . . .

6

2.2.2

Detecting Gravitational Waves with Laser Interferometry . . .

7

2.2.3

The Laser Interferometer Gravitational Wave Observatory, LIGO

8

2.2.3.1

Calibration of the LIGO Detectors . . . . . . . . . .

92.2.3.2

Calibration Lines . . . . . . . . . . . . . . . . . . . .

11

Antenna Response and Effective Distance . . . . . . . . . . .

12Sources of Gravitational Waves . . . . . . . . . . . . . . . . . . . . .

13

2.3.1

Binary Coalescence . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.2

Unmodeled Bursts . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3.3

Periodic Sources . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2.4
2.3

ix
2.3.4

Stochastic Background . . . . . . . . . . . . . . . . . . . . . .

3 Black Holes

16
18

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.2

Theoretical Black Holes. . . . . . . . . . . . . . . . . . . . . . . . .

18

3.3

Quasi-Normal Modes of Black Hole Oscillation . . . . . . . . . . . . .

20

3.3.1

Schwarzschild Black Holes . . . . . . . . . . . . . . . . . . . .

20

3.3.2

Kerr Black Holes . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.3.3

The l = m = 2 Mode . . . . . . . . . . . . . . . . . . . . . . .

24

3.3.3.1

The Black Hole Physical Parameters . . . . . . . . .

24

3.3.3.2

The Ringdown Waveform . . . . . . . . . . . . . . .

26

3.3.3.3

The Ringdown Peak Amplitude . . . . . . . . . . . .

27

Energy Emitted as Gravitational Waves . . . . . . . . . . . .

28

3.4

Astrophysical Black Holes . . . . . . . . . . . . . . . . . . . . . . . .

28

3.5

Previous Searches for Gravitational Waves from Perturbed Black Holes 30

3.3.4

4 Matched Filtering

32

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

4.2

Wiener Optimal Filtering. . . . . . . . . . . . . . . . . . . . . . . .

33

4.3

Detection Statistic for Gravitational Wave Searches . . . . . . . . . .

35

4.4

Discrete Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.5

Templated Matched Filtering Searches . . . . . . . . . . . . . . . . .

39

5 The Search Pipeline

42

5.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

5.2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

5.2.1

Segment Lists . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

5.2.2

Create the Veto Lists . . . . . . . . . . . . . . . . . . . . . . .

44

5.2.3

Create the Configuration File . . . . . . . . . . . . . . . . . .

44

5.3

Launch the Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

5.4

The Filtering Section of the Pipeline . . . . . . . . . . . . . . . . . .

45

x

5.5

5.4.1

Read In and Condition the Data

. . . . . . . . . . . . . . . .

46

5.4.2

Calculate the Response Function . . . . . . . . . . . . . . . .

46

5.4.3

Calculate the Power Spectrum . . . . . . . . . . . . . . . . . .

46

5.4.4

Calibrate and Fourier Transform the Data . . . . . . . . . . .

48

5.4.5

Generate the Template Bank. . . . . . . . . . . . . . . . . .

48

5.4.6

Create the Template . . . . . . . . . . . . . . . . . . . . . . .

49

5.4.7

Filter the Data . . . . . . . . . . . . . . . . . . . . . . . . . .

49

5.4.8

Cluster the Filter Output . . . . . . . . . . . . . . . . . . . .

50

5.4.9

Calculate and Record the Trigger Parameters . . . . . . . . .

51

Coincidence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

5.5.1

Apply Category 2 and 3 Vetoes . . . . . . . . . . . . . . . . .

52

5.5.2

Cluster the Single Detector Triggers Across the Template Bank

53

5.5.3

Implement the Time Coincidence Test . . . . . . . . . . . . .

53

5.5.4

Implement the Parameter Coincidence Test

. . . . . . . . . .

53

5.5.5

Implement a H1H2 Distance Cut . . . . . . . . . . . . . . . .

54

5.5.6

Cluster the Coincidences . . . . . . . . . . . . . . . . . . . . .

55

5.5.7

The Detection Statistic . . . . . . . . . . . . . . . . . . . . . .

56

6 The S4 Data Set

57

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

6.2

The Fourth LIGO Science Run . . . . . . . . . . . . . . . . . . . . .

57

6.3

Data Quality and Veto Categorization . . . . . . . . . . . . . . . . .

57

6.4

Horizon Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

6.5

Detection of Binary Compact Coalescence . . . . . . . . . . . . . . .

64

6.6

Predicted Ringdown Rates . . . . . . . . . . . . . . . . . . . . . . . .

66

6.6.1

Stellar Mass Black Holes . . . . . . . . . . . . . . . . . . . . .

67

6.6.2

Intermediate-Mass Black Holes . . . . . . . . . . . . . . . . .

68

7 Tuning the Search

70

7.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

7.2

Implementing an Injection Analysis . . . . . . . . . . . . . . . . . . .

71

xi
7.2.1

Creating the Injection File . . . . . . . . . . . . . . . . . . . .

71

7.2.2

Adding the Injection to the Data . . . . . . . . . . . . . . . .

72

7.2.3

Identifying the Injection Triggers . . . . . . . . . . . . . . . .

75

7.3

Background Estimation via Timeslides . . . . . . . . . . . . . . . . .

75

7.4

Tuning the Constraints . . . . . . . . . . . . . . . . . . . . . . . . . .

76

7.4.1

SNR Threshold, Template Bank Limits, and Sampling Rate .

77

7.4.2

Clustering the Filter Output . . . . . . . . . . . . . . . . . . .

78

7.4.3

Coincidence Windows. . . . . . . . . . . . . . . . . . . . . .

79

7.4.4

H1H2 Distance Cut . . . . . . . . . . . . . . . . . . . . . . . .

80

7.4.5

Detection Statistic . . . . . . . . . . . . . . . . . . . . . . . .

81

Playground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

7.5

8 Parameter Estimation

85

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

8.2

Recovery of Simulated Signals . . . . . . . . . . . . . . . . . . . . . .

85

8.2.1

Single Interferometer Injections . . . . . . . . . . . . . . . . .

86

8.2.2

Coincident Injections . . . . . . . . . . . . . . . . . . . . . . .

87

8.2.2.1

Missed and Found Injections. . . . . . . . . . . . .

88

8.2.2.2

Efficiency of Finding Triple Coincidences . . . . . . .

89

8.3

8.4

8.5

Comparison of Injected and Detected Parameters for Injections Foundin Triple Coincidence . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

8.3.1

Time of Arrival . . . . . . . . . . . . . . . . . . . . . . . . . .

92

8.3.2

Metric Distance . . . . . . . . . . . . . . . . . . . . . . . . . .

93

8.3.3

Effective Distance . . . . . . . . . . . . . . . . . . . . . . . . .

94

Comparison of Recovered Parameters Between Detectors . . . . . . .

99

8.4.1

Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

8.4.2

Metric Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.4.3

Effective Distance . . . . . . . . . . . . . . . . . . . . . . . . . 102

Comparison of the Background Estimation via Time-slides with thePlayground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xii

8.6

8.5.1

Triple Coincidences . . . . . . . . . . . . . . . . . . . . . . . . 106

8.5.2

Double Coincidences . . . . . . . . . . . . . . . . . . . . . . . 106

Comparison of Recovered Injection Parameter Accuracy for Timeslides
found in Double Coincidence . . . . . . . . . . . . . . . . . . . . . . . 111
8.6.1

Metric Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.6.2

Effective Distance . . . . . . . . . . . . . . . . . . . . . . . . . 112

9 Results

115

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.2

Opening the Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.3

9.4

9.2.1

Triples in Triple Time . . . . . . . . . . . . . . . . . . . . . . 116

9.2.2

H1L1 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 116

9.2.3

H1H2 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 116

9.2.4

H2L1 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 116

9.2.5

H1L1 Doubles in Double Time . . . . . . . . . . . . . . . . . . 117

9.2.6

H1H2 Doubles in Double Time . . . . . . . . . . . . . . . . . 117

9.2.7

H2L1 Doubles in Double Time . . . . . . . . . . . . . . . . . . 117

Following Up on the Loudest Candidates . . . . . . . . . . . . . . . . 117
9.3.1

Follow-Up Procedures . . . . . . . . . . . . . . . . . . . . . . 117

9.3.2

H1L1 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 124

9.3.3

H1H2 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 127

9.3.4

H2L1 Doubles in Triple Time . . . . . . . . . . . . . . . . . . 130

9.3.5

H1L1 Doubles in Double Time . . . . . . . . . . . . . . . . . . 132

9.3.6

H1H2 Doubles in Double Time . . . . . . . . . . . . . . . . . 132

9.3.7

H2L1 Doubles in Double Time . . . . . . . . . . . . . . . . . . 133

The Upper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4.1

9.4.2

Bayesian and Frequentist Approaches to the Upper Limit . . . 134
9.4.1.1

Frequentist . . . . . . . . . . . . . . . . . . . . . . . 135

9.4.1.2

Bayesian . . . . . . . . . . . . . . . . . . . . . . . . . 136

Cumulative Luminosity . . . . . . . . . . . . . . . . . . . . . . 136

xiii
9.4.2.1

Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 137

9.4.3

Calculating the Upper Limit . . . . . . . . . . . . . . . . . . . 138

9.4.4

Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.4.5

Including Doubles in the Upper Limit Calculation . . . . . . . 143

10 Search with Simulated Inspiral-Merger-Ringdown Signals

145

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.2 Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.3 Creating the IMR Waveform . . . . . . . . . . . . . . . . . . . . . . . 148
10.4 Single Detector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.5 Coincidence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
11 The Future for Ringdown Searches

160

11.1 Notes for Future Searches with the current Pipeline . . . . . . . . . . 160
11.1.1 Searches for Triple Coincident Events . . . . . . . . . . . . . . 160
11.1.2 Searches for Double Coincident Events . . . . . . . . . . . . . 161
11.1.3 Coincidence Test . . . . . . . . . . . . . . . . . . . . . . . . . 161
11.1.4 Extending the Template Bank . . . . . . . . . . . . . . . . . . 162
11.2 Future Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12 Summary and Concluding Remarks

166

12.1 Aim of the Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.2 The Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.3.1 Opening the Box . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.3.2 Calculation of the Upper Limit . . . . . . . . . . . . . . . . . 168
12.3.3 Parameter Accuracy . . . . . . . . . . . . . . . . . . . . . . . 168
12.3.4 Background Estimation . . . . . . . . . . . . . . . . . . . . . . 169
12.3.5 Inspiral-Merger-Ringdown . . . . . . . . . . . . . . . . . . . . 169
12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

xiv
Bibliography

171

A Ringdown Search Configuration File178

B Segmentation of the Data

185

C Loudest Ten Events in Double Coincidence

188

D Projects Undertaken at the 40 m Interferometer

194

xv

List of Figures
2.1

Illustration of the effect of a gravitational wave with plus polarization
on a ring of test particles when the direction of propagation of the gravitational wave is orthogonal to the plane of the particles. . . . . . . . .

2.2

7

Illustration of the effect of a gravitational wave with cross polarization
on a ring of test particles when the direction of propagation of the gravitational wave is orthogonal to the plane of the particles. . . . . . . . .

7

2.3

Schematic of the LIGO interferometer. Picture courtesy of LIGO lab. .

9

2.4

Best LIGO strain sensitivity curves for the science runs S1 to S5. The
LIGO design sensitivity curve is shown in black. . . . . . . . . . . . . .

10

2.5

Detector control loop . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.6

Illustration of source (x, y, z), radiation (x00 , y 00 , z 00 ) and detector (x0 , y 0 , z 0 )13

2.7

frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p
The plus, cross, and unpolarized combination F+2 + F×2 antenna patterns for the LIGO detectors. (This figure was taken from [8].) . . . . .

14

3.1

The imaginary part of the frequency ωI versus the real part ωR for
overtones of the l = 2 and l = 3 modes of a Schwarzschild black hole.
The most slowly damped mode has l = 2, n = 0 is marked by a blackbox. (The data in the plot is from [40] and [41].) . . . . . . . . . . . .

3.2

22

A demonstration of how spin removes the degeneracy in m for the l = 2
mode. The cyan, magenta, green, red, and blue sets of points correspondto m = −2, m = −1, m = 0, m = 1, and m = 2, respectively. The
frequency where the points converge corresponds to â = 0. (The datain this plot is from [44].) . . . . . . . . . . . . . . . . . . . . . . . . . .

23

xvi
3.3

The real part of Mω as a function of spin for the l = 3 (upper group)and l = 2 (lower group) modes. (The data in this plot is from [44].) . .

23

3.4

Quality factor versus dimensionless spin factor for the l = m = 2 mode.

25

3.5

Frequency versus mass for â = 0 (blue line), â = 0.5 (green dashes), andâ = 0.98 (red dash-dot) for the l = m = 2, n = 0 mode. . . . . . . . . .

3.6

25

The ringdown waveform produced by a source radiating 1% of its massin gravitational wave located at a distance of 100 Mpc with frequencyof 100 Hz and quality factor of 2 (blue), 10 (green), and 20 (red). . . .

3.7

27

M ω as a function of time for the gravitational wave emitted as an equalmass black hole binary undergoes coalescence. The energy emitted isindicated by the annotations. ωc and ωλ correspond to different methods
for extracting the frequency from the numerical waveforms. This figure
was taken from Buonanno et al. [56]. . . . . . . . . . . . . . . . . . . .

29

5.1

The ringdown search pipeline. . . . . . . . . . . . . . . . . . . . . . . .

43

5.2

The response function (scaled by dyn) for H1 between GPS times 793165201
and 793167377. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

The inverse calibrated spectrum for H1 between GPS times 793165201
and 793167377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

47

48

An example of single-template clustering with a window of 1 s. The
threshold is marked by a horizontal green line; triggers between the twolines are below threshold. The red circles mark the loudest trigger inthe cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

51

Contours of ds2 = 0.03 for the portion of the template bank between
f = 1600 Hz and 2100 Hz and Q = 2 to 30. . . . . . . . . . . . . . . .

54

6.1

Time in hours of analysable S4 data . . . . . . . . . . . . . . . . . . .

58

6.2

S4 strain sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

6.3

S4 ringdown horizon distance versus mass and frequency for â = 0.9 and = 1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

xvii
6.4

Illustration of the galaxies with ∼ 300 Mpc of the Earth. Picture Credit:
Richard Powell [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

62

S4 ringdown horizon distance versus time for an optimally located andoriented black hole of mass 250 M , spin of 0.9 and  = 1% producing
an SNR of 8 in the detector. . . . . . . . . . . . . . . . . . . . . . . . .

6.6

63

A plot of S4 strain sensitivity versus frequency (red). The blue lines
represent the inspiral phase of a binary coalescence for binaries with
masses (from bottom to top): 1.4-1.4, 1.4-3, 3-3, 10-10, 20-20, 40-40,
and 100-100, in units of M . The green dots represent the ringdown(for the same list of binaries) for a final black hole with a spin of 0.7 andassuming 1% of the mass has been radiated.

6.7. . . . . . . . . . . . . .

65

The horizon distance as a function of mass for the S4 inspiral and ringdown searches up to 100 M . A final spin of 0.7 and  = 1% are assumed
in the calculation of the ringdown horizon distance. . . . . . . . . . . .

6.8

66

The horizon distance as a function of mass for the S4 inspiral and ringdown searches, covering the full mass range of the ringdown search. A
final spin of 0.7 and  = 1% are assumed in the calculation of the ringdown horizon distance. . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9

67

The cumulative blue light luminosity as a function of horizon distance.
This figure was taken from [72]. . . . . . . . . . . . . . . . . . . . . . .

68

7.1

Distribution of injections in frequency over nine injection runs. . . . . .

72

7.2

Distribution of injections’ sky location over nine injection runs. . . . .

73

7.3

Demonstration of the addition of an injection to the raw data; amplitude(in arbitrary units of countes) versus time for 70 ms of raw data (blue)
and raw data plus an injection (red). The green line marks the time of
the injection according to the injection file. The inset zooms in on theinjection, tinj = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

xviii
7.4

Cartoon illustrating timeslides; data is time-shifted such that any coincidences are accidental, providing an estimate of the rate of false alarms.
(Picture from R. Tucker.) . . . . . . . . . . . . . . . . . . . . . . . . .

7.5

76

The template bank for the S4 ringdown search which, with the parameters Qmin = 2, Qmax = 20, fmin = 50 Hz, fmax = 2000 Hz, and ds2max =0.03 contained 584 templates. . . . . . . . . . . . . . . . . . . . . . . .

7.6

78

Dead-time before triggers in H1 due to clustering of the filter output.
The upper panel shows all the triggers while the lower panel only shows
those that are not cut by category 2 or 3 vetoes. . . . . . . . . . . . . .

7.7

79

Dead-time after triggers in H1 due to clustering of the filter output. The
upper panel shows all the triggers while the lower panel only shows thosethat are not cut by category 2 or 3 vetoes. . . . . . . . . . . . . . . . .

7.8

80

The effective distance of H2 versus H1 for injections (red) and timeslides(blue). The green line marks the distance cut; all triggers below this line
are discarded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9

81

The normalized κ = def fH1 /def fH2 histograms of injections (red) andtimeslides (blue). The green line marks the distance cut; all triggers to
the right of this line are discarded. . . . . . . . . . . . . . . . . . . . .

7.10

82

A contour plot of the L1 signal-to-noise ratio versus the H1 signal-tonoise-ratio for double coincident timeslide event. The colour-bar represents log10 (N ) where N is the number of triggers. . . . . . . . . . . . .

7.11

83

The H1L1 SNR distribution for timeslides (black) and injections (red).
Contours of constant values of the detection statistic with a = 2 andb = 2.2 are marked in green. . . . . . . . . . . . . . . . . . . . . . . . .

8.1

84

Hanford effective distance versus frequency for missed (red) and found(blue) injections in the H2 single interferometer analysis. The cyan lineis the horizon distance for a source with a spin of 0.98 producing an
SNR of 5.5 in the detector, assuming that 1% of its mass is radiated as
gravitational waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

xix
8.2

Hanford effective distance versus frequency for injections missed (redcircles) and found in coincidence. Injections found in triple coincidence
are marked as blue crosses, injections found in double coincidence areshown as green (H1H2) cyan (H1L1) and magenta (L1H2) stars andthose that were vetoed are also marked with a black circle. . . . . . . .

8.3

89

Hanford effective distance versus frequency for injections found in double coincidence. The coloured stars represent each of the detector pairs(H1H2 doubles are marked in green, H1L1 in cyan and L1H2 in magenta). The black circles mark those doubles that were vetoed in thethird detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4

The efficiency of finding injections in triple coincidences as a function ofphysical distance for injections made between 45 Hz and 2.5 kHz. . . .

8.5

92

Difference between L1 detected and injected time versus injected frequency for injections found in triple coincidence.

8.7

91

Histogram of difference between detected and injected time of arrival for
H1, H2, and L1 injections found in triple coincidence. . . . . . . . . . .

8.6

90

. . . . . . . . . . . .

93

Histogram of the mismatch ds2 between injections and the templates
they were found with for H1, H2, and L1 triple coincidence injections.
The black vertical line marks ds2 = 0.0075, the template bank maximum
mismatch. The plot shows that ∼ 70% of the injections found in triple
coincidence were found with the correct template. . . . . . . . . . . . .

8.8Plot of ds2 versus central frequency for the L1 component of injections
found in triple coincidence . . . . . . . . . . . . . . . . . . . . . . . . .

8.9

95

95

Plot of ds2 versus quality for the L1 component of injections found in
triple coincidence. The red lines mark the five values of Q in the templatebank and the black line marks the upper Q boundary of the templatebank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

xx
8.10

Plot of quality versus frequency for the L1 component of injections foundin triple coincidence. The colour scale is the mismatch between theinjected and recovered parameters ds2 . Each horizontal line denotes thefrequency range of the templates for each of the five values of Q. . . . .

8.11

96

Histogram of the fractional difference between detected and injected effective distance in H1, H2 and L1 for injections found in triple coincidence. 97

8.12

Plot of the fractional difference between detected and injected effective
distance versus frequency in H2 for injections found in triple coincidence. 98

8.13

Plot of the fractional difference between detected and injected effective
distance versus frequency in H1 for injections found in triple coincidence. 98

8.14

Histogram of difference between H1 and H2 recovered times for injections
found in triple coincidence. . . . . . . . . . . . . . . . . . . . . . . . . 100

8.15

Histogram of difference between H1L1 (red) and H2L1 (blue) recovered
times for injections found in triple coincidence. . . . . . . . . . . . . . 100

8.16

Plot of difference between H1 and H2 recovered times as a function of
H1 recovered frequency for injections found in triple coincidence. . . . 101

8.17

Plot of difference between H1L1 (blue) and H2L1 (red) recovered times
versus frequency for injections found in triple coincidence. . . . . . . . 101

8.18

Histogram of ds2 between recovered parameters in H1 and L1 for injections found in triple coincidence. . . . . . . . . . . . . . . . . . . . . . 103

8.19

Frequency versus effective distance for H1 with the colour-bar displaying
ds2 (H1,L1), the metric distance between injections recovered in H1 andin L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.20

Histogram of the fractional difference in H1H2 (green), H1L1 (red), and(H2L1) blue recovered effective distance for injections found in triple
coincidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.21

Plot of the fractional difference in H1 and H2 recovered effective distanceversus H1 frequency for injections found in triple coincidence. . . . . . 105

8.22

L1 SNR versus H1 SNR for background triggers found in triple coincidence.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xxi
8.23

The signal-to-noise distribution of playground and timeslides found indouble coincidence in L1 and H1. . . . . . . . . . . . . . . . . . . . . . 108

8.24

Histogram of number of H1L1 double coincidences per timeslide and in
the playground scaled to the full data set. . . . . . . . . . . . . . . . . 108

8.25

The signal-to-noise distribution of playground and timeslides found indouble coincidence in H2 and H1 prior to the implementation of the
distance cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.26

Histogram of number of H1H2 double coincidences per timeslide and in
the playground scaled to the full data set. . . . . . . . . . . . . . . . . 109

8.27

The signal-to-noise distribution of playground and timeslides found indouble coincidence in L1 and H2. . . . . . . . . . . . . . . . . . . . . . 110

8.28

Histogram of number of H2L1 double coincidences per timeslide and in
the playground scaled to the full data set. . . . . . . . . . . . . . . . . 110

8.29

The normalized distribution of the mismatch between H1 and L1 forinjections (red) and double coincident timeslides (blue).

8.30. . . . . . . . 112

The normalized distribution of the fractional difference in effective distance between H1 and H2 for injections (red) and double coincident
timeslides (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.31

The normalized distribution of the fractional difference in effective distance between H1 and L1 for injections (red) and double coincident
timeslides (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.32

The normalized distribution of the fractional difference in effective distance between L1 and H2 for injections (red) and double coincident
timeslides (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.1

Cumulative histogram of zero-lag and background coincidences: H1L1
doubles in triple time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.2

Scatter plot of L1 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 118

xxii
9.3

Cumulative histogram of zero-lag and background coincidences, H1H2
doubles in triple time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.4

Scatter plot of H2 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 119

9.5

Cumulative histogram of zero-lag and background coincidences, H2L1
doubles in triple time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

9.6

Scatter plot of L1 versus H2 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 120

9.7

Cumulative histogram of zero-lag and background coincidences, H1L1
doubles in double time. . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9.8

Scatter plot of L1 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 121

9.9

Cumulative histogram of zero-lag and background coincidences, H1H2
doubles in double time. . . . . . . . . . . . . . . . . . . . . . . . . . . 122

9.10

Scatter plot of H2 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 122

9.11

Cumulative histogram of zero-lag and background coincidences, H2L1
doubles in double time. . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.12

Scatter plot of L1 versus H2 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The
inset is an enlargement of the region ρ = 5 to 200. . . . . . . . . . . . . 123

9.13

The whitened spectrogram of the H1 gravitational wave channel showing
a hardware injection of an inspiral-merger-ringdown signal during S5. . 125

9.14

A qscan of the gravitational wave channel at GPS time 794949585, theloudest H1L1 candidate event in triple time. . . . . . . . . . . . . . . . 126

xxiii
9.15

A qscan of the gravitational wave channel at GPS time 793829533, thesecond loudest H1L1 candidate event in triple time. . . . . . . . . . . . 126

9.16

A qscan of the gravitational wave channel at GPS time 794291462, thethird loudest H1L1 candidate event in triple time. . . . . . . . . . . . . 127

9.17

A qscan of gravitational wave channel at GPS time 793253792, the loudest H1H2 candidate event in triple time. . . . . . . . . . . . . . . . . . 128

9.18

A qscan of the gravitational wave channel at GPS time 794654729, thesecond loudest H1H2 candidate event in triple time. . . . . . . . . . . . 129

9.19

A qscan of two environmental channels triggered at 794654729. . . . . 129

9.20

A qscan of the gravitational wave channel at GPS time 795398069, thethird loudest H1H2 candidate event in triple time.

9.21. . . . . . . . . . . 130

A qscan of the gravitational wave channel at GPS time 794966223, theloudest H2L1 candidate event in triple time. . . . . . . . . . . . . . . . 131

9.22

A qscan of the gravitational wave channel at GPS time 794490884, thesecond loudest H2L1 candidate event in triple time. . . . . . . . . . . . 131

9.23

A qscan of gravitational wave channel at GPS time 794490884, the third
loudest H2L1 candidate event in triple time. . . . . . . . . . . . . . . . 131

9.24

A qscan of the gravitational wave channel at GPS time 793258551, theloudest H1L1 candidate event in triple time. . . . . . . . . . . . . . . . 132

9.25

A qscan of the gravitational wave channel at GPS time 793589170, theloudest H1H2 candidate event in triple time. . . . . . . . . . . . . . . . 133

9.26

A qscan of the gravitational wave channel at GPS time 794432410, theloudest H2L1 candidate event in triple time. . . . . . . . . . . . . . . . 134

9.27

The efficiency of detecting triples versus physical distance for injections
made in the 45–100 Hz band. . . . . . . . . . . . . . . . . . . . . . . . 139

9.28

The efficiency of detecting triples versus physical distance for injections
made in the 100–200 Hz band.

9.29. . . . . . . . . . . . . . . . . . . . . . 140

The efficiency of detecting triples versus physical distance for injections
made in the 200–500 Hz band.

. . . . . . . . . . . . . . . . . . . . . . 140

xxiv
9.30

The efficiency of detecting triples versus physical distance for injections
made in the 500–1000 Hz band. . . . . . . . . . . . . . . . . . . . . . . 141

9.31

The efficiency of detecting triples versus physical distance for injections
made in the 1000–2500 Hz band. . . . . . . . . . . . . . . . . . . . . . 141

10.1

A preliminary assessment of the performance of a phenomenological template bank (red line) for use in IMR searches compared to searches using only inspiral (blue dots) and only ringdown (red dashes) templates.
The ringdown curve assumes  = 0.7% and the shaded area represents
0.18% ≤  ≤ 2.7%, and can be compared with figure 6.3 which displays
the ringdown horizon distance for  = 1%. This figure is taken from [90]. 147

10.2

The inspiral-merger-ringdown for a binary black hole system with component spins of 0.88 and 0.84 and masses of 8.9 M and 6.3 M (blue).
Also shown is the ringdown-only waveform for the same system (red). . 150

10.3

The transition portion of the inspiral-merger-ringdown waveform (dots)
and the ringdown-only waveform (circles) from figure 10.2. . . . . . . . 150

10.4

The frequency time series of the coalescence. The inset zooms in on theinspiral-merger transition and ringdown phase. . . . . . . . . . . . . . 151

10.5

M ωR versus (unitless) time for the coalescence of a black hole binary.
The final black hole has a spin of 0.9 and an M ωR of 0.68, which agrees
with figure 3.3 for the l = m = 2 mode. . . . . . . . . . . . . . . . . . . 151

10.6

Number of IMR and ringdown-only injections recovered in H1 as a function of the frequency of the final ringdown. The vertical lines denote thetemplate bank boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . 153

10.7

Detected versus injected ringdown frequency for IMR and ringdown-only
waveforms in H1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xxv
10.8

Frequency versus time for the templates that rang up around the time ofa ringdown-only injection. The colour of the data points represents thequality factor of the template. The black lines represent the frequency
and time of the injection, and the red lines represent the frequency andtime of the template with the largest signal-to-noise ratio. . . . . . . . 155

10.9

Frequency versus time for the templates that rang up around the time of
an IMR injection. The colour of the data points represents the quality
factor of the template. The black lines represent the frequency and timeof the injection, and the red lines represent the frequency and time of
the template with the largest signal-to-noise ratio. . . . . . . . . . . . . 155

10.10 Initial masses of the binary components for IMR injections found bycorrect (red) and incorrect (green) ringdown templates. The black line
represents the upper limit to the mass range of the S4 binary black holeinspiral search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.11 Hanford effective distance versus injected ringdown frequency for ringdownonly injections. The black vertical lines denote the template bank boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.12 Hanford effective distance versus injected ringdown frequency for IMR
injections. The black vertical lines denote the template bank boundaries. 158
11.1

A demonstration of the high SNR “wings” observed when the templatebank was extended to include frequenies between 40 Hz and 4 kHz.
This plot shows the SNR versus frequency for a ringdown injection withcentral frequency of 200 Hz.

. . . . . . . . . . . . . . . . . . . . . . . 163

xxvi
11.2

Plot of horizon distance (distance to which a ringdown signal from an
optimally oriented and located source will produce an SNR of 8 in thedetector) versus mass for Initial LIGO (blue), Enhanced LIGO (green)and Advanced LIGO (red) in a low-power configuration. We have assumed that 1% of the mass is radiated as gravitational waves. The
curves in each group are for spins of 0, 0.49 and 0.98 going from left toright. The upper x-axis is the frequency for a spin of 0.49. (Plot from
A. Weinstein.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

B.1

An illustration of how science segment number 3 (yellow box) is divided
into analysis segments (red, blue, and green dotted lines). The solidboxes denote the times that are analysed in each analysis segment. . . 187

xxvii

List of Tables
3.1

Quasi-normal modes of oscillation for a non-spinning black hole [40]. .

21

5.1

A sample of the H1 S4 segment list . . . . . . . . . . . . . . . . . . . .

44

9.1

Upper limit for triples in triple time . . . . . . . . . . . . . . . . . . . 139

9.2

Upper limit for doubles in triple time . . . . . . . . . . . . . . . . . . . 144

9.3

Upper limit for doubles in double time . . . . . . . . . . . . . . . . . . 144

C.1

The ten loudest candidate events from H1L1 doubles in triple time . . 188

C.2

The ten loudest candidate events from H1H2 doubles in triple time . . 189

C.3

The ten loudest candidate events from H2L1 doubles in triple time . . 190

C.4

The ten loudest candidate events from H1L1 doubles in double time . . 191

C.5

The ten loudest candidate events from H1H2 doubles in double time . 192

C.6

The ten loudest candidate events from H2L1 doubles in double time . . 193

1

Chapter 1
Introduction
Black hole ringdowns are amongst the most promising sources of gravitational waves,detectable with current detectors out to very large distances, as far as 300 Mpc
from the Earth. The ringdown is the final phase of a binary black hole coalescence,
following the inspiral and merger.
A central result of general relativity is that gravitational waves are emitted froman accelerating mass. It has been established using black hole perturbation theory
that the waveform emitted by a perturbed black hole can be modeled as a superposition of quasi-normal modes, with “quasi” referring to the fact that the oscillationis damped. It is expected that at late times the oscillation will be dominated by asingle mode. Throughout this analysis we will refer to a gravitational wave emitted
from a perturbed black hole as a “ringdown waveform” or just “ringdown”.
This thesis presents the results of a search for gravitational waves from perturbed
black holes using data from the Laser Interferometer Gravitational Wave Observatory(LIGO), a project dedicated to the detection of gravitational waves. LIGO is run
jointly between Caltech and MIT, and funded by the National Science Foundation.
The observatory consists of three detectors at two sites; Hanford, WA hosts a 4 km
interferometer (H1) and a 2 km interferometer (H2), and Livingston, LA is home to asecond 4 km interferometer (L1). Construction of the interferometers began in 1996,
starting the Initial LIGO phase. After several engineering runs the first LIGO sciencerun (S1) began on August 23rd, 2002 and lasted a little over two weeks. Between
then and mid-2005 three more science runs took place at a rate of about one per2year with S2 and S4 lasting approximately one month each and S3 collecting threemonths of data. By late 2005, LIGO reached its initial design sensitivity, and on
November 4th, 2005, S5 began. The goal of S5 was to collect one year’s worth of
triple coincidence data; this was achieved by September 30th, 2007. At the time ofwriting, the LIGO detectors are undergoing significant upgrades for the Enhanced
LIGO phase [1], which will implement many of the Advanced LIGO technologies andsee a factor of 2–3 increase in sensitivity. This will culminate in the S6 run, scheduled
to start in the autumn of 2009.
We begin in chapter 2 with an introduction to gravitational waves; we discuss how
they may be detected through laser interferometry and outline some possible sourcesof gravitational waves. In chapter 3 we provide the motivation for the search. We
discuss theoretical and astrophysical black holes and introduce the waveform we aresearching for. Chapter 4 describes the method of matched filtering and the templatebank used in the search. The pipeline that has been created to implement a search
for ringdowns is detailed in chapter 5. In chapter 6 we describe some of the importantdetails about the S4 science run. We describe the tools used to tune the search in
chapter 7, and explain how the final values of the constraints were arrived at. In
chapter 8 we describe the results of a large scale Monte Carlo run. We evaluate theefficiency of the search, compare the expected and detected waveform parameters,
and compare the recovered parameters between pairs of detectors. We also estimate
the background and compare it to a subset of the data. In chapter 9 we describe theresults of the search. We did not find any plausible gravitational wave candidates in
the S4 data set. We place an upper limit on the rate of ringdowns and investigate
some of the loudest candidate events. In chapter 10 we investigate the effect thatthe presence of an inspiral and merger preceding the ringdown would have on ourability to detect and estimate the parameters of ringdowns. In chapter 11 we makesome recommendations for future ringdown searches and document some issues weencountered in the course of the search. We give a brief summary of our results anda final conclusion in chapter 12.

3

Chapter 2
Gravitational Waves
In this chapter we introduce gravitational waves and outline how they may be detectedusing laser interferometry, in particular by the LIGO detectors. We also describe someof the likely sources of gravitational waves.

2.1

Gravitational Waves in General Relativity“Spacetime grips mass, telling it how to move, and mass grips spacetime, telling it
how to curve”, a famous quote of John Wheeler’s summarizing the mutual dependenceof mass and spacetime in the theory of general relativity. This was a central resultof Einstein’s theory of general relativity, expressed mathematically by Einstein inwhat is now known as the Einstein equation, a set of ten nonlinear partial differential
equations for ten metric coefficients, gαβ (x), relating the Einstein curvature tensor
Gαβ (a measure of local spacetime curvature) to the stress-energy tensor of matterTαβ (a measure of matter energy density),Gαβ (gαβ ) =

8πG
Tαβ ,
c4

(2.1)

where G is Newton’s constant and c is the speed of light. A general solution forthis equation has not been found, however various techniques exist for solving theequations under particular circumstances. One such case is weak time-varying fields
producing “ripples in spacetime” or gravitational waves.

4
Under the assumption that the gravitational waves produced by the source are
weak, the metric can be written as a small perturbation hαβ of the flat spacetime
metric in Minkowski coordinates ηαβ = diag(−1, +1, +1, +1), where |hαβ |  1,gαβ (x) = ηαβ + hαβ (x).

(2.2)

In the weak field limit the non-linear Einstein equation can be approximated as linear,
and with the choice of the transverse-traceless gauge, is given by the wave equation



→2
−
1 ∂2
− 2 2 + ∇ hαβ (x) = 0.
c ∂t

(2.3)

The solution to this equation ishαβ (x) = aαβ eik·x ,(2.4)

where aαβ is a symmetric 4 × 4 matrix of constants giving the amplitudes of thevarious components of the wave, and k is the wave vector such that→ →−k · x = −k t t + k · −
x.

(2.5)

Substituting equation (2.4) into equation (2.3) gives the conditionkα k α = 0,

(2.6)

showing that the wave propagates at the speed of light. Our choice of gauge gives us
the following constraints:
kα aαβ = 0(2.7)

aαα = 0(2.8)

aαβ uβ = 0,

(2.9)

5
where uβ is some fixed four-velocity. The first constraint restricts aαβ to be orthogonalor transverse to kα , the second requires that the matrix is traceless and the third,
if we orient the coordinate axes such that the direction of propagation is along thez-axis, implies that aαz = 0. These conditions reduce the number of components of
aαβ from ten to just four,


0 0
0


 0 axx axyaαβ = 
 0 axy −axx

0 0
0

0





0 
.

0 

0

(2.10)

We write the final form of the solution to the source-free, linearized Einstein equation
as



0 0
0


 0 h+ h×hαβ = 
 0 h× −h+

0 0
0

0





0  iω(z−t)
e
.

0 

0

(2.11)

Thus the transverse traceless gravitational wave travels at the speed of light and iscomposed of two independent polarizations; h+ is known as the plus polarization andh× is the cross-polarization.
The energy density is given by the stress-energy tensor
T GW ==

1 c2 X T T T T
hi,j,0 , hi,j,0
32π G i,j
1 c2|h+,0 |2 + |h×,0 |2 ,16π G

where h...i denotes an average over several wavelengths [2].

(2.12)

6

2.2

Detection of Gravitational Waves

2.2.1

Effects of Gravitational Waves on Test Masses

The effects of a gravitational wave cannot be seen in isolated bodies, but only byobserving the change in separation between pairs of masses. Take as an example apair of test masses separated by a distance L (as measured in the unperturbed flatspacetime) along the x-axis, and a gravitational wave propagating along the z-axis.
In the perturbed spacetime the distance between the test masses L0 is
Z L

0L (t) =

[1 + hxx (t, x)]1/2 dx(2.13)

0which, in the long wavelength approximation can be expressed as1
L (t) ≈ L 1 + hxx (t, 0)20(2.14)and thus for a change in distance between the two test masses δL0 = L0 −L, the strain
produced by the gravitational wave is1δL0 (t, 0)= hxx (t, 0) .
L
2(2.15)

An illustration of the effects of a gravitational wave is shown in figures 2.1 and2.2. These show a circular configuration of free test masses in the z = 0 plane. Fromthe view point of the central test mass the gravitational wave manifests itself bystretching space between it and the other test masses in one direction transverse to
the direction of propagation and contracting in the orthogonal direction in the sameplane, changing the circular pattern of the test masses to an elliptical configuration.
Half a period later the effect is reversed; those masses which were displaced furthestfrom the central test mass are now brought closest and vice versa. The gravitational
wave polarization which causes maximal stretching along the x and y axes is known
as the plus polarization. Rotating the coordinate axes by 45◦ in the z = 0 planedemonstrates the cross polarization. The most general gravitational wave traveling7in the z direction is a linear superposition of these two polarizations.

Figure 2.1: Illustration of the effect of a gravitational wave with plus polarization ona ring of test particles when the direction of propagation of the gravitational wave isorthogonal to the plane of the particles.

Figure 2.2: Illustration of the effect of a gravitational wave with cross polarization ona ring of test particles when the direction of propagation of the gravitational wave isorthogonal to the plane of the particles.

It is evident from figures 2.1 and 2.2, the gravitational wave is invariant under arotation of 180◦ about its direction of propagation, θinv = 180◦ . This is related to thespin S of the zero-mass particle associated with the field, which in the case of gravity
is the graviton, by the relation S = 360◦ /θinv , and thus is a consequence of the factthat the graviton is spin-2 (the quantum analogue of a classical rank-2 tensor field)
[3].

2.2.2

Detecting Gravitational Waves with Laser Interferometry

The detection of gravitational waves through laser interferometry takes advantage ofthe effects just described, and uses laser light as a displacement measuring device. We
replace the central test mass in the configuration above with a 50% reflecting mirror
known as a beam splitter (BS), and replace the ring of test masses with two highlyreflecting mirrors placed at an equal distance from, but in orthogonal directions to the8
BS. These mirrors are referred to as end test masses in the x and y direction, ETMXand ETMY. The BS directs an input beam of laser light towards the ETMs. If the
distance between the ETMs and the beamsplitter is the same in the two arms, thephase of the light reflected from the ETMs is the same. A gravitational wave travelingin a direction orthogonal to the plane of the detector will increase the distance between
the BS and the ETM in one arm and decrease the distance in the other arm. This
will produce a phase difference on the light received by a photodiode at the outputof the interferometer. The longer the distance the light has to travel the greater thephase shift will be. This optical configuration describes a Michelson interferometer
[4].

2.2.3

The Laser Interferometer Gravitational Wave Observatory, LIGO

The LIGO detectors are Michelson interferometers with the additional feature of
Fabry-Perot arms. These are resonant cavities, formed by placing an additional mirror
just after the laser in both arms at an integral number of wavelengths from the ETM.
These mirrors are known as input test masses, ITMX and ITMY. These resonantcavities allow the light to circulate many times, effectively increasing the length of
the arms. The ITMs and ETMs are separated by 4 km in H1 and L1, and by 2 km in
H2. The light circulates in the resonant cavities approximately 200 times. LIGO also
employs power recycling, in which an additional mirror (the power recycling mirror
PRM) is placed between the laser and the BS, resonantly enhancing the light stored
in the interferometer. A schematic of the LIGO detectors is shown in figure 2.3.
As mentioned in chapter 1 LIGO has successfully completed six science runs since
2002. In figure 2.4 we show the best strain sensitivity curves from each of the fivescience runs, along with the design sensitivity curve. The plot demonstrates the large
increases in sensitivity achieved between runs to the point where, in S5, the LIGO
detectors achieved design sensitivity. The plot also shows that the LIGO detectors
are most sensitive to gravitational waves between ∼ 40 Hz and 2 kHz.

9

Figure 2.3: Schematic of the LIGO interferometer. Picture courtesy of LIGO lab.

2.2.3.1

Calibration of the LIGO Detectors

Calibration of the data is essential to determining the sensitivity to distant sourcesof gravitational radiation [5, 6]. Here we describe how it is achieved.
The data to be analysed is taken from the gravitational wave channel DARM ERR,
q(t). This is the error signal on the feedback loop which is used to control the differential motion of the interferometer arms (DARM). It is related to the gravitational
wave strain in the Fourier domain h(f ) by the response function R(f )h(f ) = R(f )q(f ).

(2.16)

Accurate reconstruction of the strain from the error signal, i.e., determination of the
response function R(f ), is essential and is done through the process of calibration[5, 6].
The response of the interferometer to a gravitational wave strain can be characterized by a loop gain function G(f ), which is parameterized by three functions, asensing function C(f), an actuation function A(f), and a digital filter function D(f),by
G(f ) = C(f )D(f )A(f ).

(2.17)

10

Figure 2.4: Best LIGO strain sensitivity curves for the science runs S1 to S5. The
LIGO design sensitivity curve is shown in black.

The control loop is shown in figure 2.5. A control strain sc subtracts from the gravitational wave strain plus noise, h = sGW + n, leaving a residual strain sres . The
sensing function C(t, f ) = γ(t)C0 (f ) consists of a reference sensing function C0 and aloop gain γ(t), which depends on the light power stored in the arms. C0 is dominatedby the cavity pole frequency response (1 + f /fp )−1 , with fp ≈ 90 Hz. As describedin the next section, γ(t) is a relative measurement which changes over time as thealignment of the mirrors varies. The sensing function converts the residual strain
into a digital error signal q which is read out by the channel DARM ERR in arbitrary units of counts at a rate of 16384 Hz. The digital filter D(f ) converts the error
signal to a control signal d that is sent to the mirrors as an actuation. This quantity
is known precisely. The actuation function converts the control signal to strain bysending a current to coils surrounding magnets which are attached to the mirrors.
This produces a force and hence a displacement of the mirrors, adjusting the lengths

11of the cavities. From the figure we can see thatq(f ) = γ(t)C0 (f )sres ,(2.18)

h = sres + sc ,(2.19)

sc = sres C0 γ(t)D(f )A(f ).

(2.20)

andSubstituting these quantities in equation (2.16) and solving for R gives the response
functionR(f ) =

1 + γ(t)G0 (f ),γ(t)C0 (f )(2.21)

where G0 = C0 (f )D(f )A(f ) is the reference open loop gain.

Figure 2.5: Detector control loop

2.2.3.2

Calibration Lines

As was mentioned above, the calibration coefficient γ is a function of time. In practiceit changes on the order of minutes. In order to track these changes, sinusoidal signalsof known frequency are added to the control signal. For the S4 run these were at
46.7, 393.1, and 1144.3 Hz for H1; 54.1, 407.3, and 1159.7 Hz for H2; and 54.7, 396.7,

12
and 1151.5 Hz for L1. By digitally heterodyning the error signal, the control signals,
and the excitation signal with the injected sinusoid, the calibration coefficients canbe found.

2.2.4

Antenna Response and Effective DistanceUp until now we have been concerned with gravitational waves propagating fromdirectly above (or below) the interferometer, so-called “optimally positioned and oriented” sources that give the maximum response in the interferometer. However, in
reality not only can a gravitational wave come from any direction in the sky but its
orientation may be such that the detectors can only capture some portion of it. In
calculating the strain produced by a given source these considerations need to beaccounted for. Figure 2.6 displays coordinates for the emission, propagation, andreception of a gravitational wave. The source has axes (x, y, z), with the z-axis in the
direction of the angular momentum. The line between the source and the detector r
makes an angle ι with the z-axis. This is the angle of inclination. At the detector the
local coordinate axes are (x0 , y 0 , z 0 ), with x0 and y 0 along the arms of the interferometer. r makes an angle θ with the z 0 axis and an angle φ with the x0 axis. In between
we have the propagation coordinates, (x00 , y 00 , z 00 ) such that z 00 lies along r and the x00
and y 00 axis make an angle Ψ with the x0 and y 0 axes. This is the polarization angle.
Thus the strain produced at the detector is given byh(t) = h+ (t)F+ (θ, φ, Ψ) + h× (t)F× (θ, φ, Ψ),(2.22)

where the plus and cross detector beam functions F+ and F× [7] are given by
1
F+ = − (1 + cos2 θ) cos 2φ cos 2Ψ − cos θ sin 2φ sin 2Ψ
2
1(1 + cos2 θ) cos 2φ sin 2Ψ − cos θ sin 2φ cos 2Ψ.
F× =2
The detector plus, cross, and unpolarized combination(2.23)(2.24)

p
F+2 + F×2 are shown in figure

2.7. The figures show that there is a null point in the antenna pattern. If a gravita-

13

Figure 2.6: Illustration of source (x, y, z), radiation (x00 , y 00 , z 00 ) and detector (x0 , y 0 , z 0 )frames.

tional wave is traveling in a direction orthogonal to the plane of the detector and thepolarization angle is at 45 degrees to the x0 and y 0 axes the effect of the gravitational
wave will be the same in both arms and no phase shift will be produced.
We define the effective distance Def f as the distance to an optimally positioned
and oriented source that produces the same strain in the detector as a source at agiven position, polarization, and inclination at a distance D,
Def f = q

2.3

D

.

(2.25)

F+2 (1 + cos2 ι)2 /4 + F×2 cos2 ι

Sources of Gravitational Waves

Even though gravitational waves have not yet been directly observed, their existence
has been inferred through careful monitoring of the orbital period of the binary pulsar
PSR 1913+16, discovered by Hulse and Taylor in 1974 [9]. They observed that theorbital period of the binary system was decreasing in a manner precisely consistent14

p
Figure 2.7: The plus, cross, and unpolarized combination F+2 + F×2 antenna patterns
for the LIGO detectors. (This figure was taken from [8].)with the loss of energy and angular momentum due to gravitational radiation. For
this they were awarded the Nobel Prize in 1993.
Below we briefly outline the main sources of gravitational waves, categorized by
waveform type. The LIGO Scientific Collaboration (LSC) data analysis efforts are
structured around searches for these different waveform morphologies.

2.3.1

Binary Coalescence

A system composed of either two neutron stars, two black holes, or one of eachbound together by gravity forms a binary system. According to general relativity
the objects will lose energy through the emission of gravitational radiation. As aresult their orbits shrink and the two stars spiral in towards one another eventuallycombining to form a single star, most likely a black hole. This process is called binary
coalescence. The coalescence can be divided into three phases according to how well
we can model the waveform at different times. The “inspiral phase” is defined asthat time while the two stars are distinct objects orbiting around one another andthe gravitational waveform emitted can be well approximated by the post-Newtonianmodel (i.e., the velocities are low). The post-Newtonian approximation breaks down
as the stars begin their final few orbits and plunge in towards one another. We referto this as the “merger” phase. Although numerical simulations are telling us more

15
about the waveform produced at this stage (see chapter 10) it is still not represented
by an analytic waveform. We refer to this waveform as an unmodeled burst. (Searches
for this type of waveform will be discussed in the next section.) After the plunge,
the resulting star tries to return to a stable configuration by emitting gravitational
waves in a series of quasi-normal modes. These are also well modeled and this phaseis known as the “ringdown phase”.
The search for gravitational waves from the ringdown phase is the subject of thisthesis, thus we dedicate chapter 3 to a discussion of this waveform and black holes ingeneral.
The waveform produced during the inspiral phase is colloquially known as a chirp
waveform, because the frequency and amplitude of the signal increases rapidly withtime. For a binary of total mass M , separation a, and orbital period T at a distancer, the characteristic strain expected from the inspiral can be approximated as [10]
h∼

G Ek
,
c4 r

(2.26)

where Ek = M (πa/T )2 is the kinetic energy of an equal mass binary due to nonspherical motion. Employing Kepler’s third law T 2 = 4π 2 a3 /GM we can estimate
the strain as−20h ∼ 10

6.3 kpc
r



M
2.8 M

5/3 

T
1s

−2/3
.

(2.27)As the signal is well known it can be searched for using the method of matchedfiltering (introduced in chapter 4). Inspiral searches on LIGO data over the past fivedata runs have targeted binaries containing neutron stars, stellar mass black holes,
and primordial black holes. Details of these analyses may be found in the following
papers [11, 12, 13, 14, 15, 16, 17].

2.3.2

Unmodeled Bursts

There are many astrophysical sources which are likely to emit what is best described
as a burst of gravitational waves whose exact form is not well known. This includes

16gravitational waves from the merger of two stars described above, supernova explosions, gamma ray burst (GRB) engines, and possibly sources we are not even awareof. Data analysis algorithms capable of identifying short-duration excesses of strain
power and correlating these between detectors are employed to search for unmodeled sources. A selection of papers describing results of LIGO burst analyses are[18, 19, 20, 21, 22].

2.3.3

Periodic Sources

The mechanism by which a rapidly spinning neutron star is most likely to emit gravitational waves occurs if its shape deviates from axisymmetry. This deviation isexpressed as the ellipticity ε of the neutron star, ε = (Ixx − Iyy ) /Izz , where Ijj represent the moments of inertia about the principle axes. The resulting gravitational
wave has a frequency twice the rotational frequency frot . The expected strain for aneutron star at a distance r is
4π 2 G Izz frot ε
c4
r

2 

frot
10 kpc  ε −26= 2 × 10
1 kHz
r
10−6

h ∼

(2.28)(2.29)[10]. Both all-sky and targeted searches have been undertaken within the LSC, details
may be found in the following papers: [23, 24, 25, 26, 27, 28].

2.3.4

Stochastic Background

Analogous to the cosmic microwave background of electromagnetic radiation is thestochastic background of gravitational radiation. This may be composed of gravitational waves of cosmological origin as well as of astrophysical origin. The latter is arandom superposition of weak signals from supernovae, binary coalescences, and rotating neutron stars. Detection of gravitational waves of a cosmological origin would
provide a unique opportunity to explore the early universe, as other forms of radiation,such as electromagnetic or neutrino, cannot probe such early times. Several searches

17
for a stochastic gravitational background with LIGO data have been completed; see[29, 30, 31, 32, 33] for more details.

18

Chapter 3Black Holes
3.1

Introduction

This chapter is concerned with theoretical and astrophysical black holes. We discuss the solution to the Einstein equation for perturbed black holes and the analytic
waveform of the emitted gravitational radiation far from the source, the – ringdown
waveform. This motivates the search for ringdowns in LIGO data described in later
chapters. We discuss astrophysical black holes and outline previous searches for ringdowns.

3.2

Theoretical Black HolesThe first reference to objects now known as black holes came from the British geologist
John Michell in 1784 [34]. In a letter to Henry Cavendish describing a method ofdetermining a star’s distance, magnitude, and mass, he discusses the possibility of astar with such a large gravitational force that light would be prevented from escaping
its surface:“If the semi-diameter of a sphere of the same density as the Sun were to
exceed that of the Sun in the proportion of 500 to 1, a body falling froman infinite height towards it would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted
by the same force in proportion to its vis inertiae (inertial mass), with19
other bodies, all light emitted from such a body would be made to returntowards it by its own proper gravity.”
It took another hundred and thirty two years for this notion to be revisited. Soon after
Einstein published his theory of general relativity in 1915 Karl Schwarzschild foundan exact solution to the Einstein equation for the geometry outside of a non-spinningspherically symmetric star [35]. He found that if a star of mass M is confined to aradius R = 2GM/c2 , electromagnetic radiation is infinitely red-shifted and the starappears dark. This radius later became known as the Schwarzschild radius. In 1939,
Oppenheimer and Snyder performed the first rigorous calculation demonstrating theformation of a black hole from the implosion of an idealized star using the formalismof general relativity [36]. The name black hole itself was coined by John Wheeler in1968 [37].
General relativity tells us that a black hole is a region of spacetime where thegravitational field is so powerful that nothing, not even light can escape. At thecenter is the singularity, a point of zero volume and infinite density where all of
the black hole’s mass is located. Spacetime is infinitely curved at this point. The
singularity is enclosed by the event horizon. A black hole can be completely specified
by three parameters: its mass, spin and charge. All observable properties of the blackhole depend only on those three parameters; this is the so-called “no hair” theorem[38].
The geometry outside of a non-spinning black hole is given by the Schwarzschild
metric,2GM
ds = − 1 − 2
cr
2





2GM
c dt + 1 − 2
cr
2

2

−1

dr2 + r2 dθ2 + r2 sin2 θdφ2

(3.1)

and the geometry outside of an uncharged spinning black hole with angular momentum J is described, in terms of the Boyer-Lindquist coordinates (t, r, θ, φ), by the

20
Kerr metric,ds2

2GM r 2 2 4GM ar sin2 θρ2 2= − 1− 2 2c dt −
dφdt
+
dr
cρ
cρ2
4


2GM ra2 sin2 θ2222+ρ dθ + r + a +sin2 θdφ2 ,c 2 ρ2(3.2)

where the spin a ≡ J/cM , ρ2 ≡ r2 + a2 cos2 θ and 4 ≡ r2 − 2GM r/c2 + a2 .
From this point forth when talking about spin we refer to the dimensionless spinparameter â = Jc/GM 2 . This ranges between 0 for a Schwarzschild black holeand 1 for an extreme Kerr black hole. It is related to the spin a defined above byâ = ac2 /GM .

3.3

Quasi-Normal Modes of Black Hole Oscillation

An astrophysical black hole can become perturbed by a number of processes, forexample by a massive object falling into it, by the merger of two black holes, or inits formation through the asymmetric core collapse of a massive star. In this section
we discuss the emitted gravitational waveform.

3.3.1

Schwarzschild Black HolesIn 1957 Regge and Wheeler [39] investigated the stability of the Schwarzschild blackhole to small perturbations. Their study found that a disturbance of the black holefrom sphericity would not grow with time, but would oscillate about the equilibrium
configuration in a superposition of quasi-normal modes. They found that the solution
to the linearized Einstein equation could be expressed in terms of spherical harmonics
Ylm . Each mode has a characteristic complex angular frequency ωlm ; the real part isthe angular frequency and the imaginary part is the inverse of the damping time τ .
In subsequent sections we will express ωlm in terms of the oscillation frequency flm

21
and the quality factor Qlm :−1
ωlm = 2πflm − iτlm
= 2πflm − i

πflm
.
Qlm

(3.3)

In 1985 Leaver [40] determined the fundamental l = 2 and l = 3 modes, as wellas the first 62 overtones, indexed by n. The first of these are listed in table 3.1 [40].
The most slowly damped mode (i.e., that with the lowest value of the imaginary
part of the frequency) was found to be the l = 2, n = 0 mode. Figure 3.1 [40, 41]
shows the real part versus the imaginary part of the frequency, for a selection of
the l = 2 and l = 3 modes. The figure demonstrates that the imaginary part of
the frequency grows very quickly with n indicating that higher-order modes do notcontribute significantly to the emitted gravitational radiation. In contrast, the realpart of the frequency asymptotes to a constant value. The l = 2, n = 0 mode ismarked with a box in the figure. It was verified in 1993 that an infinity of thesemodes exist [42].

Table 3.1: Quasi-normal modes of oscillation for a non-spinning black hole [40].
l nMω
2 0 0.3737 − ı0.0890
1 0.3467 − ı0.2739
2 0.3011 − ı0.4783
3 0 0.5994 − ı0.0927
1 0.5826 − ı0.2813
2 0.5517 − ı0.4791

3.3.2

Kerr Black HolesIn 1973 Teukolsky [43] addressed the problem of perturbations of a rotating black
hole. In this case the linear equations describing the gravitational perturbations were
decoupled into spin-weighted spheroidal harmonics s Slm , where the spin weight s is
-2 for gravitational perturbations. In the same study as referenced above, Leaver

22

Figure 3.1: The imaginary part of the frequency ωI versus the real part ωR for
overtones of the l = 2 and l = 3 modes of a Schwarzschild black hole. The mostslowly damped mode has l = 2, n = 0 is marked by a black box. (The data in theplot is from [40] and [41].)

presented the l = 2 modes for different m and spin, showing that spin removed the
2l + 1 degeneracy in m. This is demonstrated in figure 3.2 [44] where the spin of thel = 2, n = 0 mode (shown as a single point in figure 3.1) is allowed to vary from0 ≤ â < 1 resulting in five different quasi-normal frequencies for each value of â. The
point where the five lines converge has â = 0. In figure 3.3 we plot the real part of
the frequency as a function of the spin for the l = 3 and l = 2 modes. Note that for
the l = m = 2 mode M ωR ranges from 0.37 for a non-spinning black hole to 0.9 fora maximally spinning black hole. In figure 3.3 we plot the real part of the frequency
as a function of the spin for the l = 3 and l = 2 modes.
Other major contributions to the understanding of the quasi-normal modes of a
black hole came from Vishveshwara [45], Zerilli [46], Press [47], Price [48], Chandrasekhar and Detweiler [49], Ferrari and Mashhoon [50]. A nice review of quasinormal modes can be found in [51].

23

Figure 3.2: A demonstration of how spin removes the degeneracy in m for the l = 2
mode. The cyan, magenta, green, red, and blue sets of points correspond to m = −2,m = −1, m = 0, m = 1, and m = 2, respectively. The frequency where the points
converge corresponds to â = 0. (The data in this plot is from [44].)

Figure 3.3: The real part of Mω as a function of spin for the l = 3 (upper group) andl = 2 (lower group) modes. (The data in this plot is from [44].)

24

3.3.3

The l = m = 2 Mode

3.3.3.1

The Black Hole Physical Parameters

Echeverria [52] found an analytic fit to Leaver’s calculations, relating the complex
frequency of the l = m = 2 mode to the black hole’s physical parameters mass M
and dimensionless spin parameter â,
f220
Q220

i
3
1 c3 h10=
1 − 0.63 (1 − â)
2π GM
9= 2 (1 − â)− 20 .

(3.4)
(3.5)

The inverse of these equations is given by" 23 #
1 c3
2
M =
1 − 0.63
2π Gf220
Q220

 209
2
.
â = 1 −
Q220

(3.6)
(3.7)

Note that the spin of the black hole depends only on the quality factor, as shownin figure 3.4, whereas the mass depends on both quality and frequency. Figure 3.5
shows the frequency as a function of mass for three spin values, â = 0, 0.5, 0.98. The
mass range reflects the sources that LIGO is most sensitive to.
We can use a simple model from continuum wave mechanics to predict the frequency for a given mass. Taking the wavelength λ to be the circumference of theblack hole at the Schwarzschild radius, λ = 2π (2GM/c2 ), the frequency is given by
f0 =

c
c3
=
,
λ
4πGM

(3.8)

which is the same order of magnitude as equation (3.4). The quality factor is related
to the rate of dissipation of energy,
d/dt
2πf0
=
.
Q


(3.9)

Energy is lost from the perturbation due to gravitational waves escaping to infinity or

25

Figure 3.4: Quality factor versus dimensionless spin factor for the l = m = 2 mode.

Figure 3.5: Frequency versus mass for â = 0 (blue line), â = 0.5 (green dashes), andâ = 0.98 (red dash-dot) for the l = m = 2, n = 0 mode.

26
falling in to the hole. We see from equation (3.5) that the quality factor grows withthe spin â. One explanation for this is that as the black hole spins, the spin energy
couples to the perturbation, amplifying it and decreasing the damping time. This isanalogous to the r-mode instability in rotating neutron stars [53].
From this point on we assume that the gravitational wave far from the source isdominated by the most slowly damped mode, l = m = 2 and neglect any contributions
from higher-order modes. We write the central frequency of the waveform as f0 andthe quality as Q.
3.3.3.2

The Ringdown Waveform

Far from the source the waveform can be approximated byh0 (t) = <A −ıωt
e
r




=<

A −ı(2πf0 −ıπf0 /Q)ter
(3.10)

where A is the amplitude of the l = m = 2 mode and r is the distance from thesource. This is usually expressed ash0 (t) =A − πfQ0 tecos (2πf0 t)
r(3.11)and this is the form we will use in subsequent chapters. An example of three ringdown
waveforms with f0 = 100 Hz and Q = 2, 10, 20 for a source at a distance of 100 Mpc
is shown in figure 3.6. The plus and cross polarizations of the wave areh+ (t) =
1 + cos2 ι h0 (t)h× (t) = 2 cos ι h0 (t)(3.12)(3.13)

where ι is the inclination angle of the source. The strain produced in the detector isthenh(t) = h+ (t)F+ (θ, φ, Ψ) + h× (t)F× (θ, φ, Ψ)(as described in section 2.2.4).

(3.14)

27

Figure 3.6: The ringdown waveform produced by a source radiating 1% of its massin gravitational wave located at a distance of 100 Mpc with frequency of 100 Hz andquality factor of 2 (blue), 10 (green), and 20 (red).

3.3.3.3

The Ringdown Peak Amplitude

We can evaluate A from the stress-energy tensor, equation (2.12). If  is the fractionof the black hole’s mass radiated as gravitational waves, then1 c2
M c =
T00 dV =
16π G
V
2

Z

Z

dV (ḣ2+ (t) + ḣ2× (t)).

(3.15)V

Solving this equation for A gives
r
A=

5

2



GM
c2



1

1

where
F (Q) = 1 +1

Q− 2 F (Q)− 2 g(a)− 2724Q2g(a) = 1 − 0.63 (1 − a)3/10 .

(3.16)

28

3.3.4

Energy Emitted as Gravitational Waves

The amount of energy emitted as gravitational waves during the ringdown phase, ,depends on the magnitude of the perturbation. For the example of a mass m falling
into a black hole of mass M it is reasonable to expect that the energy released is
proportional to some function of the ratio m/M . This was first calculated by Davis,
Ruffini, Press, and Price in 1971 [54], for the case of a mass m  M falling into a
non-spinning black hole. They found that the energy emitted was given by [54], m 2 ≈ 0.0104,
M(3.17)

with ∼ 90% of the radiation emitted in the l = 2 mode and ∼ 8% in the l = 3 mode.
Flanagan and Hughes [55] estimate an upper limit of 3% on the energy emitted
in the l = m = 2 mode for the binary coalescence of equal-mass black holes byconsidering the mode’s amplitude when the distortion of the horizon of the black holeis of order unity. For an unequal-mass binary they assume that the amount of energy
emitted is reduced by the factor (4µ/MT )2 , where µ is the reduced mass of the binary
and MT is the total mass.
Numerical simulations of binary coalescence can tell us how much energy is radiated at various stages of the evolution. Figure 3.7 [56] shows a plot of M ω as afunction of time for an equal-mass non-spinning binary. The plot is annotated withthe amount of energy emitted. From this we can clearly see that the value of  depends
on how we define the start point of the ringdown. For our purposes, an estimate of = 1% is reasonable.

3.4

Astrophysical Black Holes

Black holes do not emit electromagnetic radiation, and thus cannot be observed witha telescope. However the influence of their strong gravitational field on nearby mattercan be observed electromagnetically and it is by this indirect means that astronomers
can infer the presence of a dark compact object. If there is sufficient evidence to29

Figure 3.7: M ω as a function of time for the gravitational wave emitted as an equalmass black hole binary undergoes coalescence. The energy emitted is indicated by the
annotations. ωc and ωλ correspond to different methods for extracting the frequency
from the numerical waveforms. This figure was taken from Buonanno et al. [56].

rule out alternative sources such as a cluster of neutron stars or brown dwarfs it iscalled a black hole candidate. Only the detection of gravitational waves will provide
unambiguous evidence for the presence of a black hole. However the gravitational
wave community can benefit from astronomers’ observations of the electromagnetic
signature of an event that could be accompanied by gravitational waves such as asupernova or gamma ray burst.
Astrophysical black holes have been divided by mass into three categories. Stellar
mass black holes, which are believed to form as the end points of stellar evolution, lie
in the range 3 ≤ M/M ≤ 20; 3 M is the upper limit on the mass of the neutron star[57]. Supermassive black holes, the engines behind radio galaxies and quasars, areobserved to have masses in excess of 106 M and little is known about their formation

30[58]. Recent claims of evidence of black holes with masses in between these categories
prompted the creation of a third category, intermediate-mass black holes, with a massinterval roughly defined as 102 ≤ M/M ≤ 105 . It remains to be seen whether these
are the high (low) mass end of the stellar mass (supermassive) black hole population
or distinct population. An excellent review of intermediate-mass black holes may befound in [59].

3.5Previous Searches for Gravitational Waves from
Perturbed Black HolesThe first LSC ringdown search was carried out by Creighton [60] on data from the
LIGO 40 m prototype in 1994 using the GRASP software [61]. A single filter wasused to demonstrate the implementation of matched filtering in a gravitational wave
search. Although the poor detector sensitivity made detection extremely unlikely,
the methods used laid the foundations for subsequent searches, including the analysis
described in later chapters. The method of coincidence analysis was demonstrated
by dividing the data set in two and treating one half as if it originated from a second
detector located 3000 km from the first.
In 2004 Adhikari [62] performed a matched filter ringdown search with coincidence
analysis from the 300-hour-long second LIGO science run (S2) using the LIGO Algorithm Library (LAL) [63] software package. In this analysis simulated signals were
injected (in software) into the data stream and a detection efficiency was calculated
as a function of strain. For the most sensitive band of the detector at Livingston,
150–450 Hz, the 50% efficiency was located at a peak strain of 5 × 10−20 .
The TAMA collaboration also carried out a search for ringdowns using data fromtheir 300 m interferometer [64]. In total they analysed 1000 hours of data fromtheir sixth and eighth science runs in 2001 and 2003, respectively. The detectors were
maximally sensitive to ringdowns at a frequency of ∼ 1 kHz at a strain of ∼ 5×10−21 .
This was approximately a factor of 2 more sensitive at 1 kHz than the S2 LIGO data,31
but at least an order of magnitude less sensitive than LIGO at 200 Hz for the S2 run.
The search described in this thesis was carried out on data from the fourth LIGO
science run (S4) which amounted to ∼ 360 hours of triple coincident data at significantly improved strain sensitivity than any of the previous searches. Matchedfiltering was implemented using the LAL software. We performed a coincidence analysis between each of the three LIGO detectors and, using a study of simulated signals,
present an upper limit on the rate of ringdowns in the local universe.

32

Chapter 4
Matched Filtering
4.1

Introduction

In chapter 2 we described the generation and propagation of gravitational waves fromperturbed black holes and the general form of the wave far from the source. We
discussed the output of the detectors which may or may not contain a signal buriedin the noise. This chapter is concerned with how to uncover such a signal. When thesignal is known, the optimal method of extracting the signal from Gaussian noise ismatched filtering [65, 66, 67]; we will demonstrate this in section 4.2.
The convention used for the Fourier transform of a signal w(t) in this analysis isZ ∞w̃(f ) =w(t)e−2πıf t dt

(4.1)

w̃(f )e2πıf t df.

(4.2)

−∞

and for the inverse Fourier transform is
Z ∞w(t) =
−∞

33

4.2

Wiener Optimal Filtering

Consider a detector output s(t) which may or may not contain a weak signal of known
form h(t) superimposed on the noise n(t)s(t) =

 n(t)

signal absent(4.3)

 n(t) + h(t) signal present.
We assume without loss of generality that the signal, if present, occurs at t = 0. We
also assume that the detector output is a stationary random process with zero mean,
Gaussian probability distribution, and a one-sided power spectrum Sn (f ) defined by1
δ(f − f 0 )Sn (f ) = hñ(f )ñ(f 0 )i .
2

(4.4)

As we know the form of the signal we are looking for, the best way to ascertain
whether or not it is present in the data is to pass the detector output through a filter
K(t). The output Z of the filter is a number given byZ ∞Z≡

K(t)s(t)dt.

(4.5)

−∞

Our aim is to choose K(t) such that Z will have a large value if the signal is present
and a small value if it is not. We define
Z ∞
H≡

Z ∞
K(t)h(t)dt

and

−∞

N≡

K(t)n(t)dt

(4.6)

−∞

where H is the filtered signal, N is the filtered noise and
Z =H +N(4.7)

34
if a signal is present. Note however that whereas h(t) is a well-defined signal withfinite duration, n(t) is a random process
Z ∞N (t) ≡K(t − t0 )n(t0 )dt0

(4.8)

−∞

and thus we average over an ensemble of instantiations of the noise,
N

2

Z ∞
=

|K̃(f )|2 Sn (f )df.

(4.9)

0

Using the convolution theorem and the fact that K(t) and h(t) are real (and soK̃(−f ) = K̃ ∗ (f ) and h̃(−f ) = h̃∗ (f )) we can write the filtered signal asZ ∞H=2K̃ ∗ (f )h̃(f )df.

(4.10)

0

Next we define a statistic, the signal-to-noise ratio (SNR) ξ, and aim to find a filter
K(t) that maximizes this quantity:
H2
hN 2 i2R∞
4 0 K̃ ∗ (f )h̃(f )df
= R
2
∞
K̃(f
)
Sn (f ) df
0
ih
i 2
h
p
p
R∞
∗
4 0 K̃ (f ) Sn (f ) h̃(f )/ Sn (f ) df
=
.
2
R∞
K̃(f
)
S
(f
)
df
n
0

ξ ≡

(4.11)(4.12)(4.13)The Cauchy-Schwarz inequality tells us that for two arbitrary functions A(f ) and
B(f )
2

Z
A(f )B(f )df

Z≤

2|A(f )| df

Z|B(f )|2 df.

(4.14)

pp
Identifying K̃(f ) Sn (f ) with A(f ) and h̃(f )/ Sn (f ) with B(f ) we can see that to
attain the maximum value of ξ, we need the equality in equation 4.14 to hold, which35
only occurs when A and B are equal up to a constant C, thus,ph̃(f )
K̃(f ) Sn (f ) = C p
Sn (f )

(4.15)orK̃(f ) = Ch̃(f ).
Sn (f )

(4.16)

Thus the optimal filter for detecting signals of known form in coloured Gaussian
noise is the Fourier transform of the signal h̃(f ) weighted by the inverse of the power
spectrum.

4.3

Detection Statistic for Gravitational Wave Searches

We can employ the method of matched filtering in the search for gravitational waves
from perturbed black holes, as the waveform is known; it is a damped sinusoid,πf0h(t) = cos (2πf0 t) e− Q t .

(4.17)In this section we derive the statistic to be employed in the ringdown search.
Returning to our initial equation for the filter output equation (4.5), and allowing
the signal to occur at some unknown time, the filter output isZ ∞K(t − t0 )s(t0 )dt0 .

Z(t) = C(4.18)−∞

Using the convolution theorem this can be expressed as
Z ∞Z(t) = Cs̃(f )K̃ ∗ (f )eı2πf t df

(4.19)−∞and substituting in the expression for the optimal filter, equation (4.16), we get
Z ∞Z(t) = Cs̃(f )h̃∗ (f )eı2πf t
df.
Sn (f )−∞

(4.20)

36
At this stage we could choose a threshold value of |Z| above which a signal would be
defined as being present and below which the signal is absent. However rather thanthresholding directly on the filter output we first normalise by the variance of the
optimal filter σ 2 ,
2

Z ∞

σ =2
0

h̃(f )h̃∗ (f )df,
Sn (f )

(4.21)

for which C = 2. We define a statistic ρ, the SNR of the normalize output of the
optimal filter, as
ρ(t) =|Z(t)|
σ(4.22)and choose a value ρ∗ on which to threshold. Thusif
 < ρ∗ the signal is absentρ ≥ ρ∗ the signal is present.

(4.23)

With this comes the possibility of false alarm and false dismissal; the former occurs
when ρ ≥ ρ∗ and no signal is present and the latter occurs when ρ < ρ∗ and a signal
is present. Thus ρ∗ must be chosen carefully so as to minimize the rate of false alarms
and false dismissals. As will be described in chapter 6, the data we are dealing within this search is non-Gaussian and non-stationary and so we need to apply further
measures to minimize the rate of false alarms and false dismissals.
It is convenient to define the inner product of a and b asZ ∞ã(f )b̃∗ (f )(a|b) = 2
df = 4<
Sn (f )
−∞

"Z
0

∞

#
ã(f )b̃∗ (f )df ,Sn (f )

(4.24)

where < [X] denotes the real part of X. This allows us to express σ 2 as
σ 2 = (h|h)(4.25)

1(s|h(t))(s|h(t)) = p
.
σ
(h|h)

(4.26)and the SNR as
ρ(t) =

37

4.4

Discrete Quantities

As described in chapter 2.2.3.1 the output of the detector is not continuous but adiscrete time series sampled every ∆t seconds. Thus, in order to filter the data weneed to modify the expressions described above. First consider time; we have N data
points (where we assume N is even) sampled over a time T . The discretized timeseries can be expressed as
j = 0, 1, . . . , N − 1

tj = j∆t(4.27)

and a function of time w(t) when discretized is denoted by w(tj ). When we Fouriertransform a function we seek a discrete frequency array
fk =

k
,
N ∆t

k=−

N
N
,...,
2
2

(4.28)

where k is an integer. A continuous function of frequency w(f ), once discretized, iswritten as w(fk ). Thus the Fourier transform in equation (4.1) can be approximatedby the sumw̃(f ) ≈N−1X

w(tj )e−ı2πfk tj ∆t

j=0

=

N
−1
X

wj e−ı2π(k/N ∆t)∆tj ∆t

j=0

= ∆t

N
−1
X

wj e−ı2πjk/N

j=0= w̃k ∆t,(4.29)

38
where the quantity w̃k is the discrete Fourier transform. Similarly the inverse Fouriertransform, equation (4.2), can be approximated by the sum
N−1X

w(t) ≈w̃(fk )eı2πfk tj ∆f

k=0
N
−1
X

=

∆tw̃k eı2π(k/N ∆t)∆tj

k=0

1
N ∆t

N −1

1 X
w̃k eı2πjk/N ,=N k=0(4.30)

where we have used that∆f = fk+1 − fk =

k+1
k
1
−
=
.
N ∆t
N ∆t
N ∆t

(4.31)The discrete form of the waveform we are searching for ish(tj ) = cos (2πf0 j∆t) e−πf0 j∆t/Q .

(4.32)In a similar manner we can express the filter output and the estimated variance as asum of discrete quantities;
N

Z(tj ) = 4−12Xs̃∗ (fk )h̃(fk )Sn (fk )k=0e−ı2πfk tj ∆f

N
−1
2

= 4

X ∆ts̃∗ ∆th̃kk

Sn (fk )k=0N

e−ı2π(k/N ∆t)∆tj

1
N ∆t

−1

2
4∆t X
s̃∗k h̃k −ı2πjk/N=e
N k=0 Sn (fk )(4.33)39andN

σ2 = 4−12Xh̃∗ (fk )h̃(fk )Sn (fk )

k=0

∆f

N

= 4

−1
2
X
∆th̃∗ ∆th̃k1
Sn (fk ) N ∆t
k

k=0
N

−1

2
4∆t X
h̃∗k h̃k=.
N k=0 Sn (fk )(4.34)

We will use these equations in chapter 5 when describing the pipeline.

4.5

Templated Matched Filtering Searches

In section 4.2 we demonstrated that the optimum filter to use in extracting a signal
from noise when the signal is known is the matched filter. However it is often the case,particularly in the gravitational wave searches that we are concerned with here, that
although the form of the signal is known, the exact values of the intrinsic parameters
λi (these parameters are central frequency and quality in the ringdown search) are
unknown. To overcome this we can create an array of filters, a template bank, suchthat each template has a different value of the intrinsic parameters covering the spaceof parameters of interest, and filter the data with each one. Of course, given thatthese are discretely placed over the parameter space, it is not likely that one of thefilters will have the exact parameters of the waveform we are looking for, but if theparameters are close enough, then the SNR will be high and may exceed the threshold.
In practice, templates are laid out in a very specific way, covering the entire space withas few templates as possible. A nice discussion of template spacing for gravitational
wave searches can be found in [68]. Some of the main points are illustrated here. We
define the match M between two templates ũ(f ; µ, λ) and ũ(f ; µ + ∆µ, λ + ∆λ) as theinner product between the two templates maximized over the extrinsic parameters µ40(such as time of arrival and phase),
M (λ, ∆λ) ≡ maxµ,∆µ (u(µ, λ)|u(µ + ∆µ, λ + ∆λ)) .

(4.35)

This is the fraction of the maximum SNR achieved by filtering a signal with a templatewith the same form but slightly different parameters. Expanding M in a power series
about ∆λ = 0 gives1
M (λ, ∆λ) ≈ 1 +
2



∂2M
∂∆λi ∂ ∆λj

∆λi ∆λj(4.36)∆λk =0from which we can define the metric1
gij (λ) = −
2



∂2M
∂∆λi ∂∆λj(4.37)∆λk =0so that the mismatch (1 − M ) between two nearby templates is equal to the squareof the proper distance between themds2ij = gij ∆λi ∆λj .

(4.38)For the ringdown templates the mismatch between two templates differing in frequency by df0 and in quality by dQ is given by [61]ds2"
3 + 16Q4
3 + 4Q2
1
2dQ−
2dQ df0=
8 Q2 (1 + 4Q2 )2f0 Q(1 + 4Q2 )#3 + 8Q2 2
+
df0 .
f02

(4.39)

We will find it useful when laying out the template bank to define φ = log(f0 ), asthen the metric coefficients no longer depend on f0 ,
ds2

"
#
2

1
3 + 16Q4
3+4Q
dQ2 − 2=dQ dφ + 3 + 8Q2 dφ2 (4.40)
8 Q2 (1 + 4Q2 )2
Q(1 + 4Q2 )= gQQ dQ2 + gQφ dQ dφ + gφφ dφ2 .

(4.41)

41
The templates form a two-dimensional lattice whose unit cell has sides of proper
length dl2 . The highest mismatch will occur for a signal whose parameters lie in the
middle of the cell, that is for ds2 = dl2 /2.

42

Chapter 5
The Search Pipeline
5.1

Overview

The set of steps we take in analysing data output from the interferometers in order todetect gravitational waves is known as a search pipeline. As discussed in chapter 4,
for the case where the waveform is known we implement the method of matched filtering. However, as the noise in the data stream is non-stationary and non-Gaussian,matched filtering alone is not enough to extract a gravitational wave from the noise.
Noise can often mimic the signal we are searching for, and so a large effort goes into
characterizing the noise to best separate it from a potential gravitational wave signal.
We implement several consistency checks on any candidate events to increase ourdetection confidence.
The ringdown search pipeline is summarized in figure 5.1. Each step will be
explained in detail in this chapter, but in brief the main steps are as follows:• Data from each detector is read in from frame files and conditioned.
• The template bank is generated and the data is filtered, yielding a set of trigger
files for each detector.
• The triggers from each detector are then brought together and compared (i.e.,
put through a coincidence test).
• Those triggers failing the coincidence test are discarded and those that pass arefollowed up on as candidate events.

43

Figure 5.1: The ringdown search pipeline.

The pipeline is run on large computer clusters using the Condor high-throughput
computing system [69], a specialized workload management system for computeintensive jobs. The steps in the pipeline can be broken into individual jobs which arescheduled by DAG Man (Directed Acyclic Graph Manager) and submitted to Condor
to run in an order specified by a directed acyclic graph (DAG). Condor places thejobs in a queue, chooses what cluster node to run them on and allows the user tomonitor their progress.

5.2

Preliminaries

Prior to launching the pipeline three files are required: list of times to be analysed, alist of times to be vetoed, and a configuration file containing arguments required by
the search pipeline.

44

5.2.1

Segment Lists

We begin by generating a list of times for which data is available for each detector.
This segment file contains the GPS start time, GPS end time, and duration of eachinterval of time for which the detector was taking science quality data. A sample
of a segment file is shown in table 5.1. In generating the list we apply category 1
vetoes which preclude times when the quality of the data was unacceptable from thesegment list (see section 6.3 for more details).

Table 5.1: A sample of the H1 S4 segment list
segment number GPS start time GPS end time Duration
1
793154953
793155973
1020
2793162453
793162693
240
3793166413
793170673
4260
4793171813
793175893
4080
5793176613
793179853
3240

5.2.2

Create the Veto Lists

We also create a list of times when the quality of the data was suboptimal due
to a known source of noise making the detection of gravitational waves impossible.
Triggers during these times are removed before the coincidence stage of the pipeline.
These are known as category 2 and 3 vetoes (and are described further in section 6.3).

5.2.3

Create the Configuration File

The configuration file lists all of the arguments needed by the pipeline. All of theparameters mentioned in the following sections, such as thresholds, coincidence windows, clustering windows, and template bank boundaries are specified in this file. An
example of the configuration file used in the S4 search can be found in appendix A.

45

5.3

Launch the Pipeline

The pipeline is launched by running the python script lalapps inspiral hipe.
This reads in the parameters listed in the configuration file, and generates a set offiles with instructions on how the different parts of the pipeline are to run.
The first step is the segmentation of the data. The input lists of variable-lengthscience segments are subdivided into contiguous 2176-s-long analysis segments overlapping each other by 128 s on each end, and written to a file. If there is a non-integer
number of 2176 s analysis segments in the science segment, with the remainder n s
in length, then the final analysis segment begins (2176-n) s earlier, overlapping theprevious analysis segment, but only the previously unanalysed data is analysed (seeappendix B for an example of segmentation).
Files containing instructions on how each of the main jobs, “datafind”, “ringdown”, and “rinca” is to run are also created;• The datafind job runs LSCdataFind to get the location of the frame files on
disk.
• The ringdown job runs the main data conditioning and filtering code
lalapps ring.
• The rinca job runs the coincidence step of the analysis lalapps rinca.
The DAG file is submitted to the Condor pool with the condor submit dag command. The DAG specifies that the datafind jobs run first, followed by the ringdown
jobs and then the rinca jobs.

5.4

The Filtering Section of the Pipeline

Each of the steps described in this section are run on an individual analysis segment
basis.

46

5.4.1

Read In and Condition the Data

The uncalibrated data read in from frame files is sampled at 16384 s−1 ; however toreduce the computational cost we re-sample to 8192 s−1 , and a Butterworth low-passfilter is used to remove any power above the new Nyquist frequency, 4096 s−1 . The
data is then high-pass filtered to remove power below the frequency range of interest,
40 Hz. Although our basic analysis segment is 2176 s in length, an additional 8 s ofscience data is read in before the start time of the segment and after the end time
defining each analysis segment. This padding data is used for these data conditioning
steps in order to avoid any corruption of the data in the analysis segment. Once thesedata conditioning steps have been completed, the 16 s of padding data is removed.
No data between contiguous segments is lost, however the first 72 s at the start of ascience segment and the last 72 s at the end of the science segment are not used in
the search. (An example of segmentation is given in appendix B.)

5.4.2

Calculate the Response Function

Next the response function R(f ) for the segment is calculated. As discussed in section
2.2.3.1 the output of the gravitational wave channel is converted to strain via the
response function using the calibration coefficients read in for that particular epoch.
The numerical value of the response function is very small, ∼ 10−15 , and so to savethe computational cost of extra precision we scale this quantity by the dynamical
range factor dyn = 1020 . The scaled response function, shown in figure 5.2 has unitsof strain counts−1 .

5.4.3

Calculate the Power Spectrum

The interferometer noise is characterized by the one-sided power spectral density
Sn (f ) introduced in chapter 4. For this search it is calculated using the medianmean method. In each 2176 s segment the first and last 64 s are discarded and theremaining 2048 s is split into sixteen 256 s blocks which overlap each other by 128 s.
(This segmentation is discussed in more detail in appendix B.) These are divided (by47

Figure 5.2: The response function (scaled by dyn) for H1 between GPS times
793165201 and 793167377.

block number) into even and odd groups and are transformed to the Fourier domain.
The bin-by-bin median of the even blocks is calculated, as is the bin-by-bin medianof the odd blocks. The uncalibrated spectral density Sv (f ) in units of s counts2 isthen the bin-by-bin mean of these two medians. In practice it is the inverse of thisquantity that is required and so we invert it to get the inverse uncalibrated power
spectrum Sv−1 , with units of s−1 counts−2 . We calibrate the spectrum by dividing
by the square of the (scaled) response function. Finally, the inverse calibrated power
spectrum shown in figure 5.3 in units of s−1 strain−2 can be written as1
1=
S −1 .
Sn (f )
dyn2 R2 v

(5.1)

48

Figure 5.3: The inverse calibrated spectrum for H1 between GPS times 793165201
and 793167377

5.4.4

Calibrate and Fourier Transform the Data

Next, 2176 s of data s(t) is read in in units of ADC counts, and divided into sixteen 256
s overlapping blocks. Each block is Fourier transformed and calibrated, converting it
into units of s strain. Finally we multiply the data by the inverse calibrated spectrum,
giving the frequency series
Sv−1∆t s̃k ,
dyn R

(5.2)

with units of strain−1 .

5.4.5

Generate the Template Bank

Five user specified parameters are required to lay out the template bank; the maximum mismatch ds2max (the maximum value of the mismatch between a signal andthe nearest template that will be tolerated in the search, see section 4.5), and thefrequency and quality boundaries, fmin , fmax , Qmin , and Qmax (the tuning of these49
constraints is discussed in section 7.4.1). Recall from section 4.5 that if we define φ = log(f0 ) the metric coefficients no longer depend on f0 and so we use
φmin = log(fmin ) and φmax = log(fmax ). Then, starting at the point (Qmin , φmin )p
one moves across the Qmin line incrementing φ in steps of 2ds2max /gφφ until φmax ispreached. Then Q is incremented by 2ds2max /gQQ and the process is repeated until
the point (φmax , Qmax ) is reached. The tuning of these parameters is discussed in
section 7.4.1 and the final template bank is shown in figure 7.5.

5.4.6

Create the Template

As discussed in section 4.4, the template used in this search is given in its discrete
form byhj = e−

πf0 j∆tQ

cos(2πf0 j∆t),(5.3)

where, as mentioned earlier ∆t = 1/8192 s. The length of the template was set to ten
e-folding times, tmax = 10 τ where τ = Q/πf0 . The template time series is Fouriertransformed to giveh̃(fk ) = ∆t h̃k(5.4)

with units of strain/Hz.

5.4.7

Filter the Data

The filtering step is done template by template for each of the sixteen 256 s overlapping blocks of data in a 2176 s analysis segment. We multiply the template and thedata weighted by the power spectrum and inverse Fourier transform to get the timeseries

N

−1
2


−1
X
2
Sv
<
(∆t s̃∗k ) (∆t h̃k )eı2πjk/N .
z(tj ) =
∆t N  k=0 dyn R

(5.5)

50
The variance of the template σT is evaluated according to equation (4.34) as
N

σT2 =

−1

2
i
4 X
Sv−1 h22<(∆t h̃k ) + =(∆t h̃k ) dyn2 .
22N ∆t k=0 dyn |R|(5.6)

Next, using the user-defined threshold on the SNR, ρ∗ , we calculate the equivalentthreshold on the filter output, z ∗ , for a given template, T ,z ∗ = ρ∗

σT
2 dyn

(5.7)

and compare the series z(t) to z ∗ . Note that this is done on a template by templatebasis for each 256 s block of data.

5.4.8

Cluster the Filter Output

For data handling purposes it is preferable at this stage to cluster those triggers above
threshold in time. This is achieved with a sliding window of 1 s in width; starting at
the beginning of the time series, the loudest trigger is temporarily assigned to zmax
and the window is moved so that the left edge coincides with the time of zmax , t(zmax ).
If a trigger within this new window exceeds zmax , then this becomes the new zmax andthe window moves once more. In order for this to be a symmetric window, the sliding
is continued until a window is reached where there are no further triggers above zmax .
When this occurs, t(zmax ) is deemed to be the time of the trigger and written out
to a file. The window is shifted by 1 s and the process starts again, continuing until
the end of the block is reached. This is illustrated by figure 5.4; the threshold z ∗ isdenoted by the green horizontal lines, and it is clear that there are four groups ofdata points with z > z ∗ which are separated from one another by more than 1 s. The
loudest data point in each of these clusters, denoted by a red circle, is recorded as a
trigger. This process is repeated for each template. The effectiveness of this methodof clustering in reducing the level of background is discussed in section 7.4.2.

51

Figure 5.4: An example of single-template clustering with a window of 1 s. The
threshold is marked by a horizontal green line; triggers between the two lines are
below threshold. The red circles mark the loudest trigger in the cluster.

5.4.9

Calculate and Record the Trigger Parameters

The SNR of the clustered triggers is calculated asρ(tj ) = z(tj )

2 dyn
.
σT

(5.8)

The amplitude A is calculated from the template parameters f0 and Q according toequation (3.16) with  = 0.01. The sensitivity σ of the detector to a signal at 1 Mpc
is evaluated as
σ 2 = (1 Mpc) σT2 A2 ,(5.9)

from which the effective distance of the trigger is calculated
Def f = (1 Mpc)σ.
ρ

(5.10)

52
All of the parameters mentioned here are recorded for each trigger. The triggers fromall of the templates in each of the 256 s blocks are written out to a file.

5.5

Coincidence Analysis

Prior to comparing triggers from multiple detectors, category 2 and 3 vetoes areapplied, and the triggers are clustered across all templates in the bank. Then, theremaining triggers from all three detectors are brought together and compared. Thosethat fail the coincidence test are discarded and those that pass are written out to a
file to undergo further examination. There are four possible types of coincidence:a triple coincidence consisting of triggers from H1, H2, and L1 and three types ofdouble coincidence, H1H2, H1L1, and H2L1.

5.5.1

Apply Category 2 and 3 Vetoes

Contiguous trigger files written out from the filtering stage are read in together by
the coincidence code lalapps rinca for each of the detectors. Category 2 and 3vetoes are applied to the data at this stage. The reason these times were not vetoedat the segment selection stage is because the science segments would be interrupted
further and the likelihood of data being lost increases (recall a minimum science
segment length of 2176 s is required in the analysis). The data during times flagged
as category 2 and 3 is not so bad that including it adversely effects the calculationof the power spectrum. (This is not true for category 1 times and they are removed
at the segment selection stage.) Thus, as the detection of gravitational waves duringcategory 2 or 3 times would be very difficult, these times are removed before wecompare data from multiple detectors.

53

5.5.2

Cluster the Single Detector Triggers Across the Template Bank

The most information that we need at the end of the coincidence stage is whether
or not triggers from different detectors were coincident in f0 and Q at a particulartime, i.e., coincidence of a single pair of templates is sufficient to draw our attention
to a particular time; we do not need to know about every pair of templates that were
found in coincidence at that time. In theory the template closest in f0 and Q to theactual signal will ring off the loudest; in practice noise will change this somewhat,
however we expect the values of these parameters to be close in different detectors.
Thus, recording the loudest trigger in a time interval shorter than the duration of
the gravitational waves we are sensitive to is sufficient. Therefore, before we compare
triggers, we first cluster over all templates in a (fixed) time window of 1 ms, retaining
the loudest trigger in that interval.

5.5.3

Implement the Time Coincidence Test

The time coincidence test requires that triggers be seen within a given time window
δt of each other for co-located detectors, and δt + 10 ms for the Hanford-Livingston
pairs, to allow for the extreme case where gravitational waves are emitted from a
distant source along the line connecting the two detectors. As will be explained in
section 7.4.3, the value of δt is determined by evaluating how wide this window needs
to be in order to recover as many simulated signals added to the data as possible,
while keeping the rate of accidental coincidences low. The optimal value for thissearch was found to be δt = 2 ms. (Note that the window is applied to each trigger
in a pair; thus triggers can be a total of 2δt s apart.)

5.5.4

Implement the Parameter Coincidence Test

For triggers surviving the time coincidence test the next consideration is the waveform
parameters. As demonstrated in section (5.4.5) the templates are not distributed

54

Figure 5.5: Contours of ds2 = 0.03 for the portion of the template bank between
f = 1600 Hz and 2100 Hz and Q = 2 to 30.

uniformly throughout the bank. Thus the use of fixed windows δf0 and δQ is not asuitable coincidence test. A more appropriate test is based on the metric distance ds2
as this quantity depends on both f0 and Q. Thus it may be used to define a windowthat essentially varies the size of δf0 and δQ depending on the region of the bankunder investigation. This is illustrated in figure 5.5. This window can be described
as a contour of a constant ds2 about each template.

5.5.5

Implement a H1H2 Distance Cut

If a gravitational wave ringdown is detected in both H1 and H2, then given that H1
is twice as long as H2 and they have correlated displacement noise, we expect that22
ρH1 ≈ 2ρH2 . Furthermore, σH1≈ 2σH2and thus we would expect the effective dis-

tance measured by each instrument to be consistent. Inconsistent effective distancesfrom co-located detectors would suggest that the coincident pair was not due to a
gravitational wave signal but caused by noise. Thus a coincidence including a H1H2

55
trigger is retained only if
def fH1> κ,
def fH2(5.11)

where κ has a user-specified value. As will be discussed in section 7.4.4, the conservative value of κ = 2 was used in the search.
This test is not possible to implement in L1 primarily because the arms of thisdetector are not aligned with those of the Hanford detectors, nor do the arms lie inthe same plane. Thus, one could imagine a situation where a source was optimallyaligned and oriented for L1 producing a loud trigger, whereas in the Hanford detectors
the signal would be weaker by a factor γ. Assuming comparable sensitivities of thetwo 4 km interferometers, this would mean that the effective distance calculated forH1 would be a factor of γ higher than that calculated for L1.
After the time coincidence, parameter coincidence, and distance cut are imposed,
the resulting list of coincident groups (i.e., H1H2L1 triple coincidences, H1L1 double
coincidences, H1H2 double coincidences, and H2L1 double coincidences) are timesorted, with the members of each group listed alphabetically by interferometer name.
This list of coincidences (which contains all the information from the original trigger
files) is written out to a table.

5.5.6

Cluster the Coincidences

The output coincidence files generally have multiple groups of coincident triggers
lying in a short time window, as a noise event (or indeed a gravitational wave signal)
will ring up several templates, a number of which may be found in coincidence with
one or more templates in the second and possibly the third interferometer. As justone group is sufficient for drawing our attention to a particular time we can cluster
the coincident groups of triggers within a short time window, retaining the most“significant” group, using the LAL program lalapps coincringread. A (fixed)
window of 10 s was used in this analysis. Ranking the significance of the groups isachieved by defining a detection statistic ρDS , discussed below. The final clusteredgroups were once more written out to a sngl ringdown table and followed up as

56
detection candidates.

5.5.7

The Detection Statistic

Defining a detection statistic is a mechanism for ranking groups of coincidences. It
is a process whereby a single number is assigned to a coincident group of triggers
describing the collective significance of the group. In choosing a detection statistic theaim is to find one which will best discriminate between signal and background. We use
the detection statistic in a number of stages throughout the analysis: for clustering,for comparison of foreground and background, and in deciding which coincidences tofollow up at the end of the pipeline.
There are a number of valid criteria that one could impose in the choice of detection
statistic, for example one could choose the group of coincident triggers whose elements
are closest in time, or which have the smallest ds2 . The parameter we choose to use
for the selection process is SNR, i.e., we measure the loudness of the group. For triple
coincidences a good statistic to discriminate between signal and background is thesum of the squares of the SNRs,ρDS = ρ2H1 + ρ2H2 + ρ2L1 .

(5.12)For doubles, the above statistic was impractical because of the high level of nonGaussian noise. Instead, we required a ρDS that prevents a high-SNR glitch in one
interferometer from (unfairly) influencing which group is chosen as the most significant. Hence for a double coincidence in interferometers 1 and 2 the detection statistic
used is the chopped-L;1
ρDS = min{ρifo1 + ρifo2 , aρifo1 + b, aρifo2 + b},(5.13)

where a and b are parameters that are tuned for the particular search. A discusssionof the detection statistics and the tuning of a and b is given in section 7.4.4.
1This is similar to the bitten-L used in the inspiral analysis, but with a flat “bite”.

57

Chapter 6
The S4 Data Set
6.1

Introduction

In this chapter we discuss general aspects of the S4 run.

6.2

The Fourth LIGO Science Run

The fourth LIGO Science (S4) run took place between February 22nd and March 24th ,2005, — GPS times 793130413 to 795679213, a total of 708 hours. This yielded 567.4
hours of analysable data from H1, 571.3 hours from H2, and 514.7 hours from L1.
In this analysis we require that data be available from at least two detectors at any
given time. This results in approximately 364 hours of triple coincidence and 210
hours of double coincidence as shown in figure 6.1.
At their best during S4, H1 and L1 had comparable sensitivities at high frequencies, and H1 was more sensitive below ∼ 200 Hz. H2 was about a factor of two lesssensitive than the 4 km interferometers. Figure 6.2 displays the best noise curves foreach of the interferometers during the S4 run.

6.3

Data Quality and Veto Categorization

There are times during each science run when some component of the instrument
malfunctions or an external disturbance couples to the gravitational wave channel

58

Figure 6.1: Time in hours of analysable S4 data

Figure 6.2: S4 strain sensitivity

59
introducing short bursts of noise into the data stream. This is troublesome for data
analysis as these glitches can often match many templates in a matched filter search,
producing high SNR triggers. This is of particular concern to the ringdown searchas the templates are short in duration, just like the glitches. This increases the false
alarm rate making it more difficult to detect a gravitational wave. In many cases
glitches can also be seen in auxiliary channels, for example, a glitch in the gravitational wave channel due to a seismic disturbance can also be seen in the seismometerchannels. During and after a science run, times during which it was known that thequality of the data was compromised because of noise are flagged. We refer to theseas data quality flags.
Members of the inspiral and burst analysis groups within the LIGO Scientific Collaboration have used information from auxiliary channels to flag times when the levelof noise in the detector was unacceptably high or the quality was questionable, andgrouped these times into four categories depending on the severity of the disturbance.
The categorization of data quality flags depends primarily on two quantities: (i) the
efficiency, that is the number of noise triggers above a particular SNR that it vetoes,and (ii) the dead-time, the total science time cut by applying the veto. Ideally wewant to veto times with maximum efficiency and minimum dead-time.
The first category of data quality flags vetoes time when the data is of very poor
quality and should not be analysed. These are applied at the segment selection
stages. We try to keep this set to a minimum because, as described in section 5.3, aminimum analysis segment length of 2176 s is required for filtering. Thus we want toavoid unnecessarily breaking up the data.
Data during which detection of a gravitational wave is not possible because ofa known correlation between an environmental or instrumental disturbance and thegravitational wave channel is marked by category 2 data quality flags. Examples
include times during which hardware injections (an actuation on one of the test masses
simulating a gravitational wave) are being performed (accounting for approximately
1% of the data) and overflows in any of the digital signals used in feedback loops
controlling the detector lengths. Category 2 vetoes effect 4% of the triple coincidence60
data, 56% of the H1L1 data, 3% of the H1H2 data, and 27% of the L1H2 data [70].
The third category of data quality flags includes those times during which there
was a statistical correlation between the gravitational wave channel and an external
disturbance. A gravitational wave may be searched for, but caution must be taken
when determining confidence levels. Data quality flags under this category include
elevated levels of dust reported by a particle counter in the vicinity of photo-detectors.
These times may be associated with glitches in the gravitational wave channel due toa dust particle passing across the beam.
For S4, triggers from the binary neutron star inspiral search were used to evaluate
efficiency and dead-times which led to the data quality flag categorization. This
categorization was then implemented in the binary black hole inspiral search (BBH)[16] and the ringdown search. In the inspiral analysis, data with category 2 data
quality flags is searched over for gravitational wave candidates but not included in theupper limit calculation, while category three data is included. The inspiral analyses
implement various tests to check the consistency of the data around a given trigger
with nearby templates, so-called signal-based vetoes. In the ringdown analysis we
do not implement signal-based vetoes and for this reason category 2 and 3 vetoes
were combined and no data lying in either category was used in the calculation of theupper limit. This affected 12% of triple time, 8% H1-H2 time, 62% of H1-L1 time,and 29% of H2-L1 time.
Times flagged as category four showed a weak but positive correlation with false
alarms. Data from these times were not vetoed, but the presence of such a flag on a
candidate event decrease our confidence in it being caused by a gravitational wave.

6.4

Horizon Distance

Using the noise curves discussed in section 6.2 we can get a sense of how sensitive
we are to gravitational waves from a particular black hole. The horizon distance DH
is the distance at which we can detect a ringdown from an optimally oriented and61
located black hole with signal-to-noise ratio (SNR) of 8, and is given by1
DH (f ) =82
Sn (f )

 21
hrss (1 Mpc)

(6.1)

where Sn (f ) is the average power spectral density of the detector noise and hrss isthe root sum squared of the strain for a signal with optimal orientation at 1 Mpc,
h2rss =



A
1 Mpc

2 

2
πf0




Q

1 + 2Q2
1 + 4Q2

(6.2)

and A is given by equation (3.16). The physical distance of the source is always lessthan or equal to the horizon distance. Figure 6.3 shows that in S4 a gravitational
wave from an optimally oriented black hole of mass 250 M at a distance of 350 Mpc
directly above (or below) the detector, will produce an SNR of 8 in H1. The source
would need to be at 140 Mpc and 270 Mpc to produce the same SNR in H2 and L1
respectively. In this calculation we have assumed that the black hole is spinning withâ = 0.9 and that 1% of the mass is radiated as gravitational waves. To put this into
an astrophysical context consider figure 6.4. This shows the universe out to ∼ 300
Mpc. According to [71], there are approximately three million large galaxies in thisregion of the sky. From [72] we can estimate the number of sources as 2.2 × 106 L10
or 1.3 × 106 MWEG.
A standard figure of merit used in the control room is the “inspiral horizon distance”; this is the distance out to which an optimally oriented and located 1.4 M −1.4 M binary neutron star system can be seen with an SNR of 8. This quantity has
been used throughout the five science runs as a means of evaluating the sensitivityof the detector during a run and as a comparison between different science runs. We
can make an analogous plot for ringdowns; we choose as our standard candle a sourcethat gives the maximum horizon distance in the above plot. This corresponds to an
optimally located and oriented black hole with mass 250 M and spin â = 0.9 with
 = 1%. Using the value of σ output from the filter with these parameters we canplot the horizon distance as a function of time for the duration of the run. As can

62

Figure 6.3: S4 ringdown horizon distance versus mass and frequency for â = 0.9 and = 1%.

Figure 6.4: Illustration of the galaxies with ∼ 300 Mpc of the Earth. Picture Credit:
Richard Powell [71].

63
be seen from figure 6.5, H2 was the most stable of the three detectors, particularly
during the first half of the run, with an average horizon distance of ∼ 150 Mpc.
During the last 15 days the horizon distance fluctuated about this value. H1 had thelargest horizon distance for this source, averaging ∼ 320 Mpc. The small-scale fluctuations can be attributed to the diurnal activities in the area with a minimum in thehorizon distance usually occurring around 10am local time, and the quietest time of
day occurring around 11pm. The larger dips were attributed instrumental problems
such as the beam drifting in the interferometer causing mis-alignment, or prolonged
environmental disturbances such as high wind (in excess of 15 mph) causing increased
seismic noise. The L1 range was consistently lower than H1 for the run, reaching atbest ∼ 320 Mpc. This dropped to 120 Mpc for days 6–15 because of lower light powerin the interferometer.

Figure 6.5: S4 ringdown horizon distance versus time for an optimally located andoriented black hole of mass 250 M , spin of 0.9 and  = 1% producing an SNR of 8
in the detector.

64

6.5

Detection of Binary Compact Coalescence

As mentioned in section 2.3.1, a promising source of gravitational waves is expected
to be from compact binary coalescences, in which case the ringdown will be precededby an inspiral and merger. In this section we examine which of the searches, theinspiral or ringdown, is more sensitive to gravitational waves from a given binary.
Compact binaries can be composed of two neutron stars (NS-NS), two black holes(BH-BS), or a neutron star and a black hole (NS-BH). As discussed in section 3.4,
neutron stars have a maximum mass of 3 M and stellar mass black holes are believed
to lie in the range 3 M< M < 20 M . Black holes with masses above 20 M

are referred to as intermediate-mass black holes, however, their existence is still inquestion.
In figure 6.6 we plot the strain of a number of compact binaries at an arbitrary
distance as the frequency of their emitted radiation passes through the Initial LIGO
band. The binaries considered are the NS-NS pair, 1.4 M –1.4 M , a NS-BH binary
with component masses 1.4 M –3 M , and BH-BH pairs with equal mass components
of 10 M , 20 M , 40 M , and 100 M . The blue line describes the inspiral, as it
sweeps through a range of frequencies. The end-point of the inspiral for the four least
massive binaries is at the inner-most stable circular orbit (ISCO), at which point the√frequency is fISCO = c3 /(6 6πGM ). The plot shows that for these binaries a large
proportion of the inspiral is in the LIGO band. In order to detect the inspiral fromthe higher mass binaries with a matched filter however, it is necessary to evolve thewaveform further, to the light-ring, at which point the frequency of the gravitational√
waves is fLR = c3 /(3 3πGM ). It is clear from the figure that as the mass of thebinaries increases, less and less of the inspiral is in-band. The ringdown on theother hand, which appears as a dot marking the single frequency of the gravitationalradiation emitted, is out of the LIGO band for the low mass binaries, but as themasses increase, the ringdown frequency decreases, bringing it into the LIGO band.
For the 100 M pair the ringdown is at the most sensitive frequency of the LIGO
detectors. In calculating the ringdown strain, we assume that the final black hole65
has a spin of 0.7 (this is the expected spin for a the black hole formed following themerger of two equal-mass, non-spinning compact objects) and that 1% of the mass isradiated.

Figure 6.6: A plot of S4 strain sensitivity versus frequency (red). The blue lines
represent the inspiral phase of a binary coalescence for binaries with masses (frombottom to top): 1.4-1.4, 1.4-3, 3-3, 10-10, 20-20, 40-40, and 100-100, in units of M .
The green dots represent the ringdown (for the same list of binaries) for a final blackhole with a spin of 0.7 and assuming 1% of the mass has been radiated.

We can estimate how sensitive each of the searches is to a given source by calculating the horizon distance. As before, the horizon distance is the maximum distanceto which an optimally oriented and located black hole will produce an SNR of 8 in thedetector. In this section we calculate the ringdown horizon distance for a black holewith spin â = 0.7. Figure 6.7 shows the H1 inspiral and ringdown horizon distances
for binaries with (total) mass of up to 100 M , using the S4 noise curve. The plotshows that the maximum inspiral horizon distance, ∼ 95 Mpc is attained for a binary
with a total mass of ∼ 30 M . For this mass, the ringdown search is only able to seeto ∼ 10 Mpc, and therefore, we are much less sensitive to the ringdown than we are to

66
the inspiral of a black hole binary with that mass. As the mass increases however, theringdown search becomes more sensitive and the inspiral search becomes less sensitive, with the cross-over point occurring at ∼ 62 M . Figure 6.8 shows the ringdown
horizon distance for the entire ringdown mass range along with the inspiral horizon
distance, demonstrating how much wider the ringdown mass range and further the
distance is for ringdowns.

Figure 6.7: The horizon distance as a function of mass for the S4 inspiral and ringdown
searches up to 100 M . A final spin of 0.7 and  = 1% are assumed in the calculationof the ringdown horizon distance.

6.6

Predicted Ringdown Rates

Here we discuss the predicted rate of ringdowns for the S4 search.

67

Figure 6.8: The horizon distance as a function of mass for the S4 inspiral and ringdown
searches, covering the full mass range of the ringdown search. A final spin of 0.7 and = 1% are assumed in the calculation of the ringdown horizon distance.

6.6.1

Stellar Mass Black HolesBlue light luminosity is a tracer of massive star formation, and therefore is also
expected to scale linearly with the binary coalescence of massive stars [73]. We
measure luminosity in terms of the blue light luminosity, in units of L10 which isdefined as L10 = 1010 LB, , and LB,= 2.16 × 1033 erg s−1 is the solar blue lightluminosity. Population synthesis models predict that the merger rate, R, of 10 M −1
10 M black hole binaries is R = 0.4 L−1[74]. Figure 6.7 showed that for a
10 Myr20 M black hole the ringdown search can see to approximately 3 Mpc. We can see
how many sources we are sensitive to at that distance with figure 6.9, the cumulative
blue light luminosity CL as a function of distance. This plot shows that at 3 Mpc we
can see approximately 5 L10 . Therefore the expected rate of stellar mass black holecoalescences detectable at S4 sensitivity, RS4 , is RS4 = RCL = 0.4 L10 Myr−1 × 5
L10 = 2 Myr−1 . Thus, given the sensitivity (and thus the distance reach) of S4, the68detection of a stellar mass black hole ringdowns is unlikely. However at these low
masses the S4 ringdown search is sensitive to the local group of galaxies, and thus
should an event have occurred there during the run, the ringdown search would becapable of detecting it.

Figure 6.9: The cumulative blue light luminosity as a function of horizon distance.
This figure was taken from [72].

6.6.2Intermediate-Mass Black HolesThe most likely formation scenario for an intermediate-mass black hole (IMBH) is bycore collapse of a very massive star [75]. Recent simulations have indicated that starclusters with binary fractions larger than 10% will produce two IMBHs which will
form a binary (IMBHB), which will eventually merge [76]. A model to predict therate of ringdowns from IMBHB merger for a given detector sensitivity [77] is
4π
RS4 = R
3



DH
2.26

3
ρGC g,(6.3)

69
where R is the rate of IMBHB merger per cluster, taken to be the age of the universe(as this is expected to occur just once in each cluster) (13.7)−1 Gyr−1 ; DH is thehorizon distance, which, from figure 6.8 is 300 Mpc for a final black hole mass of 200
M ; ρGC is the number density of star clusters sufficiently massive to produce an
IMBH, taken to be the current density of globular clusters, ρGC = 8h3 Mpc−3 ; andg, the fraction of globular clusters with a binary fraction high enough to produce an
IMBHB, is the most uncertain term in the model and is taken to be 10%. With thesevalues the predicted rate is RS4 = 1 × 10−4 yr−1 . Therefore, once again we see that it
is not expected that this type of event would occur within the range of the S4 search.
However, our knowledge of IMBHs from electromagnetic observations is very poor,and hence there is a large uncertainty associated with the assumptions made in thecalculation just described. Gravitational wave searches could provide the evidence
needed to affirm the existence of this population of black holes.

70

Chapter 7
Tuning the Search
7.1

Overview

In chapter 5 we outlined the analysis pipeline, the set of steps that are followed toanalyse data with the aim of detecting any gravitational wave ringdown signals that
may be present. We described a number of thresholds, cuts, coincidence windows,
and clustering techniques that are implemented to isolate the most likely gravitational
wave candidates. In LIGO matched filtering searches we adopt the “blind search”
philosophy, whereby the constraints are decided upon prior to looking at the full data
set. In this chapter we discuss methods used to determine optimal values for eachof these constraints in the S4 search, i.e., we tune the search. To achieve this we
employ the following three tools, each of which will be described in more detail inthis chapter:• Monte Carlo simulations of the ringdown waveform (injections),• time-shifted data sets (timeslides),• a representative subset of the data (playground).
To tune the search we run the pipeline several times on injections, timeslides andplayground each time modifying the constraints on the pipeline to get the desired
result. Once the tuning has been finalized the data set is unblinded, or in LIGO
language, the box is opened. Implementing a blind search prevents any bias on thepart of the analyst when tuning the search from influencing the result.

71

7.2

Implementing an Injection Analysis

Injections are simulations of the signals we are trying to detect whose parameters are
randomized within a given distribution. These simulated signals are added to thedata stream and the pipeline is used to recover them from the noise. We employ
injections for many purposes in the course of the analysis; in this section we discuss
the use of injections to tune the search. We utilize injections and timeslides in tandem to find a balance between recovering as many simulated signals in coincidence
between multiple detectors as possible while keeping the rate of false coincidences
to a minimum. (Background estimation via timeslides will be discussed in the next
section.)

7.2.1

Creating the Injection File

A table of injection parameters is created using the LAL program lalapps rinj
based on a set of input arguments.
• Time: Injections are placed at a random time within an interval of 250 s every
7000/π seconds. This ensures that (i) there is not more than one injection per
analysis segment, (ii) the injection does always occur at the same number of
seconds after the start of the analysis segment, and (iii) the injection does notoccur at the same fraction of a second each time.
• Frequency and Quality: For the purpose of tuning we want to cover the parameter space available to the search (this is dependent on the detector sensitivity).
Injections are made uniformly in quality factor Q between values of 2 and 22 andin logarithmic frequency log10 (f0 ) with 45 Hz ≤ f0 ≤ 2500 Hz. The distributionof injections in frequency is shown in figure 7.1.
• Sky location and source orientation: The injections are placed uniformly in
logarithmic distance between 1 kpc and 200 Mpc and uniformly in sky position(right ascension α and declination δ), as shown in figure 7.2. The inclination ι

72and polarization Ψ angles and initial phase φ0 of the injection are also uniformly
distributed.
• From these parameters the amplitude A, mass, spin, and the arrival time andeffective distance at each site are calculated using equations (3.16), (3.6), (3.7),
and (2.25) . The percentage of mass radiated as gravitational waves  is fixed
at 1%.

Figure 7.1: Distribution of injections in frequency over nine injection runs.

7.2.2

Adding the Injection to the Data

In the context of the pipeline described in chapter 5, the injection list is read in
directly after the raw data. The injection parameters are passed to the ringdown
waveform generation code which creates a time series array for the plus and cross

73

Figure 7.2: Distribution of injections’ sky location over nine injection runs.

amplitudes, a+ and a× , the frequency f and phase φ, where
πf0

a+ (t) = A(1 + cos2 ι) e− Q t ,
πf0

(7.1)

a× (t) = A(2 sin ι) e− Q t ,(7.2)

f (t) = f0 ,(7.3)

φ(t) = 2πf0 t + φ0 .

(7.4)

From these the plus and cross polarizations of the gravitational wave are created:h+inj (t) = a+ cos(2πf0 t + φ0 ),(7.5)

h×inj (t) = a× sin(2πf0 t + φ0 ).

(7.6)

Using information about the source and detector positions, the detector antenna
patterns F+ (α, δ, Ψ) and F× (α, δ, Ψ) are calculated and the injection waveform is

74
then given byhinj (t) = h+inj (t)F+ (α, δ, Ψ) + h×inj (t)F× (α, δ, Ψ).

(7.7)

The transfer function (the inverse of the response function) is applied to the
waveform so that it has the same units as the raw data, counts. The waveform isthen added data point by data point to the raw data as shown in figure 7.3, and thepipeline continues as normal. (As can be seen from figure 7.3 there is a discontinuity
where the injection starts. This did not have an impact on our ability to recover the
injections or on the accuracy of the parameter estimation.)

Figure 7.3: Demonstration of the addition of an injection to the raw data; amplitude(in arbitrary units of countes) versus time for 70 ms of raw data (blue) and raw dataplus an injection (red). The green line marks the time of the injection according to
the injection file. The inset zooms in on the injection, tinj = 0.

75

7.2.3

Identifying the Injection Triggers

The output of the pipeline is a list of coincident triggers consisting of the signals thatwere injected and coincidences in the data. In order to keep the search blind, at thisstage we only want to examine the times at which we added a simulated signal. We
do this by invoking the injection options in lalapps coincringread. The input
list of injections is read in along with the coincident triggers, and only those groupsof coincidences within a 100 ms time window of each injection are examined further.
As before, the coincident groups are clustered according to the detection statistic. It
is important to note that, even though additional information about the waveform
such as the waveform parameters, we do not utilize this as we want to apply a methodthat can also be applied to the zero-lag data where this additional information is notavailable. We can, however, use this knowledge after the fact to evaluate how well
the injections are recovered (that is the subject of chapter 8). The parameters of thesuccessfully recovered injections were written to a “found” file, while those that were
not recovered were written to a “missed” file. Injections in the list for times wheredata was not available are discarded, reserving the missed category for injections thatwere added to the data but were not recovered.

7.3

Background Estimation via Timeslides

As with any search it is important to get an estimate of the background or false alarm
rate. Unlike a particle detector we cannot simply turn off the source, or change theorientation of the instrument like one would do in astronomy. Because we requirecoincidence in time between triggers in multiple detectors, an alternative is to takethe trigger files from the filtering stage of two instruments and shift one set in timewith respect with the other by a time much longer than the expected length of thesignal (and light travel time for separated detectors) and look for coincidences. This is
illustrated by the cartoon in figure 7.4, where we can see that the timeslides are doneon a ring ensuring that each timeslide contains the same duration of data. In pairs

76
of detectors with uncorrelated noise (H1L1 and H2L1) this provides a good estimate
rate of accidental coincidences of noise triggers as the noise sources are completely
independent. In the case of H1H2 this is not such a good estimate because by sliding
the data sets we are removing known noise correlations from the background andactually under-estimating the rate of false coincidences. We slide L1 fifty times insteps of 5 s and fifty times in steps of −5 s, H2 in steps of ±10 s, while H1 stays
in place. Timeslides are implemented at the coincidence stage of the pipeline andare initiated by including the “timeslide” option in the configuration file. Just as
with the zero-lag and injection runs we cluster the coincidences and write them to asngl ringdown table.

Figure 7.4: Cartoon illustrating timeslides; data is time-shifted such that any coincidences are accidental, providing an estimate of the rate of false alarms. (Picture fromR. Tucker.)

7.4

Tuning the Constraints

Next we describe the process involved in tuning the various constraints using injections and timeslides and the final values chosen.

77

7.4.1

SNR Threshold, Template Bank Limits, and SamplingRate

On the first run through the data the SNR threshold was set to 6 in each detector. The
template bank frequency limits reflected the sensitivity range of the interferometers,
fmin = 50 Hz, fmax = 2000 Hz, and the quality factor range Qmin = 2 and Qmax = 20,
was chosen based on the likely range of spins from black holes 0 ≤ â ≤ 0.98. The
maximum mismatch of the template bank was set to 3%. We downsampled from 16384
s−1 to a new sampling rate of 4096 s−1 . As this yielded a very low rate of background
events it was decided to lower the threshold to 5.5 and extend the template bank tosearch between 40 Hz and 4 kHz. This new fmax necessitated a higher sampling rate
and thus 8192 s−1 was used. The result was a factor of two increase in the number oftriggers in H1 and L1 and a factor of four increase in the number of H2 triggers, giving
approximately 106 triggers from each interferometer in the playground. However, forreasons outlined in section 11.1.4 the increased scope of the template bank was not
feasible and it so was returned to its previous frequency range. The threshold andsampling rate remained at 5.5 and 8192 s−1 , respectively. This resulted in a dropof about 10% in the number of triggers in H1 and L1 and about 20% from H2. Tosummarize, the final values were:• threshold: ρ∗ = 5.5,
• sampling rate: 8192 s−1 ,• maximum mismatch: ds2max = 0.03,• template bank boundaries: fmin = 50 Hz, fmax = 2000 Hz, Qmin = 2, Qmax = 20.
With these values the final bank, shown in figure 7.5 consisted of 583 templates. The
same template bank was used for each detector throughout the run.

78

Figure 7.5: The template bank for the S4 ringdown search which, with the parameters
Qmin = 2, Qmax = 20, fmin = 50 Hz, fmax = 2000 Hz, and ds2max = 0.03 contained
584 templates.

7.4.2

Clustering the Filter Output

In section 5.4.8 we described a method of clustering the matched filter output using asliding window. This method is implemented to reduce the number of output triggers.
It assumes that any triggers occurring within 1 s of each other are correlated. Of
course there is the possibility that a noisy block of data could be clustered entirely,
giving one trigger for a particular template out of the 256 s of data, but our thresholdis sufficiently high that this does not occur. We call the length of time clustered overthe dead-time. The upper panel of figure 7.6 displays the dead-time before each of thetriggers in H1 during S4 and the upper panel of figure 7.7 displays the dead-time after
the trigger. The plots show that a high proportion of triggers have dead-times longer
than 5 s associated with time. However looking just at the triggers that survive thecategory 2 and 3 vetoes (the lower panels in figures 7.6 and 7.7), it is apparent thatmany of these long dead-times occurred during noisy times, and thus the clustering

79
successfully reduces the number of noise triggers output from each filter.

Figure 7.6: Dead-time before triggers in H1 due to clustering of the filter output. The
upper panel shows all the triggers while the lower panel only shows those that are
not cut by category 2 or 3 vetoes.

7.4.3

Coincidence Windows

To tune the coincidence windows described in section 5.5 we ran the pipeline on
injections and timeslides several times, starting with very wide windows and eachtime narrowing the windows and comparing the results. Starting with ds2coinc = 1(recall that in addition to being used for template placement the mismatch is also
used to define the parameter coincidence test) and time window δt = 5 ms (oneither side of the trigger) these parameters were decreased until we reached the point
where we started to lose injections. It was observed throughout this iterative processthat as one window was tightened the accuracy of the other generally increased.
This demonstrates that the clustering techniques described in chapter 5 were not too
stringent. The final tuned values were ds2coinc = 0.05 and δt = 2 ms for H1H2, and80

Figure 7.7: Dead-time after triggers in H1 due to clustering of the filter output. The
upper panel shows all the triggers while the lower panel only shows those that are
not cut by category 2 or 3 vetoes.

δt = 12 ms for Hanford-Livingston pairs to allow for the maximum gravitational wave
travel time between the sites.

7.4.4

H1H2 Distance Cut

When the coincidence windows had been fixed, the effective distance distributions ofinjections and timeslides were considered. As discussed in section 5.5.5 we expect thevalues of effective distance found by H1 and H2 to be similar for real signals whereas
for false coincidences they should be more randomly distributed. Figure 7.8, a plot of
H1-recovered effective distance for injections and timeslides, shows that even though
the distributions overlap, there is some portion of the timeslides that can be isolated
from the injections. We choose a value of κ = def fH1 /def fH2 = 2, denoted in the plot
by a green line. With this value we easily retained all of our injections and were able
to reduce the background by 15%. Figure 7.9 displays a normalized histogram of

81
κ = 2 for injections and timeslides.

Figure 7.8: The effective distance of H2 versus H1 for injections (red) and timeslides(blue). The green line marks the distance cut; all triggers below this line are discarded.

7.4.5

Detection Statistic

As discussed in section 5.5.7, the detection statistic is a ranking mechanism using
the SNRs of coincident triggers. The exact form of the detection statistic for a given
population (doubles or triples) depends on the properties of the SNR distributionsof simulated signals and background signals. For triple coincidences the level ofbackground is very low and the SNRs of the false coincidences all tended to have low
values of SNR. As distant simulated signals have a similar distribution the most logicaldetection statistic is the sum of the squares of the individual triggers. This, however,is not so for double coincidence background events; these tend to have long “tails”,
that is, coincidences with a very loud SNR in one detector and a much lower SNR
in the other. The contour plot of the H1L1 background SNR distribution shown infigure 7.10 illustrates this. Injections on the other hand generally lie on the diagonal,

82

Figure 7.9: The normalized κ = def fH1 /def fH2 histograms of injections (red) andtimeslides (blue). The green line marks the distance cut; all triggers to the right ofthis line are discarded.

— the component SNRs are comparable. If we implemented the triples’ detection
statistic for doubles then the background trigger at the point (5.5,1000) in figure 7.10
would be given the same significance as a trigger at (250,250) (250 = 102.4 ). Clearly
this does not make sense. Thus, while we cannot rule out the possibility that a real
signal lies in the tails, we do not want to give it a high significance. Our choice ofdetection statistic should reflect this.
A simple statistic to do this is the “chopped-L” statistic; for a coincidence withsignal-to-noise ratio (SNR) ρifo1 in detector 1 and ρifo2 in detector 2 the ranking insignificance isρDS = min{ρifo1 + ρifo2 , a1 ρifo1 + b1 , a2 ρifo2 + b2 }.

(7.8)

The values of a and b were chosen by considering both the injections and the timeslide
distributions. For simplicity we choose a symmetric statistic a1 = a2 = a, b1 =

83

Figure 7.10: A contour plot of the L1 signal-to-noise ratio versus the H1 signal-tonoise-ratio for double coincident timeslide event. The colour-bar represents log10 (N )where N is the number of triggers.

b2 = b. Figure 7.11 shows the SNR distribution of H1L1 timeslides and injections
with detection statistic contours of constant values of the detection statistic marked.
Tuning the values of a and b essentially amounts to varying where the horizontal andvertical lines cross the diagonal for a given contour. The aim is to find a balance
between having as many injections lie on the diagonal while keeping the horizontal
and vertical lines forward of the tails. Suitable values for this search were found to
be a = 2, b = 2.2.

7.5

PlaygroundA trigger at GPS time t is in the playground if it satisfies
mod(t − tS2 , 6370) < 600

(7.9)

84

Figure 7.11: The H1L1 SNR distribution for timeslides (black) and injections (red).
Contours of constant values of the detection statistic with a = 2 and b = 2.2 aremarked in green.

where tS2 is the GPS start time of the second LIGO science run 729273613. This
constitutes approximately 9.5% of the data set. In the development of the pipeline
and in the tuning process we are allowed to look at this representative subset andleave the remaining 90.5% blind until all the parameters have been decided upon. It
is, of course, possible that a gravitational wave may lie in the playground; this by
no means invalidates the detection. However information from the playground is not
used in determining an upper limit, in order to avoid bias on the part of the analyst
from influencing cuts that determine the upper limit.
In this analysis we did not use the playground directly for tuning, however it
was used in the development of the pipeline for testing code. The distribution ofcoincident events found in the playground is discussed in section 8.5.

85

Chapter 8
Parameter Estimation
8.1

Introduction

In the previous chapter we discussed how the search was tuned, with our goal being
to recover as many injections as possible while keeping the rate of false alarms to aminimum. Once all of the constraints were decided upon we ran a large scale injection
run; the findings of the injection run are described in this chapter. In particular
we discuss the efficiency of the search and the accuracy with which we recover the
injected parameters. We also compare recovered parameters between detectors andthe parameters of accidental coincidences. In addition we unblind a fraction of thedata, the playground, and compare this to our estimated false alarm rate.

8.2

Recovery of Simulated Signals

Nine injection runs were made into times when all three interferometers were in sciencemode (hereafter referred to as triple time). As discussed in section 7.2, injections weremade at a maximum rate of one per 2176 s analysis segment, totaling 5142 injections
over the nine runs. The primary purpose for such a large scale Monte Carlo simulation
is to evaluate the efficiency, which is needed to calculate an upper limit. As we intend
to place an upper limit on the rate of triple coincidences, we do not perform injections
into times when data was available from only two instruments (double time).

86

8.2.1

Single Interferometer Injections

Before delving into the results of the coincidence analysis it is interesting to consider the single detector results; we choose H2 as an example. We can compare
the triggers from the output of the filtering stage to the list of injections using
lalapps ringread (in much the same way as we use lalapps coincringread
for the coincidence analysis, as described in section 5.5.6) to ascertain how many injections were missed and how many were found. As we discussed in relation to thecoincidence analysis, this process does not make use of any information regarding
the injections, apart from the time interval into which they were injected. As a consequence, a trigger due to noise occurring within that time interval with a higher
signal-to-noise ratio (SNR) than the injection would be erroneously identified with
the injection. In such a situation the recovered waveform parameters (frequency andquality) will most likely not be close to those injected.
In figure 8.1 we plot the effective distance versus frequency of the injections missed(in red) and found (in blue) in the nine injection runs. The plot shows that, for agiven frequency, there is a distance at which we begin to miss injections. As we moveacross in frequency from 50 Hz this distance increases to a maximum of 200 Mpc atapproximately 90 Hz and then falls off again to 300 kpc at the high-frequency endof the template bank. Superimposed is the expected horizon distance (discussed in
section 6.4) for a ringdown with an SNR of 5.5 and spin of 0.98 created using thebest S4 H2 noise curve. The boundary between missed and found injections follows
the horizon distance curve nicely, illustrating how features in the noise curve effectthe distance out to which an injection can be recovered. (Such features include thecalibration line at 54 Hz, the power line at 60 Hz and its harmonic at 120 Hz, and theviolin mode resonances of the test mass suspensions at 340 Hz.) This plot shows that
we find a trigger within 100 ms of every injection that we would expect to recover (i.e.,
below the horizon distance line). In addition we find some distant injections, however
in most of these cases, it is spurious noise in the detector that we are finding andnot the injection itself. We will see in the next section that this effect is dramatically87
reduced by requiring coincidence between detectors, illustrating of the power of thecoincidence test.

Figure 8.1: Hanford effective distance versus frequency for missed (red) and found(blue) injections in the H2 single interferometer analysis. The cyan line is the horizon
distance for a source with a spin of 0.98 producing an SNR of 5.5 in the detector,
assuming that 1% of its mass is radiated as gravitational waves.

8.2.2

Coincident Injections

Running lalapps coincringread on the output of the coincidence step gives us
a list of the triggers identified with the injections, one coincidence for each injection(as discussed in section 5.5.6). These can be categorized as injections found in triple
coincidence (referred to as triples in triple time, or simply triples) and injections foundin coincidence in two detectors (doubles in triple time, or doubles). The doubles arefurther divided into those that were missed in the third detector because that time
was vetoed and those that were simply not seen in the third detector. Note thatintervals within triple time when one detector was vetoed are still regarded as triple
time; we consider this a loss of efficiency (of detecting triples) rather than a loss

88
of live time. One could, in theory, account for the lost analysis time, however the
extra complexity this entails in terms of bookkeeping is not justified given the small
difference this makes.
8.2.2.1

Missed and Found Injections

Figure 8.2 shows a plot similar to that discussed above, the effective distance versus
frequency of missed and found injections from the coincidence analysis. The injections
found in triple coincidence are marked in blue and those injections found in two
detectors are denoted by green, cyan, and magenta stars for H1H2, H1L1, and H2L1
doubles, respectively. The injections vetoed in one detector and found in the othertwo have a black circle surrounding the star to emphasize the fact that technicallythey were not missed in the third detector (although they are treated that way in thecalculation of the efficiency). The doubles are shown on their own in figure 8.3. Asbefore, the missed injections are marked in red. Our sensitivity to triples is limitedby the least-sensitive instrument, H2, and thus the distance out to which we see
triples depends on how far H2 can see. For that reason the distance at which we nolonger find injections in triple coincidence is approximately the same as the distancethat we start missing H2 injections in figure 8.1. Beyond this limit H1 and L1 arestill sensitive enough to detect ringdowns, and so we see a thin line of H1L1 double
coincidences beyond the distance at which the triples end. At high frequencies there
is also a band of H1H2 injections mixed in with the triples. This is because during
the S4 run the sensitivity in L1 decreased as the laser power was lower (as discussed
in section 6.2), and thus injections made at large distances during this time were
missed in L1 while those made when L1 was running with full power were found.
The remaining uncircled doubles (i.e., those missed in the third detector) scatteredthroughout the predominantly blue area should have been found in the third detector
and were followed up on an injection-by-injection basis. Further investigation showedthat these were predominantly due to excess noise in the third detector, causing the
SNR to peak at a frequency other than the injected frequency, and as a result thisdetector failed the coincidence test. However, with these exceptions, this plot shows

89
that we are recovering the all injections we would expect to recover, and this reassures
us that the tools we are employing in the analysis are sound.

Figure 8.2: Hanford effective distance versus frequency for injections missed (redcircles) and found in coincidence. Injections found in triple coincidence are marked
as blue crosses, injections found in double coincidence are shown as green (H1H2)
cyan (H1L1) and magenta (L1H2) stars and those that were vetoed are also marked
with a black circle.

8.2.2.2

Efficiency of Finding Triple Coincidences

We evaluate the efficiency ε of finding triples, that is the fraction of injections found in
triple coincidence, as a function of injected (physical) distance. This is implemented
by binning the injections in logarithmic distance and calculating the efficiency in eachbin. A plot of efficiency versus distance is shown in figure 8.4. The uncertainty in

90

Figure 8.3: Hanford effective distance versus frequency for injections found in double
coincidence. The coloured stars represent each of the detector pairs (H1H2 doubles
are marked in green, H1L1 in cyan and L1H2 in magenta). The black circles markthose doubles that were vetoed in the third detector.

the efficiency is assumed to be binomial,σε2 =ε(1 − ε)
,
N

(8.1)

where N is the total number of injections made.
The first thing to note is that the efficiency is never unity, even at small distances,because, as mentioned in section 5.5 we apply category 2 and 3 vetoes, and thus
some injections that otherwise may have been found as triples are found as doubles
or not at all. The second feature of note is the gradual slope. This is because theplot encompasses all frequencies and, as is obvious from figure 8.2, the efficiency isa strong function of frequency. In chapter 9 we present analogous plots for smaller

91

Figure 8.4: The efficiency of finding injections in triple coincidences as a function ofphysical distance for injections made between 45 Hz and 2.5 kHz.

frequency bands. Considering injections at all values of the central frequency, we see
from figure 8.4 that the 50% efficiency point lies at a distance of approximately 4
Mpc.

8.3

Comparison of Injected and Detected Parameters for Injections Found in Triple Coincidence

Comparing the recovered parameters to the injected parameters gives us a sense of
the accuracy with which we can expect to recover the parameters of a gravitational
wave ringdown. Here we compare the injected and detected time, metric distance,and effective distance of injections found in triple coincidence.

92

8.3.1

Time of Arrival

Figure 8.5 shows a histogram of the difference between the detected and injected
times δt for H1, H2, and L1. This distribution is highly asymmetric for all threedetectors with a peak close to zero and a tail extending to negative δt. Figure 8.6
shows that the accuracy of the time of the injection is a strong function of frequency(L1 is shown, H1 and H2 displayed a similar trend), with largest δt occurring at low
frequencies and decreasing as the frequency increases. Further investigation showedthat this is because the injections turn on suddenly at t = 0 with a random phaseas demonstrated in figure 7.3. The templates have a phase of zero, and thus the
maximum time difference is inversely proportional to the frequency. The spread in δt
represents the randomness of the initial phase of the injection.

Figure 8.5: Histogram of difference between detected and injected time of arrival for
H1, H2, and L1 injections found in triple coincidence.

93

Figure 8.6: Difference between L1 detected and injected time versus injected frequency for injections found in triple coincidence.

8.3.2

Metric Distance

We can calculate the metric distance between the injection and the template thatrang off with the loudest SNR. Recall from section 5.4.5 that the template bank islaid out such that the maximum metric distance between any point within the f0
and Q boundaries and the nearest template is less that ds2max , where in this analysis
we have set ds2max = 0.03. We can calculate this distance between an injection andthe template that recovered it using equation (4.39). In the implementation of thecoincidence test an error was made with the result that the metric distance is underestimated by a factor of 4. Therefore, in the context of the following discussion thevalue of the maximum mismatch in the layout of the template bank is ds2 = 0.03/4 =0.0075. A histogram of ds2 calculated between the injected and detected quantities
for H1, H2, and L1 triple coincident injections is shown in figure 8.7. The maximum
distance between a template and any point in the space is marked by a black verticalline at ds2 = 0.0075; to the left of this line are all the injections that were found94
by the correct (i.e., the closest) template. This accounts for approximately ∼ 70%of the injections and gives us a high level of confidence in our ability to recover theparameters of ringdowns.
Figure 8.8 shows how ds2 varies with frequency in L1 (H1 and H2 displayed similar
structure). Not surprisingly, the points follow the general trend of the noise curve(see figure 6.2) with the lower values of ds2 lying in the most sensitive frequency bandof the detector and higher values of ds2 at the highest and lowest frequencies. The
injections made outside the template bank obviously have the highest values of ds2 ,as the nearest template is always going to have ds2 > 0.0075. The spread in ds2
for a given frequency can be attributed to the coarseness of the Q parameter in thetemplate bank. This is demonstrated by figure 8.9, a plot of ds2 versus quality factor.
Here we see that as the injected value of Q deviates from one of its five templates,
the mismatch increases until such point as the next Q template is closer than theprevious. Thus we see a series of arches, each with the same maximum height.
We can examine both f0 and Q on a scatter plot with ds2 as the colour scale; thisis shown in figure 8.10. From this we can see that the lowest value of the mismatch
between a template and an injection occurs along the lines of Q templates, indicatedby black lines. The highest values of ds2 occur, as expected, along the templateboundaries. However it is interesting to note that the frequency dependence that we
observed in figure 8.8 is most pronounced at low values of Q. Injections on the Q = 17line are mostly found by the closest template (dark blue to cyan on the colour scale).
In contrast, on the Q = 3.6 line, injections are rarely found by the closest template
above an injected central frequency of ∼ 300 Hz.

8.3.3

Effective Distance

Another check we can make is between injected and detected effective distance. We
find the best way to evaluate this is by the fractional differenceδdef f2 [def f (det) − def f (inj)]=
hdef f i[def f (det) + def f (inj)]

(8.2)

95

Figure 8.7: Histogram of the mismatch ds2 between injections and the templates they
were found with for H1, H2, and L1 triple coincidence injections. The black verticalline marks ds2 = 0.0075, the template bank maximum mismatch. The plot showsthat ∼ 70% of the injections found in triple coincidence were found with the correct
template.

Figure 8.8: Plot of ds2 versus central frequency for the L1 component of injections
found in triple coincidence

96

Figure 8.9: Plot of ds2 versus quality for the L1 component of injections found in
triple coincidence. The red lines mark the five values of Q in the template bank andthe black line marks the upper Q boundary of the template bank.

Figure 8.10: Plot of quality versus frequency for the L1 component of injections
found in triple coincidence. The colour scale is the mismatch between the injected
and recovered parameters ds2 . Each horizontal line denotes the frequency range ofthe templates for each of the five values of Q.

97
where det stands for detected and inj for injected. A histogram of this quantity
evaluated for H1, H2, and L1 detected effective distances is shown in figure 8.11. From
this we see a distribution which is sharply peaked at zero with a slight asymmetry
in the tails for all three detectors. Figure 8.12 shows that the accuracy with which
the effective distance is recovered is frequency dependent in H2. Similar plots forH1 (figure 8.13) and L1 (not shown) display a similar behaviour. Below 100 Hz andabove 1 kHz the noise increases and we become less accurate. It should be noted that
injections made below 50 Hz and above 2 kHz are outside the template bank and thus
are recovered by the “wrong” template. Therefore the SNR is less than the SNR ofan exactly matched template, and hence the effective distance is over estimated. This
explains the asymmetric tails seen in the histograms. The feature at approximately
340 Hz is believed to be due to the test mass suspension violin mode resonances.

Figure 8.11: Histogram of the fractional difference between detected and injected
effective distance in H1, H2 and L1 for injections found in triple coincidence.

98

Figure 8.12: Plot of the fractional difference between detected and injected effective
distance versus frequency in H2 for injections found in triple coincidence.

Figure 8.13: Plot of the fractional difference between detected and injected effective
distance versus frequency in H1 for injections found in triple coincidence.

99

8.4

Comparison of Recovered Parameters Between
Detectors

Comparing the recovered parameters between detectors is also a useful exercise, as
this gives us an idea how close we can expect the parameters of real signals to be. In
chapter 7 we discussed the importance of such a comparison for tuning the coincidence
test. Of course, these injections have survived the coincidence test and so we expect
the parameters to be similar in different detectors.

8.4.1

Time

The difference between recovered times in H1 and H2, δt(H1-H2), shown in figure
8.14 is a sharply peaked distribution, as expected given their co-locality, whereas thedifference between recovered times in the H1L1 δt(H1-L1) and H2L1 δt(H2-L1) pairs,
shown in figure 8.15, is a wide distribution where the light travel time of 10 ms can
be clearly seen.
The scatter plot of δt(H1-H2) shown in figure 8.16 displays interesting structure.
The data points lie on equally separated discrete (horizontal) lines in δt; the lines are
separated by the inverse of the sampling rate (8192)−1 s, showing just how well thetime parameters are recovered for the majority of the injections. The next interesting
structure is the overall shape of the population of points surrounding the δt = 0
line, with the spread in δt increasing as the frequency decreases. This is the same
phenomenon discussed in section 8.3.1 for the comparison of detected and injected
times. The third feature of note is the second population of points following the same
shape, but separated from the main group by a larger |δt|. This population is actually
the coincidence between two waves out of phase by π. In other words, the first peakof the injection in one detector was found in coincidence with the second peak in theother detector. For example, for a 300 Hz ringdown a half cycle is ±1.7×10−3 s induration, and as can be seen from the plot, this is approximately where the second
population lies for this frequency. This plot gives important information regarding the

100

Figure 8.14: Histogram of difference between H1 and H2 recovered times for injections
found in triple coincidence.

Figure 8.15: Histogram of difference between H1L1 (red) and H2L1 (blue) recovered
times for injections found in triple coincidence.

101

Figure 8.16: Plot of difference between H1 and H2 recovered times as a function of
H1 recovered frequency for injections found in triple coincidence.

Figure 8.17: Plot of difference between H1L1 (blue) and H2L1 (red) recovered times
versus frequency for injections found in triple coincidence.

102
time coincidence requirement, implying that the width of this window should dependon the frequency (this will be discussed further in section 11.1.3). Noting that a third
population does not appear, we know that either that same peak or peaks separated
by one half cycle are found in coincidence.
A scatter plot of the H1L1 time difference is shown in figure 8.17. We begin tosee a spread in δt below 200 Hz, but this is not as pronounced as in the H1H2 case.

8.4.2

Metric DistanceThe ds2 histogram in figure 8.18 summarizes the mismatch between the recovered
parameters in H1 and L1. The bar at ds2 = 0 indicates that approximately half of
the injections were found with exactly the same template in both H1 and L1. The
next bar at ds2 = 0.0075 are the injections found with the same Q and adjacent f0
values. The third highest bar at ds2 = 0.015 represent the injections found with thesame f0 but differing by one row of Q, and so on. The H1H2 and H2L1 plots show very
similar structure. In all, 37% of injections were found with the same template in all
three detectors. An interesting plot, shown in figure 8.19, is frequency versus distancefor H1 with ds2 (H1,L1) as the colour axis. One might expect that as the effective
distance increases the parameter accuracy would degrade, in particular close to themissed-found boundary seen in figure 8.2. The plot shows however that this was not
so; even weak signals can be accurately recovered. Noise at the lowest and highest
frequencies appear to have more of an effect as the plot shows an increased incidence of
higher mismatched templates at those frequencies. Note that the coincidence window
discussed in section 5.5 allows a window of ds2 = 0.05 on both templates in a pair,allowing a total window of ds2 = 0.1, as the range of the colour-bar shows.

8.4.3

Effective Distance

When comparing injected and detected quantities, effective distance was one of theparameters we looked at, as we expect to be able to reconstruct this quantity. However recall that effective distance is different for separated detectors as it depends on

103

Figure 8.18: Histogram of ds2 between recovered parameters in H1 and L1 for injections found in triple coincidence.

Figure 8.19: Frequency versus effective distance for H1 with the colour-bar displaying
ds2 (H1,L1), the metric distance between injections recovered in H1 and in L1.

104
the sky location. Thus, in this section we should only expect to see a sharp narrow
peak at zero in the H1H2 case, as co-located detectors should measure similar valuesof this quantity and this is the basis for the distance cut discussed in section 5.5.5.
However, just for comparison, we compare the effective distance in all detector combinations. Once more we evaluate the fractional difference in effective distance definedin equation (8.2)
δdef f2 [def f (det1 ) − def f (det2 )]=,hdef f i[def f (det1 ) + def f (det2 )](8.3)

where det1 and det2 refer to detectors 1 and 2. Figure 8.20 shows three distributions in δdef f / hdef f i. As expected, the H1H2 distribution is sharply peaked while
the H1L1 and H2L1 distributions are much broader. The latter two distributions are
symmetric about zero while the H1H2 shows a slight asymmetry. A scatter plot of
the H1H2 distribution revealed a frequency dependence not seen in the other combinations. Figure 8.21 shows that at low frequency δdef f / hdef f i rapidly falls off tolarge negative values, indicating that the distance measured in H2 is greater than thatmeasured in H1. This can be explained by referring back to figures 8.12 and 8.13,
which compare injected and recovered parameters. Here we see that in both plots
δdef f / hdef f i increases as the frequency reaches the low- and high-frequency ends ofthe template bank. However, in comparing these it is clear that in H2 δdef f / hdef f iincreases faster than in H1 at low frequencies; that is, H2 is overestimating the distance to a larger degree than H1 is. Thus when we take the difference between thesetwo quantities we observe an asymmetric tail.

8.5

Comparison of the Background Estimation via
Time-slides with the PlaygroundWe discussed the estimation of the rate of accidental coincidences in section 7.3 andthe definition of the playground data set in section 7.5. In this section we compare
the prediction of the background distribution of the final tuned pipeline with theplayground.

105

Figure 8.20: Histogram of the fractional difference in H1H2 (green), H1L1 (red), and(H2L1) blue recovered effective distance for injections found in triple coincidence.

Figure 8.21: Plot of the fractional difference in H1 and H2 recovered effective distanceversus H1 frequency for injections found in triple coincidence.

106

8.5.1

Triple Coincidences

The number of background events found in triple coincidence was quite small; from100 timeslides there were just 14 events, making the triple-time false alarm rate lessthan one event per run. Figure 8.22 shows the projection of the SNR distributionof triple coincident background triggers onto the H1L1 plane. The plot shows thatthe background events were found at low SNR in all three detectors. There wereno events found in triple coincidence in the playground. This is consistent with theprediction of the background from the timeslide estimate.

Figure 8.22: L1 SNR versus H1 SNR for background triggers found in triple coincidence.

8.5.2

Double Coincidences

In contrast, however, the number of background events found in double coincidence
was considerably higher. From 100 timeslides we found an average of 620, 150, and800 background events in H1L1, H1H2, and H2L1, respectively per timeslide. It

107should be noted that, as mentioned previously, the H1H2 false alarm rate is known
to be underestimated.
There were 53, 21 and 69 double coincidences found in H1L1, H1H2 and H2L1
respectively in the playground data set.
A plot of the SNR distribution for timeslides and playground found in double
coincidence in H1 and L1 is shown in figure 8.23, for H1 and H2 in figure 8.25, andfor H2 and L1 in figure 8.27. In addition to having a much larger false alarm rate
than triples, the double coincidence timeslide plots also show a different type of SNR
distribution; in addition to a central component close to the diagonal, the plots show
long tails extending to high SNR. Note that the H1H2 background distribution we
see the effects of the distance cut. The playground triggers were all found with SNRs
lower than 20.
We can compare the number of coincidences found in the playground to the estimated background. Figures 8.24, 8.26, and 8.28 show histograms of the number ofdouble coincidences in each of the one hundred timeslides, along with the numberof double coincidences found in the playground scaled to the full data set for H1L1,
H1H2, and H2L1 respectively. If the background estimate is an accurate measure of
the false alarm rate, one would expect that the scaled number of playground eventsshould be to be comparable to number of events in each timeslide. We find that,
although the scaled playground is lower than the average number of timeslide double
coincidences for H1L1 and H2L1, it is within the estimated error. As we have discussed previously the timeslide method of background estimation in H1H2 is flawed
and we expect to see a greater number of coincidences than predicted. Figure 8.26
shows that this is indeed the case.

108

Figure 8.23: The signal-to-noise distribution of playground and timeslides found indouble coincidence in L1 and H1.

Figure 8.24: Histogram of number of H1L1 double coincidences per timeslide and in
the playground scaled to the full data set.

109

Figure 8.25: The signal-to-noise distribution of playground and timeslides found indouble coincidence in H2 and H1 prior to the implementation of the distance cut.

Figure 8.26: Histogram of number of H1H2 double coincidences per timeslide and in
the playground scaled to the full data set.

110

Figure 8.27: The signal-to-noise distribution of playground and timeslides found indouble coincidence in L1 and H2.

Figure 8.28: Histogram of number of H2L1 double coincidences per timeslide and in
the playground scaled to the full data set.

111

8.6

Comparison of Recovered Injection Parameter
Accuracy for Timeslides found in Double Coincidence

We have just demonstrated that the expected rate of false alarms in double coincidence is very high. Here we investigate the possibility that by comparing the accuracy
between timeslide components to the accuracy between injection components we could
distinguish between a signal and a noise event. We look at the mismatch between
pairs of detectors and the difference in recovered effective distance.

8.6.1

Metric DistanceFirst we compare how close in parameter space the background events were in comparison to the injections by plotting histograms of ds2 normalized to the total numberof events. (The injection distribution was already discussed in section 8.4.2.) Figure8.29 shows a comparison between H1 and L1. Here again we see a series of peaks
due to the fact that the same template bank is used throughout the search. We seethat the fraction of timeslide coincidences is low, but not zero at ds2 = 0, telling usthat in double coincidence there is a non-negligible chance that two independent noise
triggers could be found with the exact same parameters in widely separated sites. In
fact the timeslide estimation of the H1L1 background predicts that in a typical S4
run we can expect 30 background events to be found in H1 and L1 with exactly thesame waveform parameters! The fraction of injections found at higher values of themismatch drops rapidly and the fraction of false alarms increases slightly, showingthat a candidate event found with a large values of ds2 is more likely to be background than a real signal. Almost identical distributions were seen for H1H2 andH2L1 combinations.

112

Figure 8.29: The normalized distribution of the mismatch between H1 and L1 forinjections (red) and double coincident timeslides (blue).

8.6.2

Effective Distance

Recall from our discussion in section 8.4.2 that we really can only make use of theH1H2 comparison of effective distances as we expect these to be very close for real
signals. Figure 8.30 shows the comparison of δdef f / hdef f i between H1 and H2 fortimeslides and injections. The plot shows that, whereas the injection distribution ispeaked at δdef f / hdef f i = 0, the timeslide distribution is peaked at δdef f / hdef f i ∼ 0.3.
We can also see the point where the distance cut was implemented, beyond where
the injection distribution ended at δdef f / hdef f i = 0.67. This shows the advantage of
having a co-located half-length interferometer; if H2 was the same length as H1, thetimeslide peak would also lie at δdef f / hdef f i = 0.
The background distributions of δdef f / hdef f i = 0.67 in H1L1 and H2L1 are interesting too. Firstly the plot shows that because the background distribution is broad
and peaked close to zero just like the injection distribution, we cannot implement a
distance cut. Both distributions display a second peak shifted from the first. This is113
because the L1 sensitivity was decreased for a period during the run, as the effective
distance is proportional to the sensitivity, the effective distance of the background
was also less.
From this study we can conclude that it is impossible to distinguish between asignal and background from the accuracy of parameters between pair of detectors.

Figure 8.30: The normalized distribution of the fractional difference in effective distance between H1 and H2 for injections (red) and double coincident timeslides (blue).

114

Figure 8.31: The normalized distribution of the fractional difference in effective distance between H1 and L1 for injections (red) and double coincident timeslides (blue).

Figure 8.32: The normalized distribution of the fractional difference in effective distance between L1 and H2 for injections (red) and double coincident timeslides (blue).

115

Chapter 9
Results
9.1

Introduction

Once we were satisfied with the tuning of the search the pipeline was applied to thefull data set and the search was unblinded. A cumulative histogram of the detection
statistic of background events was compared to that of foreground events for eachcategory of coincidences. A significant excess of the foreground distribution above
the background distribution could indicate the presence of a gravitational wave. On
opening the box we found that no triple coincidence signals survived to the end of
the pipeline. As expected there were a number of foreground triggers found in double
coincidence and while some of those categories showed a slight excess above the background we did not find sufficient evidence to claim a detection of gravitational waves.
In section 9.2 we describe our findings on unblinding each category of coincidences inthe search. In section 9.3 the double coincidences with the highest detection statistic
are followed up. In each case the presence of instrumental artifacts or environmental
disturbances were sufficient to explain the coincident triggers. The upper limit on
the rate of ringdowns is calculated in section 9.4.

9.2

Opening the Box

In each of the cumulative histograms described in this section we plot the background
distribution scaled down by a factor of 100 (as this many timeslides were performed)

116
for comparison with the zero-lag events. The histogram is a function of the detection
statistic, which for double coincidences is ρDS = min{ρ1 +ρ2 , 2ρ1 +2.2, 2ρ2 +2.2} whereρ1 and ρ2 refer to the signal-to-noise ratio (SNR of triggers from interferometers 1
and 2. The error bars on the background distribution denote a one σ error, wherepσ =N/100 and N is the number of background events per bin. In all of thescatter plots referred to in this section we plot all of the coincident events from 100
timeslides along with the zero-lag triggers. The inset in each plot zooms in on the
low-SNR region containing most of the zero-lag triggers. Tables detailing the ten
loudest candidate events in each category can be found in appendix C.

9.2.1

Triples in Triple TimeNo triple coincident zero-lag events were found.

9.2.2

H1L1 Doubles in Triple Time

Figure 9.1 shows the cumulative histogram of H1L1 doubles in triple time. The plotshows that the foreground was within the expected range of accidental coincidences.
Figure 9.2 shows the SNR from the individual zero-lag and background triggers.

9.2.3

H1H2 Doubles in Triple Time

Figure 9.3 shows the expected excess of zero-lag triggers due to our inability to predict
the H1H2 false alarm rate. There was one trigger with an exceptionally high detection
statistic, ρDS ∼ 63. Figure 9.4 shows the individual SNRs were ρH1 = 1300 andρH2 = 30.

9.2.4

H2L1 Doubles in Triple Time

Figure 9.5 shows that the H2L1 foreground was consistent with the background. A
plot of ρL1 versus ρH2 is shown in figure 9.6.

117

9.2.5

H1L1 Doubles in Double TimeFor this category we see a deficit of foreground triggers compared to background in
figure 9.7. Figure 9.8 shows the distribution of foreground and background triggers
in SNR.

9.2.6

H1H2 Doubles in Double Time

Once more we see an excess in the H1H2 foreground due to our inability to estimate
the background accurately. Figure 9.9 displays the H1H2 cumulative histogram andfigure 9.10 shows the ρH2 versus ρH1 scatter plot.

9.2.7

H2L1 Doubles in Double Time

Figure 9.11 shows that the H2L1 double time foreground was consistent with background. Figure 9.12 displays the ρL1 versus ρH2 scatter plot.

9.3

Following Up on the Loudest Candidates

Here we list a number of checks that can be applied to candidate events to increaseor decrease our confidence in each as being caused by a gravitational wave. The
histograms in the previous section showed that there were no events standing out
above the background and thus we do not believe that we have detected a gravitational
wave in this data set. However, as an exercise we apply this checklist to the threeloudest candidates in each of the triple time categories and the loudest candidate ineach of the double time categories and demonstrate how these candidates fail severalof the tests.

9.3.1

Follow-Up Procedures• Check what data quality flags (if any) were in place at the time of the candidate
events. Candidate events occurring at the same time as category 4 data quality
flags were generally downgraded in significance.

118

Figure 9.1: Cumulative histogram of zero-lag and background coincidences: H1L1
doubles in triple time.

Figure 9.2: Scatter plot of L1 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The inset is anenlargement of the region ρ = 5 to 200.

119

Figure 9.3: Cumulative histogram of zero-lag and background coincidences, H1H2
doubles in triple time.

Figure 9.4: Scatter plot of H2 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The inset is anenlargement of the region ρ = 5 to 200.

120

Figure 9.5: Cumulative histogram of zero-lag and background coincidences, H2L1
doubles in triple time.

Figure 9.6: Scatter plot of L1 versus H2 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in triple time. The inset is anenlargement of the region ρ = 5 to 200.

121

Figure 9.7: Cumulative histogram of zero-lag and background coincidences, H1L1
doubles in double time.

Figure 9.8: Scatter plot of L1 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The inset is anenlargement of the region ρ = 5 to 200.

122

Figure 9.9: Cumulative histogram of zero-lag and background coincidences, H1H2
doubles in double time.

Figure 9.10: Scatter plot of H2 versus H1 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The inset is anenlargement of the region ρ = 5 to 200.

123

Figure 9.11: Cumulative histogram of zero-lag and background coincidences, H2L1
doubles in double time.

Figure 9.12: Scatter plot of L1 versus H2 signal-to-noise ratio for double coincident
zero-lag and background (100 timeslides) triggers in double time. The inset is anenlargement of the region ρ = 5 to 200.

124
• Run a qscan1 [78] on the time under investigation. This has a twofold purpose:the first is to check if the signal in the gravitational wave channel is consistent
with what we expect for a ringdown (a short signal at a single frequency) oran inspiral (a signal whose frequency increases with time), and the second is
to see if there was excess power in any of the auxiliary channels which could
explain the coupling of a non-gravitational wave source to the gravitational
wave channel. For an example of what we expect in a qscan of the gravitational
wave channel in the presence of a gravitational wave see the qscan of an inspiralmerger-ringdown hardware injection in figure 9.13.
• Plot quantities such as the SNR time series, SNR versus frequency, and frequency time series around the time of the candidate event using the trigger files
output after the filtering stage and compare with similar plots of injections andbackground events.
• If a double coincident candidate event occurred during triple time, try to understand why it was not found in the third interferometer.
• See if the event was also found in other searches, such as the S4 binary blackhole search (S4 BBH) [16] or the S4 burst search [21].

9.3.2

H1L1 Doubles in Triple Time

Candidate number 1:• t = 794949585, ρDS = 21, ρH1 = 9.4, ρL1 = 12.5.
• Examination of the data quality flag database shows that no category 4 data
quality flags were on during the time of this candidate event.
1

Qscan is a script that creates time frequency maps of selected channels around a desired time. It
is a useful tool for obtaining an overview of excess power in the gravitational wave channel, excitation
channels, auxiliary channels, and environmental channels.

125

Figure 9.13: The whitened spectrogram of the H1 gravitational wave channel showing
a hardware injection of an inspiral-merger-ringdown signal during S5.

• If this was a real signal we would expect that ρH2 ≈ 4.7. However this is below
the threshold for H2 and therefore if this was a gravitational wave it could nothave been found as a triple coincidence.
• This candidate was not in the top ten loudest candidates in the binary blackhole search.
• A qscan of the H1 and L1 gravitational wave channels at this time (shownin figure 9.14) revealed a short broadband signal in each interferometer in anotherwise quiet time window. These characteristics indicate that the signal is
not a ringdown (or an inspiral).
Candidate number 2:• t = 793829533, ρDS = 20, ρH1 = 9, ρL1 = 11.
• The category 4 data quality flag L1:ASDC LOW THRESH was on at this time.
This flag represents times when the amount of light in the dark port exceeds a

126(a) H1(b) L1Figure 9.14: A qscan of the gravitational wave channel at GPS time 794949585, theloudest H1L1 candidate event in triple time.

(a) H1(b) L1Figure 9.15: A qscan of the gravitational wave channel at GPS time 793829533, thesecond loudest H1L1 candidate event in triple time.
certain threshold. This is usually indicates problems with alignment and raises
our suspicions about the validity of this candidate.
• The SNR of H1 indicates that this candidate was below the threshold for H2.
• This candidate was not among the BBH search loudest triggers.
• The qscan immediately eliminates this candidate. Figure 9.15 shows a longbroadband disturbance in L1.
Candidate number 3:• t = 794291462, ρDS = 19, ρH1 = 8, ρL1 = 11.

127(a) H1(b) L1Figure 9.16: A qscan of the gravitational wave channel at GPS time 794291462, thethird loudest H1L1 candidate event in triple time.
• Like the previous two candidates this also was too quiet to have been seen in
H2.
• The qscan, figure 9.16, once more indicates a noisy time in L1 during which thedetection of gravitational waves would be unlikely.

9.3.3

H1H2 Doubles in Triple Time

Candidate number 1:• t = 793253792, ρDS = 63, ρH1 = 1300, ρH2 = 30.
• The data quality flag H1H2 COHERENCE was on at the time, indicating timesof strongly coherent noise between H1 and H2. This decreases our confidence
in this coincidence as a candidate event.
• Looking at the qscan for this candidate, figure 9.17, we see a large broadband
glitch in both the H1 and H2 gravitational wave channels. While we can not
say with certainty what the cause of this glitch is, we can conclude that it does
not have the characteristics of a gravitational wave ringdown. It does appear
to be coincident in time with a glitch in a magnetometer channel, however acorrelation between these channels has not yet been firmly established.

128(a) H1(b) H2

Figure 9.17: A qscan of gravitational wave channel at GPS time 793253792, theloudest H1H2 candidate event in triple time.
Candidate number 2:• t = 794654729, ρDS = 21, ρH1 = 13, ρH2 = 10.
• Figure 9.18, the qscan of the gravitational wave channel revealed a ∼ 0.5 slong narrow-band signal in both H1 and H2. In addition, a large number ofenvironmental channels showed a significant disturbance. In the magnetometerchannels this manifested itself as a ∼ 7 s long broadband disturbance, as canbe seen in figure 9.19(a). All of the accelerometer channels that were triggered
displayed a shorter (< 0.5 s) glitch at ∼ 128 Hz; this can been seen in figure9.19(b). This also appeared in the microphone channels with the same frequency
and in some voltmeter channels at ∼ 292 Hz. Some of the magnetometerchannels also showed this line feature at ∼ 192 Hz. This candidate is clearly
not a gravitational wave.
Candidate number 3:• t = 795398069, ρDS = 15, ρH1 = 6, ρH2 = 8.
• The H1 category 4 data quality flag HIGH PIXEL FRACTION 1KHZ was on
at the time of the candidate, indicating a large deviation from Gaussianity,
making this coincidence more likely due to noise than a gravitational wave.

129(a) H1(b) H2

Figure 9.18: A qscan of the gravitational wave channel at GPS time 794654729, thesecond loudest H1H2 candidate event in triple time.

(a) The magnetometer channel H0:PEMLVEA MAGX.

(b) The accelerometer channel H0:PEMISCT4 ACCX.

Figure 9.19: A qscan of two environmental channels triggered at 794654729.

130(a) H1(b) H2

Figure 9.20: A qscan of the gravitational wave channel at GPS time 795398069, thethird loudest H1H2 candidate event in triple time.
• A plot of SNR versus frequency shows a large portion of the both banks rang
off over a small SNR range.
• For this candidate the gravitational wave channel qscan, figure 9.20, shows aseries of small glitches at various frequencies indicating a noisy period of timein the detector.

9.3.4

H2L1 Doubles in Triple Time

Because of the similarity of the loudest three candidates in this category we discussthem collectively.
• Candidate number 1: t = 794966223, ρDS = 20, ρH2 = 10, ρL1 = 10.
Candidate number 2: t = 794490884, ρDS = 17, ρH2 = 9, ρL1 = 8.
Candidate number 3: t = 793181575, ρDS = 16, ρH2 = 9, ρL1 = 7.
• The qscans of the gravitational wave channel for candidates 1, 2, and 3 areshown in figures 9.21, 9.22, and 9.23 respectively. Each demonstrates that thecoincident triggers occurred during noisy times and are most likely false alarms.
• For each of the candidates H1 was in science mode and if the H2 signal was due
to a gravitational wave then, given that H1 is approximately a factor of 2 moresensitive, the event would also have been seen in H1.

131(a) H2(b) L1Figure 9.21: A qscan of the gravitational wave channel at GPS time 794966223, theloudest H2L1 candidate event in triple time.

(a) H2(b) L1Figure 9.22: A qscan of the gravitational wave channel at GPS time 794490884, thesecond loudest H2L1 candidate event in triple time.

(a) H2(b) L1Figure 9.23: A qscan of gravitational wave channel at GPS time 794490884, the third
loudest H2L1 candidate event in triple time.

132(a) H1(b) L1Figure 9.24: A qscan of the gravitational wave channel at GPS time 793258551, theloudest H1L1 candidate event in triple time.
• Several data quality flags were on for candidates 1 and 3, some of which werecategory 4 flags reinforcing the false alarm claim. At the time of candidate 1
the H2 flag DUST ELEVATED indicating an increased particle count in thedark port (which could be responsible for glitches) and the L1 category 4 flags
ACOUSTIC ELEVATED (indicating elevated acoustic noise in the 62–188 Hzband) and ASDC LOW THRESH were on.

9.3.5

H1L1 Doubles in Double Time

Candidate number 1:• t = 793258551, ρDS = 17, ρH1 = 7, ρH2 = 10.
• A qscan of the gravitatational wave channel is shown in figure 9.24.

9.3.6

H1H2 Doubles in Double Time

Candidate number 1:• t = 793589170, ρDS = 17, ρH1 = 11, ρH2 = 8.
• The category 4 data quality flag SEISMIC 0D8 2D0 was on during this time
indicating an excess of seismic activity between 0.8 and 2.0 Hz. Such a distur-133(a) H1(b) L1Figure 9.25: A qscan of the gravitational wave channel at GPS time 793589170, theloudest H1H2 candidate event in triple time.
bance is likely to couple to both H1 and H2, producing the coincidence that wesee.
• A qscan of the gravitational wave channel at the time of this candidate event isshown in figure 9.25.
• The category 4 data quality flag SEISMIC 0D8 2D0 was on during this time.

9.3.7

H2L1 Doubles in Double Time

Candidate number 1:• t = 794432410, ρDS = 16, ρH1 = 9, ρH2 = 7.
• The qscan of this candidate, figure 9.26, is quite similar to the H2L1 double
coincident candidates in triple time; once more we find that the coincidence
occurred during a noisy time in the detector.

9.4

The Upper Limit

This goal of this search is to detect gravitational wave ringdowns. However if agravitation wave is not detected, as is the situation here, we can place an upper limiton the rate of black hole ringdowns in a volume of space. In doing this we only consider

134(a) H2(b) L1Figure 9.26: A qscan of the gravitational wave channel at GPS time 794432410, theloudest H2L1 candidate event in triple time.
times when it was possible to detect a signal in all three detectors. As we have seenin previous chapters, the level of false alarms for two-detector coincidence was very
high, making detection confidence in those times very low. As we will demonstrate at
the end of this chapter, confining ourselves to triple time does not make a significant
difference to the upper limit we calculate.
Calculating an upper limit is achieved by taking the astrophysical population of
black holes and our ability to detect ringdowns into account. For now we express
this as the cumulative luminosity CL and explain this quantity in more detail in thenext section. We use the detection statistic of the loudest triple coincident event ρmax
detected as a threshold above which we evaluate CL and calculate an upper limit. In
this search there were no triple coincident events in the zero-lag data and so we set
the ρmax to the search threshold.

9.4.1

Bayesian and Frequentist Approaches to the Upper
Limit

Two schools of thought exist on how an upper limit should be calculated; one uses
Bayesian probabilities while the other employs frequentist probabilities. The primarydifference between the two methods is that the Bayesian method uses prior knowledge
about the rate in calculating probabilities whereas the frequentist method does not. A

135nice derivation and discussion of Bayesian and frequentist upper limits in gravitational
wave searches can be found in [79], and [80] is a detailed reference on Bayesian upper
limits for LIGO searches. Here we summarize the main points as they apply to the
ringdown search.
9.4.1.1

Frequentist

The probability that there were no gravitational wave events with SNR greater than
ρ is given by
P (ρ|R, T ) = e−RT CL (ρ)(9.1)

where R is the rate of ringdown events per unit cumulative luminoisty and T is theobservation time. If there was a background B present then the probability that nobackground events were present with SNR greater than ρ, PB (ρ), may be taken into
account giving an overall probability ofP (ρ|R, T, B) = PB e−RT CL (ρ) .

(9.2)

We can choose a confidence level α at which we wish to evaluate the rate of events
above an SNR of ρmax and solve the equation 1 − α = P (ρmax |RT, B) for R. Thisgives
Rα = −ln(1 − α) − ln(PB (ρmax )).
T CL (ρmax )(9.3)

In the ringdown analysis there were no events found in triple coincidence, and so CL
was evaluated at threshold. The level of foreground was perfectly consistent withbackground (we had less than one event in both cases) and so the probability that aforeground event was associated with noise is 1. Thus PB = 1, ln(PB ) = 0, and the
90% confidence frequentist upper limit on the rate is
R90% =

2.303.
T CL (ρmax )

(9.4)

136
9.4.1.2

Bayesian

A Bayesian upper limit is calculated from the posterior probability P (R < Rα , T |ρmax )P (R < Rα , T |ρmax ) = N

−1

Z Rα
p(R, T ) p(ρmax |R, T ) dR

(9.5)

0

where p(R, T ) is the prior distribution on the event rate, N is a normalization constantand the likelihood isp(ρ|R, T, B) =dP (ρ|R, T, B)
dCL (ρ) −RT CL (ρ)= −PB RT
e
dρ
dρ

(9.6)

where the second equality makes use of equation (9.2). The upper limit is determinedby solving P (R < Rα , T |ρmax ) = α for Rα . In this search we do not have any
information on the rate of ringdowns and so we choose a uniform prior, p(R, T ) = 1,
and obtain1 − α = e−Rα T CL (ρmax ) [1 + ξ R T CL (ρmax )]
where

#−1

"

d ln(PB )ξ = 1−d ln(CL )

(9.7)

.

(9.8)

CL (ρmax )In the ringdown search the loudest event was consistent with background, ξ = 0, andthus the 90% rate upper limit is given by
R90% =

2.303.
T CL (ρmax )(9.9)

Thus, in this particular case both the frequentist and Bayesian approaches give thesame upper limit.

9.4.2

Cumulative Luminosity

In the previous section we introduced the quantity CL , the cumulative blue lightluminosity, and stated that this was a measure of our ability to detect gravitational
wave ringdowns from a given population of sources. The sources of interest are black

137
holes in the high end of the stellar mass ranges and in the lower end of the intermediate
mass range 6 M < M < 600 M . As described in chapter 3, we know very little
about the population of stellar mass black holes and even less about intermediatemass black holes, — indeed there has been no strong evidence to date for their
existence. However we do know that the formation of stars in general scales with theblue-light luminosity emitted by galaxies, and as it is expected that the rate of binary
coalescence follows the rate of star formation, we work under the assumption thatthe rate of binary coalescence also scales with blue light luminosity. In an effort tointerpret the results of LIGO binary coalescence searches a catalog of nearby galaxies
that could host compact binary systems was compiled [72]. Beyond ∼ 30 Mpc the
cumulative luminosity scales as the cube of distance; this is the regime of interest forthe current search. The relationship between cumulative luminosity and distance isgiven by
CL = ρL Vef f(9.10)

where ρL = (1.98 ± 0.16) × 10−2 L10 Mpc−3 . CL has units of L10 which is defined as
L10 = 1010 LB, , and LB, = 2.16 × 1033 erg s−1 is the solar blue light luminosity. The
effective volume Vef f is the volume of space we are sensitive to, which we quantify interms of a detection efficiency expressed as a function of distance ε(r)Z
Vef f (r) = 4πε(r)r2 dr.

(9.11)

Thus, the rate of ringdowns is given in units of yr−1 L−1
10 byR=
9.4.2.1

2.303.
T ρL Vef f(9.12)

Efficiency

We evaluate the efficiency of detecting gravitational wave ringdowns from a hypothetical population of sources by injecting simulated signals into the data stream
and searching for these signals, implementing the same pipeline used to detect a real
ringdown in the noise. This population was discussed in section 7.2, and our ability

138
to detect them was discussed in chapter 8. The ratio of the number of injections
found in triple coincidence compared to the number injected gives a measure of theefficiency for a given distance.
Evaluation of Vef f is somewhat complicated by the fact that we evaluated theefficiency as a function of logarithmic distance rather than linear distance. Thus weneed to make the substitution
u = log10 (r) =ln(r),ln(10)(9.13)

which can be expressed as
r = 10u .

(9.14)

11
drloge (10) r(9.15)

dr = loge (10)10u du.

(9.16)

This gives
du =or in terms of dr

Equation (9.11) can be rewritten asZ
Vef f (u) = 4π ln(10)ε(u)103u du.

(9.17)

In practice ε and r are discrete quantities and so Vef f is expressed as
Vef f = 4π ln(10)Xε(ui )ri3 ∆u.

(9.18)

i

9.4.3

Calculating the Upper Limit

As we saw from figure 8.2 the efficiency varies dramatically with frequency, andtherefore, in the calculation of the upper limit we divide the frequency space into
separate bands, based roughly on the different levels of sensitivity: 45–100 Hz, 100–200 Hz, 200–500 Hz 500–1000 Hz, 1–2 kHz. For each band we calculate the efficiency
as a function of effective distance shown in figures 9.27 to 9.31, the effective volume

139
Vef f and corresponding radius ref f = Vef f / 4π
3

1/3

, a measure of the sensitivity Vef f T ,and the 90% rate upper limit R90 . These quantities are displayed in table 9.1. As
mentioned above, the assumption of uniform density is only valid at large distances;in the frequency bands with lower sensitivity we are restricted to short distances, andso the calculation becomes invalid and thus we do not quote an upper limit for these.
This is denoted by “N/A” in the table.

Figure 9.27: The efficiency of detecting triples versus physical distance for injections
made in the 45–100 Hz band.

Table 9.1: Upper limit for triples in triple time
f-band (Hz)[45,100][100,200] [200,500] [500,1000] [1000,2500][260,120] [120,60][60,24][24,12][12,5]
M-band (M ), a = 0
M-band (M ), a = 0.994 [600,270] [270,140] [140,54] [54,27][27,11]366
4
2
Vef f (Mpc )2.1×10
2.3×10
3.6×10
5.8×105.1×10−1
ref f (Mpc)
79
82
20
5.2
0.49
T (yr)
0.03750.03750.03750.03750.03753
4
4
3
1
Vef f T (Mpc yr)
7.9×10
8.6×10
1.3×102.2×102×10−2−1 −1−3−3
R (yr L10 )1.5×10
1.4×10
N/A
N/A
N/A

140

Figure 9.28: The efficiency of detecting triples versus physical distance for injections
made in the 100–200 Hz band.

Figure 9.29: The efficiency of detecting triples versus physical distance for injections
made in the 200–500 Hz band.

141

Figure 9.30: The efficiency of detecting triples versus physical distance for injections
made in the 500–1000 Hz band.

Figure 9.31: The efficiency of detecting triples versus physical distance for injections
made in the 1000–2500 Hz band.

142

9.4.4

Error Analysis

Due to our lack of knowledge about the population of black holes we assign no systematic error to the astrophysical source population. Similarly, the waveform uncertainty
is unquantifiable; the waveform we use is taken to be the definition of a black hole
ringdown. We limit ourselves to evaluating systematic errors associated with theexperimental apparatus and analysis method. The only systematic error associatedwith the former is calibration of the data. The only systematic error associated withthe latter is with the limited number of Monte Carlo simulations to evaluate theefficiency. We consider only the 100–200 Hz band.
As described in section (2.2.3.1) the response function is calculated from a reference loop gain function, a reference sensing function, and calibration coefficientsrecorded every minute. Errors in the calibration can cause the SNR of a signal to
be incorrectly quantified, thus introducing inaccuracies in the distance. As the efficiency is a function of distance, care has to be taken to adjust the efficiency curve
appropriately. The fractional uncertainty in amplitude δ was found from calibration
studies [5] to be 5%. Returning to equation (9.11), this means that we are actually
evaluating ε(r[1 + δ]). Making the substitution of u for r[1 + δ],Z
Vef f = 4π

1ε(u)u2 du,(1 + δ)3(9.19)

shows we could be over- or under-estimating the effective volume by 15.67%. Thus theerror in the volume in the 100–200 Hz band due to the calibration is δVcal = 3.12×105
Mpc3 .
The second source of error is due to the limited number of injections in our Monte
Carlo (MC) simulations to evaluate the efficiency. Assuming binomial errors, the2variance of the efficiency σMC
is

2
=
σMC

ε(1 − ε)N

(9.20)

143
where N is the number of injections made. Thus the error in the effective volume,2
evaluated by multiplying σMCby the square of the volume of each bin,σV2MC =X

2
σMC(dVi )2 ,i(9.21)i

was found to be σVM C = 9.1 × 105 Mpc3 . Summing these errors in quadrature gives
a total error of σV = 9.6 × 105 Mpc3 . Multiplying by 1.64 gives the 90% confidence
interval of δV = 1.57 × 106 Mpc3 . To be conservative we apply a downward excursion
to the effective volume, giving an upper limit of R90% = 4.3 × 10−3 yr−1 L−1
10 .

9.4.5

Including Doubles in the Upper Limit Calculation

Before opening the box on the analysis the decision was made to examine all double
and triple coincident triggers for a detection but consider only triples in the calculationof the upper limit. We knew that in the absence of signal-based vetoes the level ofbackground was very high, and thus we would not gain very much by including
them. As an exercise we quantify this. The efficiency of detecting injections indouble coincidence in triple time was evaluated at the SNR of the loudest foreground
event for H1L1, H1H2, and H2L1 pairs. The calculated rate, along with intermediate
results of this calculation, are displayed in table 9.2. We did not perform injections
in double time, however we do not expect the efficiency of doubles in double time tobe significantly different from the efficiency of doubles in double time. We calculate
the efficiency of each of the three pairs at the loudest event found for each in double
time and using the double-time durations, calculated the upper limit. This is shownin table 9.3. Comparing with the 100–200 Hz band in table 9.1, the results show that
doubles in triple time contribute an additional 7% to the sensitivity, while doubles
from double time contribute just 1%.

144

Table 9.2: Upper limit for doubles in triple time
f-band (Hz)[45,100][100,200] [200,500][500,1000] [1000,2000]
M-band (M ), a = 0[260,120] [120,60][60,24][24,12][12,5]
M-band (M ), a = 0.994 [600,270] [270,140] [140,54][54,27][27,11]355
1
2
Vef f (Mpc )4.1×10
1.6×10
2.5×10
3.2×10
2.6×10−3
ref f (Mpc)
46
34
1.8
0.91
0.085
0.03750.03750.03750.03750.0375T (yr)
3
4
3−1−1
Vef f T (Mpc yr)
1.5×10
5.9×10
9.4 ×101.2×10
9.6×10−5−1
R (yr−1 L10 )7.6×10−3 2.0×10−2 1.2×102
9.8×103
1.2×106

Table 9.3: Upper limit for doubles in double time
f-band (Hz)[45,100][100,200] [200,500] [500,1000]
M-band (M ), a = 0[260,120] [120,60][60,24][24,12]
M-band (M ), a = 0.994 [600,270] [270,140] [140,54][54,27]3−1−2(Vef f T )H1H2 (Mpc yr) 1.6×10
2.2
2.8×10
3.3×10−332−33(Vef f T )H1L1 (Mpc yr)
2.2×10
8.3×10
9.4×10
6.4×10−3
3
(V
9.0
6.8
9.4×10−2 7.6×10−3
Pef f T )H2L1 (Mpc3 yr)
2.2×103
8.4×102
1.3×10−1 1.7×10−2
i (Vef f T )i (Mpc yr)
−1 −1
−2
−1
R (yr L10 )5.4×10
1.4×108.9×102
6.8×103[1000,2000][12,5][27,11]
9.3×10−6
1.2×10−6
7.4×10−6
1.8×10−5
6.5×106

145

Chapter 10
Search with Simulated
Inspiral-Merger-Ringdown Signals
10.1

Introduction

As described in previous chapters, the simulated signals on which the pipeline wastuned and the upper limit calculated consisted of isolated ringdown waveforms. However, an important source of gravitational wave ringdowns is expected to be binary
black hole coalescences, in which case the ringdown will be preceded by an inspiraland merger. The inspiral and ringdown phases are well modeled, but analytic expressions of the merger waveform do not exist. However, recent breakthroughs innumerical relativity have given us a clearer picture of what to expect from the merger
phase, and several groups are currently working on methods to utilize these results
to provide analytic waveforms for use in coherent matched-filter searches.
To complete our investigation into the presence of ringdowns in S4 data it isnecessary to check if the presence of an inspiral and merger would hamper or enhance
our ability to detect and estimate the parameters of the ringdown using a ringdownmatched filter. To that end, we create inspiral-merger-ringdown (IMR) waveforms;
as described in detail below, these waveforms consist of an inspiral waveform which is
stitched in a continuous manner to a ringdown with the intervening signal representing
the merger. The calculation of the ringdown parameters from the inspiral parameters
is guided by the recent results of numerical relativity. Our simulations cover a much

146
larger space than that for which numerical waveforms are currently available, andthus for now should be viewed as an approximation to the true waveform. In thisanalysis we inject both IMR and ringdown-only waveforms into the data and compare
the outputs of both the single and coincident detector analyses.

10.2

Numerical Relativity

Numerical relativity is a branch of computational physics concerned with solving
Einstein’s equation numerically. 2005 saw a major breakthrough in the field when
for the first time an equal mass non-spinning binary black hole system was evolved
from the last few orbits of inspiral through merger to ringdown, and the gravitational
wave was extracted [81]. These and subsequent results [82], [83], indicate that forsuch a system the final spin of the black hole is expected to be close to 0.7. Studies ofunequal mass non-spinning black holes suggest that the amplitude of the l = m = 2
mode of the ringdown waveform decreases as the mass ratio of the binary components
increases whereas the amplitude of the l = m = 3 mode increases [84]. If the black
holes are initially spinning, simulations show that the spin of the final black hole andthe amount of radiation emitted also depend on the magnitude and inclination of thespins with respect to the orbital angular momentum [85, 86, 87, 88].
Evolving a binary system from inspiral through to ringdown is computationallyexpensive; creating a bank of these waveforms for use in matched filter searches is
simply not feasible. However efforts are currently underway within the LSC andelsewhere to use the results from numerical simulations of non-spinning black holes
with mass ratios 1 : 1 to 1 : 4 in conjunction with the well-modeled inspiral waveform
to create IMR waveforms. One method [89] extracts the mass and spin of the final
black hole from the numerical waveforms, and from this calculates the fundamental
quasi-normal mode and two overtones of the ringdown. To this an inspiral waveform(given by the effective one body model) is matched, giving a complete inspiral-mergerringdown waveform. A second method [90] constructs hybrid waveforms by matchinga post-Newtonian inspiral to the merger and ringdown of numerical simulations, and147from these proposes a family of phenomenological waveforms which closely match thehybrid waveform. Both these methods of creating accurate coalescence waveforms foruse in LIGO analyses are at early stages of development but promise to give a larger
range of sensitivity than templated searches for a single phase. This is illustrated
in figure 10.1 where, by using the initial LIGO sensitivity curve, a comparison ismade between the horizon distance attainable by the inspiral and ringdown searches
and the coherent IMR search described in [90]. The figure shows that the coherent
IMR search allows us to see to much larger distances. Model waveforms for spinningbinaries and binaries with larger mass ratios are also under development.

Figure 10.1: A preliminary assessment of the performance of a phenomenologicaltemplate bank (red line) for use in IMR searches compared to searches using onlyinspiral (blue dots) and only ringdown (red dashes) templates. The ringdown curve
assumes  = 0.7% and the shaded area represents 0.18% ≤  ≤ 2.7%, and can becompared with figure 6.3 which displays the ringdown horizon distance for  = 1%.
This figure is taken from [90].

148

10.3

Creating the IMR Waveform

At the present time numerical waveforms encompassing the range of ringdown injections made into S4 data are not available. We can, however, make an approximation using the well-modeled inspiral and ringdown waveforms currently used inthose single-phase searches. Here we describe how the individual parts of the IMR
waveform were created from a set of initial conditions and stitched together to forma coherent waveform for injection into the data.
As discussed in section 7.2, to inject a signal into the data (in software), theuser supplies a list of input parameters that the appropriate waveform generation
code uses to produce a waveform structure containing the plus and cross amplitudes,
phase, and frequency for each time series data point in the waveform. For the IMR
injections we start with the initial parameters of the inspiral phase. These include thecomponent masses and spins, distance, and source position and orientation. Using
this information a spinning inspiral waveform1 is created up until the point where thebinary separation is 6GM/c2 . We denote the frequency at that point by f6M . The
next step is to estimate the parameters of the ringdown. The final spin of the blackhole â can be estimated from the masses m1 and m2 and the spins â1 and â2 of thebinary components using the results of numerical simulations as a guide [85],sâ =â21 m21 â22 m22
η
+
+
0.7
2
2
MT
MT
0.25

(10.1)

where MT is the sum of the individual masses and the last term is the contributionof the orbital angular momentum with κ representing the symmetric mass ratio η =m1 m2 / (m1 + m2 )2 . The estimation of final mass M also uses results from numerical
relativity [85],M = MT 1 − 0.01 1 + 6â2 .

(10.2)

Then, assuming that all the gravitational radiation is emitted in the l = m = 2 mode,
1

For more details about searches for gravitational waves from inspiraling spinning black holes see
[91].

149
the ringdown waveform parameters fr and Q may be calculated using equations (3.4)
and (3.6).
Now that we know both f6M and fr the challenge is to match these in a smoothand continuous manner. This is implemented in two stages in the LAL [63] code. Firstthe evolution of the frequency from the inspiral stage is continued with f˙ ∼ f 11/3 until
90% of the ringdown frequency f0.9 is reached. From there the ringdown frequency isapproached exponentiallyf (t) = fr − Ke−λt(10.3)

where K = fr − f0.9 and λ = f˙/A. In a similar manner the phase evolution from theinspiral stage is continued until f reaches f0.9 and then it is evolved asφ(t) = φ0.9 + 2πf t.

(10.4)The plus and cross amplitudes between the inspiral and ringdown are fit with aquadratic,
A = α0 + α1 ∗ t + α2 ∗ t2

(10.5)

where α0 is the amplitude at the cut off point of the inspiral, α1 = α̇0 (i.e. , the rateof change of the amplitude during the inspiral), and
α2 =(γ − 1) (α0 + τ α1 )− τ,(1 − γ) τ 2 + 2τ(10.6)

where γ is a damping factor and τ is the length of time between the end of the inspiraland the start of the ringdown. Finally the ringdown plus and cross amplitudes,
frequency, and phase proceed as described in section 7.2.
Because we want to compare a simulated IMR signal with an isolated ringdown
it is important that an identical ringdown is generated in each case. Thus, for aringdown-only injection we create the full IMR waveform and set the inspiral andmerger amplitudes to zero. To be consistent with the S4 ringdown analysis we define
the ringdown to start at the point where the frequency becomes constant. Figure

150

Figure 10.2: The inspiral-merger-ringdown for a binary black hole system with component spins of 0.88 and 0.84 and masses of 8.9 M and 6.3 M (blue). Also shownis the ringdown-only waveform for the same system (red).

Figure 10.3: The transition portion of the inspiral-merger-ringdown waveform (dots)
and the ringdown-only waveform (circles) from figure 10.2.

151

Figure 10.4: The frequency time series of the coalescence. The inset zooms in on theinspiral-merger transition and ringdown phase.

Figure 10.5: M ωR versus (unitless) time for the coalescence of a black hole binary.
The final black hole has a spin of 0.9 and an M ωR of 0.68, which agrees with figure
3.3 for the l = m = 2 mode.

152
10.2 shows one example of IMR and ringdown-only waveforms produced for a systemwith spins of 0.88 and 0.84 and masses of 8.9 M and 6.3 M respectively. The end
time of the inspiral and the start time of the ringdown are marked, and it is only after
the latter point that the amplitude of the ringdown-only injection becomes non-zero.
This is best seen in figure 10.3 which zooms in on the transition between the inspiraland ringdown. After this point the amplitudes match exactly. Figure 10.4 shows thefrequency evolution of the same IMR waveform with the inset showing just the final
20 ms of the waveform. Here we see how the inspiral frequency increases rapidly until
the constant ringdown frequency is reached. This marks the ringdown start time. In
figure 10.5 we plot the same quantity in the dimensionless quantities. The final spinof the black hole is 0.9 and the ringdown M ωR = 0.68. This is in agreement with
figure 3.3 for the l = m = 2 mode. We can also compare this to figure 3.7.
If we want to evaluate how well we are recovering the injections it is necessary
to calculate and record the ringdown parameters to compare with the output of the
ringdown filter. Solving a+ and a× for A and ι in equations (7.1) and (7.2) enables us
to calculate the effective distance and the percentage of mass radiated as gravitational
waves . For every injection these parameters are written out to a sim ringdown
table.

10.4

Single Detector Analysis

The simulated signals were created uniformly in logarithmic component mass between
4 M

and 600 M , with a maximum total mass of 650 M . The distribution in

distance was also logarithmic between 0.1 and 1 Mpc. Each component black hole
had a spin â1 , â2 whose magnitude was distributed uniformly in the range 0 and 1.
This investigation was composed of two separate runs; in the first run we injected
the full IMR waveform and in the second run only the ringdown was injected. The
same initial conditions were used in both runs ensuring that an identical ringdown
was injected each time.
Comparing the trigger files output from the pipeline with the injection file allows

153

Figure 10.6: Number of IMR and ringdown-only injections recovered in H1 as afunction of the frequency of the final ringdown. The vertical lines denote the templatebank boundaries.

Figure 10.7: Detected versus injected ringdown frequency for IMR and ringdown-only
waveforms in H1.

154
us to count how many injections were found and how well the parameters were recovered. As shown in figure 10.6, we found that approximately 18% more injections
were found in the IMR run than in the ringdown-only run, with the excess appearing
above an injected ringdown frequency finj of ∼ 600 Hz. Figure 10.7 shows that of
the injections that were found in common by the two runs, the detected frequency
fdet of injections with finj below about 200 Hz were fairly consistent between the two
runs. Those injections with finj > 200 Hz were not so consistent; the ringdown-only
injections were found with templates close to the injected ringdown parameters, but
the IMR injections were mostly found by templates in the 100–200 Hz band.
This can be explained as follows: for IMR injections with low ringdown frequency
the inspiral part of the waveform is outside the LIGO band. As we increase theringdown frequency, an increasing proportion of the inspiral and merger enters theband, matching an increasing number of templates. This is demonstrated in figures
10.8 and 10.9, which show the templates that rang off during the 120 ms around a
ringdown-only and an IMR injection, respectively, where the ringdown frequency was
finj ∼ 1500 Hz. In the ringdown-only case the only templates that ring are closein frequency to the frequency of the injection and do so right at the time of peak
amplitude of the waveform, as indicated by the dashed lines. For the IMR case,however, most of the templates in the bank ring off. The templates ring off just as
expected for the characteristic chirp frequency evolution of an inspiral; the inspiralenters the LIGO band at low frequency and its frequency increases until it reaches
the ringdown. The template that rings up the loudest (and hence is the templateassociated with the injection) in this case is at ∼ 110 Hz, as indicated in the plot by
the red horizontal line. This is far from the ringdown frequency, denoted by the black
horizontal line, but it is where the LIGO strain sensitivity is best (see figure 6.2).
In figure 10.10 we plot the initial component masses of all the IMR injections
with finj > 50 Hz that were found at the correct ringdown frequency in red, andthose that were found incorrectly in green. The plot shows that the majority of theinjections that were found incorrectly by the ringdown search fall within the scope of
the S4 BBH search, and thus that search should be able to find the signal and correctly

155

Figure 10.8: Frequency versus time for the templates that rang up around the time ofa ringdown-only injection. The colour of the data points represents the quality factorof the template. The black lines represent the frequency and time of the injection,and the red lines represent the frequency and time of the template with the largest
signal-to-noise ratio.

Figure 10.9: Frequency versus time for the templates that rang up around the timeof an IMR injection. The colour of the data points represents the quality factorof the template. The black lines represent the frequency and time of the injection,and the red lines represent the frequency and time of the template with the largest
signal-to-noise ratio.

156
identify the component masses. Most of those injections that were correctly identified
lie outside that region, and so we can conclude that between the two searches themass space is covered quite well.
Our results concur with a study by Baumgarte et al. [92] in which two numerical
waveforms — one with ringdown frequency of ∼ 80 Hz and the other with f ∼ 280 Hz— were filtered with ringdown templates. For the former waveform the best-matched
template triggered at the time of the ringdown, whereas in the latter case, when the
ringdown frequency was higher than LIGO’s most sensitive band, the best matchoccurred earlier, during the inspiral phase.

Figure 10.10: Initial masses of the binary components for IMR injections found bycorrect (red) and incorrect (green) ringdown templates. The black line represents theupper limit to the mass range of the S4 binary black hole inspiral search.

10.5

Coincidence Analysis

Thus far we have ascertained that although a ringdown may be detected with a lower
frequency template, the presence of an inspiral and merger before the ringdown does

157
not prevent it from being detected. The next important question is whether or not an
IMR injection survives the coincidence test, as this is a key test in our pipeline. Recall
from section 5.5 that for a given time window δt triggers from different detectors are
considered coincident if they lie within a specified parameter window ds2 (f0 , Q) of
each other. If multiple groups of coincidences (i.e., many triples and doubles) arefound within δt, then the group with the loudest value of the detection statistic ischosen as the “correct” coincidence.
Ringdown-only and IMR injections were made into H1, H2, and L1 and the output
coincidence files compared to the injection list; the results are shown in figures 10.11
and 10.12, respectively, in plots of effective distance as measured at Hanford versus theinjected ringdown frequency. These plots show the injections that were found in triple
coincidence, those found in double coincidence (both because of vetoes and because
they were missed in the third detector), and the missed injections. The ringdown-onlyresults shown in figure 10.11, are, as expected, similar to those discussed in section
8.2. What is interesting about the IMR coincidence, however, is that even thougha large proportion of injections were found with the wrong ringdown parameters inall detectors, these were close enough to each other to allow the injection to pass thecoincidence test! In fact, the efficiency of detection is higher in the IMR case than in
the ringdown-only case (recall that parameter accuracy is not taken into account whencalculating efficiencies). The high ringdown frequency injections that were missed in
the ringdown-only case (because of high levels of noise above 1 kHz) were found in
the IMR case because the inspiral part of the injection occurred at a less noisy (moresensitive) region of the template bank. This is a very encouraging result.

10.6

Conclusion

The calculation of the upper limit on the rate of ringdowns described in section 9.4
was based on our ability to recover injected signals. For that study we used isolated
ringdown signals. The question here was how would this change if the ringdown waspreceded by an inspiral and merger. This investigation has shown that the presence

158

Figure 10.11: Hanford effective distance versus injected ringdown frequency for
ringdown-only injections. The black vertical lines denote the template bank boundaries.

Figure 10.12: Hanford effective distance versus injected ringdown frequency for IMR
injections. The black vertical lines denote the template bank boundaries.

159
of an additional signal before the ringdown does not in any way hinder our ability
to detect the signal. In fact, this model of an IMR injection improves our ability
to detect coalescences with high ringdown frequencies i.e., low black hole masses.
This increases our efficiency, which also positively impacts the upper limit we canset. From this we can conclude that not only is the upper limit for the S4 ringdown
search presented in section 9.4 still valid, it may be regarded as a conservative upper
limit.
What is impacted, however, is our ability to correctly recover the black hole’s
physical parameters; this study demonstrated that we can only correctly identify
ringdown frequencies occurring below ∼ 200 Hz. However, this lower limit on theaccurate recovery of the mass of a black hole corresponds to the upper limit to thescope of the binary black hole inspiral search. Thus, a low-mass binary black holecoalescence will be detected by both searches and correctly parameterized by theinspiral search, while high-mass coalescences should be detected and parameterized
by the ringdown search (of course, only within the distance reach of the searches).

160

Chapter 11
The Future for Ringdown Searches
This thesis describes the first ringdown detection search in LIGO data and has demonstrated that the pipeline is an effective method of searching for triple coincident
ringdown events. However this is just the beginning; with every science run comesincreased sensitivity and the possibility of exploring a much larger population of astrophysical sources.
In the course of the analysis we have gained an understanding of the characterof ringdown waveforms in noisy data. In this chapter we list some of the unsolved
issues, lessons learned, and future recommendations for this particular search. We
discuss some new ideas for combining searches for the individual inspiral, merger andringdown phases of the binary coalescence and discuss the parameter space available
to future ringdown searches.

11.1

Notes for Future Searches with the current
Pipeline

11.1.1

Searches for Triple Coincident Events

We saw that in the S4 search the rate of false alarms in triple coincidence was lessthan one event per run. Now that we have some understanding of the characteristics
of simulated ringdown waveforms in data we can tolerate a somewhat higher levelof background and use these known features to veto false alarms. This gives us

161
leeway to loosen some of the constraints on the search and gain sensitivity. We havedemonstrated that the coincidence windows were sufficiently loose and that we did
not lose any injections because of clustering. However the search signal to noise
thresholds could be lowered further. Given that a triple coincidence search is limitedby H2 we recommend lowering its SNR threshold. Decreasing the H2 threshold to
4 would allow triple coincident signals to be seen with SNRs as low as 8 in H1 andL1 (as opposed to 11 in the current search). Given the rate of false alarms in double
coincidence, attaining this level of sensitivity without H2 is currently not possible.

11.1.2

Searches for Double Coincident Events

The results of the double coincidence analysis showed that the level of backgroundwith the current pipeline was too high to detect gravitational waves at threshold of
5.5. We are a long way from being able to claim a detection of gravitational wave
ringdown from co-located detectors however requiring two site coincidence shouldin theory provide sufficiently strong evidence. We just discussed how to increase oursensitivity to triples without changing the pipeline. However increasing our sensitivity
to doubles will require significant additions to the pipeline. We will need to work
harder at reducing the level of false coincidences. One method of doing this is byimplementing signal-based vetoes; vetoes based on our knowledge of a signals shapein the time and frequency domains [93]. These have been implemented in inspiralsearches and are effective in reducing the false alarm rate. Caution has to be exercised
however when implementing these in the ringdown search. The ringdown is a shortduration single frequency waveform and is likely (but not necessarily) to be precededby an inspiral and thus any signal-based vetoes must be tested on IMR waveforms toallow for this possibility.

11.1.3

Coincidence Test

The coincidence test described in this analysis in which we use the metric to define
coincident windows is an vast improvement on the traditional rectangular coincidence162
test. However the results of the injection (section 8.4.1) run revealed that the difference in the time of arrival of the injected waveform was a strong function of frequency,
particularly for the H1H2 pair (see figure 8.16). This plot showed that at high frequencies a much tighter time accuracy could be required. A frequency dependence
time soincidence test should be considered for future searches as it is likely to reduce
the false alarm rate considerably.

11.1.4

Extending the Template Bank

For the S4 search the region of frequency space searched over was from 50 Hz to 2
kHz. As mentioned in section 7.4.1 we had hoped to extend the template bank toencompass a frequency range of 40Hz to 4kHz. However in the course of tuning thesearch and following up on missed injections a peculiar feature was observed. In plotsof SNR versus frequency for injections, such as is shown in figure 11.1, the SNR falls
off from the injection frequency as expected, but then begins to increase on both sidesof the peak. This was observed in varying degrees of severity in every injection looked
at regardless of the frequency of the injection. This feature became problematic when
the templates far from the injection had higher SNR than those close to the injected
frequency. When this occurred in one detector the injection failed the coincidence
test and if it occurred in two or three detectors the injection was found at the wrong
frequency. In the example shown in figure 11.1 the injection was made at 200 Hz but
was found close to 4 kHz. Weeks of investigations were dedicated to this problem but
a solution was not found and so the smaller bank was reinstated. These “wings” arestill observable with the smaller bank but the effect is small enough that they do notinterfere with signal recovery.
This is an important problem to solve because the wider the frequency range we
can search over, the larger the number of black hole ringdowns we are sensitive to.
In particular, in increasing the upper frequency bound to 4 kHz we become sensitive
to gravitational waves from the entire mass range of non-spinning stellar mass black
holes. This would provide an excellent overlap with the binary black hole inspiral

163
search.

Figure 11.1: A demonstration of the high SNR “wings” observed when the templatebank was extended to include frequenies between 40 Hz and 4 kHz. This plot shows
the SNR versus frequency for a ringdown injection with central frequency of 200 Hz.

11.2

Future Searches

From an analysis point of view the hope for the future is to focus on IMR searches,bringing together the efforts of the three stand-alone pipelines: the binary blackhole inspiral search, a burst search (which is sensitive to the merger phase) andthe ringdown search. As we demonstrated in chapter 10 there is a large degree ofoverlap between the searches. This combined effort could be implemented by runningthe filtering and coincidence steps separately and then comparing coincident triggers
from the three searches. Another possibility is to combine the outputs of the filtering
stage, and require coincidence between searches for each of the detectors and thenlook for coincidences between detectors.

164
The future holds exciting prospects for ringdown searches. This is best illustrated
by figure 11.2. Recall that the horizon distance is the distance to which we can detecta ringdown from an optimally oriented and located black hole of spin â with a signal
to noise ratio of 8 in the detector. The figure shows horizon distance as a functionof black hole mass for the predicted sensitivities of Initial LIGO (the blue curve),
Enhanced LIGO (shown in green) and Advanced LIGO (shown in red) assuming that
1% of the mass is radiated as gravitational waves during the ringdown. The three
curves in each group correspond to different spins, â = (0, 0.49, 0.9) for the curves
from left to right, respectively. On the upper side of the horizontal axis the central
ringdown frequency for a spin of 0.49 is also marked.
Comparing figures 6.3 and 11.2 demonstrates that we can expect to see ∼ 100 Mpc
further for Initial LIGO at design sensitivity than we did for S4. Preparations are
currently underway to analyse data from the S5 run with the ringdown pipeline. S5
was the first science run at design sensitivity and includes one year of triple coincident
data.
Enhanced LIGO is due to come on-line by the end of 2008. As figure 11.2 shows,
we can expect a factor two increase in sensitivity as well as an extended mass range,
with the lower mass limit extending further into the stellar mass black hole range.
The Advanced LIGO sensitivity curve used here is for the low power configuration,(optimized for low frequency signals by reducing the radiation pressure quantum
noise) and even at that the prediction is that our reach will be extended by anorder of magnitude entering the regime of cosmological distances. At higher power
this would be increased even further for low mass black hole ringdowns. The lower
frequency limit in Advanced LIGO will be 12 Hz. This corresponds to a mass of 1000
M for non-spinning black holes up to 2300 M for rapidly spinning black holes. This
makes the prospects for detection fro intermediate mass black holes very promising.

165

Figure 11.2: Plot of horizon distance (distance to which a ringdown signal from an
optimally oriented and located source will produce an SNR of 8 in the detector) versus
mass for Initial LIGO (blue), Enhanced LIGO (green) and Advanced LIGO (red) ina low-power configuration. We have assumed that 1% of the mass is radiated as
gravitational waves. The curves in each group are for spins of 0, 0.49 and 0.98 goingfrom left to right. The upper x-axis is the frequency for a spin of 0.49. (Plot from A.
Weinstein.)

166

Chapter 12
Summary and Concluding Remarks
12.1

Aim of the Search

The gravitational radiation emitted by a perturbed black hole (e.g., the final stageof a compact binary coalescence (CBC)) is well modeled and can be searched forusing the method of matched filtering. However in the presence of non-Gaussian
noise, optimal filtering alone is not sufficient to uncover a weak signal in the data. A
powerful method of reducing the rate of false alarms is to require that a trigger beseen at the same time in multiple detectors with similar parameters in order to beconsidered a candidate gravitational wave event.
In this study we have described the pipeline for a matched filter search withcoincidence analysis and applied the pipeline to data from the fourth LIGO sciencerun. The aim of the search was to detect gravitational waves from perturbed black
holes and in the absence of a detection to place an upper limit on the rate of blackhole ringdowns in the nearby universe.

12.2

The Analysis Method

We ran the search as a blind analysis to prevent any bias on the part of the analyst
from influencing the outcome of the search (for example setting cuts based on triggers
in the zero-lag data in order to get a better upper limit). We tuned cuts, thresholds
and coincidence windows with

167
• a coincidence analysis of simulated ringdown signals to gauge how well we can
expect to recover the parameters of a real signal,• a coincidence analysis of time shifted data sets to estimate the rate of false
alarms,• and a representative subset of the data as a sanity check of the analysis pipeline
and a consistency check of the background estimation.
Tuning is an iterative process with the outcome of the previous run influencing theconstraints on the subsequent run until the point is reached where the maximum
number of injected signals are recovered while the rate of accidental coincidences
is kept to a minimum. Once the tuning was finalized a large scale Monte Carlosimulation of ringdown waveforms was run to evaluate the efficiency of the searchand to facilitate parameter accuracy investigations. A final background analysis was
run to evaluate the false alarm rate.
It was decided in advance of unblinding the analysis to consider all double andtriple coincidences as possible gravitational wave candidates. However in the absenceof a detection only triple coincident signals would be used in setting an upper limit.
Finally the tuned pipeline was applied to the data.
In a separate study we investigated the effects of an inspiral and merger waveform
preceding the ringdown on our ability to detect gravitational waves with a ringdown
filter and on our ability to recover the ringdown waveform parameters. We injected
inspiral-merger-ringdown waveforms into S4 data and ran with the same pipeline andtuning described above. We then repeated the run injecting only the ringdown partof the pipeline and compared the results.

168

12.3

Results

12.3.1

Opening the Box

No candidate events were found in triple coincidence. A large number of candidate
events were found in double coincidence, however the distribution was in agreementwith the expected false alarm rate to within 1 sigma. Thus there was no evidence in
the S4 data set of a detection of gravitational waves from perturbed black holes.

12.3.2

Calculation of the Upper Limit

Our Monte Carlo simulations revealed that the efficiency of detecting gravitational
wave ringdowns is highly frequency dependent. For that reason we divided the simulations into five frequency bands and evaluated the efficiency in each band. The
100-200 Hz band was found to be the most sensitive. This frequency range corresponds to a mass range of 60 M ≤ M ≤ 120 M for non spinning black holes and
150 M ≤ M ≤ 300 M for maximally spinning black holes assuming that the waveform is dominated by the l = m = 2 mode. The 90% confidence upper limit of the rateof ringdowns in the 100 − 200 Hz band was found to be R90% = 4.3 × 10−3 yr−1 L−1
10 ,where L10 is a measure of the number of potential sources, equal to 1010 solar blue
light luminosity.

12.3.3

Parameter Accuracy

We compared the injected and detected quantities and found that the accuracy withwhich the time of arrival of the injection could be determined was frequency dependent due to the uncertainty in the phase of the waveform. The waveform parameter
accuracy was very high, with 70% of the triple coincident injections found with thecorrect template. We found that the match between the injections and templatedecreased with the combination of high frequency and low quality factor values.

169

12.3.4

Background Estimation

Our background studies showed that with the chosen tuning the false alarm rate intriple coincidence was very low, less than one event per S4 run. In double coincidence
the rate was much higher, more than 600 events per S4 run for two site coincidence. Infact the false alarm rate for signals found in detectors located in two widely separated
sites with the exact same waveform parameters was 30 per S4 run. Clearly this is toohigh a rate to be able to confidently detect gravitational waves at the chosen thresholdof ρDS ∼ 12. However at a higher threshold of ∼ 16, less that 0.2 background eventsare expected in H1L1 and detection is possible.

12.3.5Inspiral-Merger-Ringdown

We found that with the inclusion of the inspiral and merger waveforms our efficiency
of detecting simulated signals increased. Isolated ringdowns at high frequency are
more difficult to recover than those at low frequency because the level of noise above
500 Hz increases rapidly and our efficiency decreases. However the inspiral and merger
preceding the ringdown sweep up through the LIGO band finding a significant matchwith ringdown templates in the process. We found that the part of the signal in
the most sensitive band of the detector, 100–200 Hz produced the loudest SNR andthe ringdown was “found” in this band. IMR injections with ringdown frequencies
above 200 Hz were generally detected with templates in the 100–200 Hz band. A
consequence of this was that our ringdown parameter estimation was not accurate for
IMRs with ringdown frequency above 200 Hz.

12.4

Conclusions• We have demonstrated using simulated signals that the pipeline presented inthis study is an effective means of detecting gravitational waves from perturbed
black holes in triple coincidence.

170• Our results verify that the timeslide method of determining the false alarm rateand distribution with SNR is accurate for detectors at widely separate sites,
but not for co-located detectors.
• We have found that the level of noise in double coincidence is too high to allow
the detection of gravitational waves with the chosen SNR threshold.
• We have shown that the accuracy with which the ringdown waveform parameters
can be recovered for ringdowns with frequency in excess of 200 Hz depends onwhether or not an inspiral and merger preceded the ringdown. If our assumptionthat the radiation is dominated by the l = m = 2 mode is correct we canestimate the mass of those ringdowns to a high degree of accuracy.
• The next few years are very promising for IMR searches. Enhanced LIGO will
come online within a year increasing our sensitivity by a factor of 2. In 2014
Advanced LIGO will push the range in which we can search out to cosmologicaldistances in addition to extending the population of black holes we are sensitive
to beyond 103 M .

171

Bibliography[1] R. Adhikari, P. Fritschel, and S. Waldman, LIGO-T060156-01-I (2006).
[2] J. B. Hartle, Gravity: an introduction to Einstein’s general relativity, Addison
Welsey (2003).
[3] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman
and Company (1970).
[4] P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detection,
World Scientific Publishing (1994).
[5] A. Dietz, J. Garofoli, G Gonzales, M. Landry, B. O’Reilly, and M. Sung, LIGOT050262-01-D (2005).
[6] X. Siemens, B. Allen, J. Creighton, M. Hewitson, and M. Landry, Class. Quantum Grav. 21, S1723 (2004).
[7] B. F. Schutz and M. Tinto, MNRAS, 224, 131 (1987).
[8] D. Sigg, LIGO-P980007-00-D (1998).
[9] R. A. Hulse and J. H. Taylor, Ap. J., 195, L51 (1975).
[10] K. S. Thorne, 300 Years of Gravitation, edited by S.W. Hawking and W. Israel,
Cambridge University Press, Cambridge, England (1987).
[11] B. Abbott et al., “Analysis of LIGO data for gravitational waves from binary
neuton stars”, Phys. Rev. D, 69, 122001 (2004).

172[12] B. Abbott et al., “Search for gravitational waves from galactic and extragalactic binary neutron stars”, Phys. Rev. D, 72, 082001 (2005).
[13] B. Abbott et al., “Search for gravitational waves from primordial black holebinary coalescences in the galactic halo”, Phys. Rev. D, 72, 082002 (2005).
[14] B. Abbott et al., “Search for gravitational waves from binary black-hole inspirals in LIGO data”, Phys. Rev. D, 73, 062001 (2006).
[15] B. Abbott et al., “Search of S3 LIGO data for gravitational wave signals fromspinning black hole and neutron star binary inspirals”, arXiv:0712.2050 (2007).
[16] B. Abbott et al., “Search for gravitational waves from binary inspirals in S3and S4 LIGO data”, arXiv:0704.3368 (2007).
[17] B. Abbott et al., “Implications for the Origin of GRB 070201 from LIGO
Observations”, arXiv:0711.1163 (2007).
[18] B. Abbott et al., “First upper limits from LIGO on gravitational-wave bursts”,
Phys. Rev. D, 69, 102001 (2004).
[19] B. Abbott et al., “Upper limits on gravitational-wave bursts in LIGO’s second
science run”, Phys. Rev. D, 72, 062001 (2005).
[20] B. Abbott et al., “Search for gravitational-wave bursts in LIGO’s third sciencerun”, Class. Quant. Grav., 23, S29 (2006).
[21] B. Abbott et al., “Search for gravitational-wave bursts in LIGO data from thefourth science run”, Class. Quant. Grav., 24, 5343 (2007).
[22] B. Abbott et al., “Search for Gravitational Waves Associated with 39 GammaRay Bursts Using data from the Second, Third, and Fourth LIGO Runs”,
arXiv:0709.0766 (2007).
[23] B. Abbott et al., “Setting upper limits on the strength of periodic gravitational
waves from PSR J1939+2134 using the first science data from the GEO 600
and LIGO detectors”, Phys. Rev. D, 69, 082004 (2004).

173[24] B. Abbott et al., “Upper limits from the LIGO and TAMA detectors on therate of gravitational-wave bursts”, Phys. Rev. D, 72, 102004 (2005).
[25] B. Abbott et al., “Limits on gravitational wave emission from selected pulsars
using LIGO data”, Phys. Rev. Lett., 94, 181103 (2005).
[26] B. Abbott et al., “Coherent searches for periodic gravitational waves fromunknown isolated sources and Scorpius X-1: results from the second LIGO
science run”, Phys. Rev. D, 76, 082001 (2007).
[27] B. Abbott et al., “Upper Limits on Gravitational Wave Emission from 78 Radio
Pulsars”, Phys. Rev. D, 76, 042001 (2007).
[28] B. Abbott et al., “All-sky search for periodic gravitational waves in LIGO S4data”, Phys. Rev. D, 77, 022001 (2008).
[29] B. Abbott et al., “Analysis of first LIGO science data for stochastic gravitational waves”, Phys. Rev. D, 69, 122004 (2004).
[30] B. Abbott et al., “Upper limits on a stochastic background of gravitational
waves”, Phys. Rev. Lett., 95, 221101 (2005).
[31] B. Abbott et al., “Searching for Stochastic Background of Gravitational Waves
with LIGO”, Ap. J., 659, 918 (2007).
[32] B. Abbott et al., “Upper limit map of a background of gravitational waves”,
Phys. Rev. D, 76, 082003 (2007).
[33] B. Abbott et al., “First Cross-Correlation Analysis of Interferometric andResonant-Bar Gravitational-Wave Data for Stochastic Backgrounds”, Phys.
Rev. D, 76, 022001 (2007).
[34] J. Michell, Philos. Trans. R. Soc. London, 74, 35 (1784).
[35] K. Schwarzschild, Sitzungsber. Gottingen Math. Phys. Kl. 424 (in German),arXiv:physics/9912033v1 (in English) (1916).

174[36] J. R. Oppenheimer and H. Snyder, Phys. Rev., 56, 455 (1939).
[37] J. A. Wheeler, Am. Sci., 56, 1 (1968).
[38] J. A. Wheeler and R. Ruffini, Phys. Today, 24, 30 (1971).
[39] T. Regge and J. A. Wheeler, Phys. Rev., 108, 1063 (1957).
[40] E. W. Leaver, Proc. R. Soc. Lond., 402, 285 (1985).
[41] E. Berti, arXiv:gr-qc/0411025v1 (2004).
[42] A. Bachelot and A. Motet-Bachelot, Ann. Inst. Henri Poincaré, 59, 3 (1993).
[43] S. A. Teukolsky, Ap. J., 185, 635 (1973).
[44] E. Berti, V. Cardoso, and C. Will, Phys Rev. D, 73, 064030 (2005).
[45] C. V. Vishveshwara, Nature, 227, 936 (1970).
[46] F. J. Zerilli, Phys. Rev. D, 2, 2141 (1970).
[47] W. H. Press, Ap. J. 170, L105 (1971).
[48] R. H. Price, Phys. Rev. D, 5, 2439 (1972).
[49] S. Chandrasekhar and S. Detweiler, Proc. R. Soc. Lond., 344, 441 (1975).
[50] V. Ferrari and B. Mashhoon, Phys. Rev. D, 30, 295 (1984).
[51] K. D. Kokkotas and B. G. Schmidt, Living Rev. Relativity, 2, 2,
http://www.livingreviews.org/lrr-1999-2 (1999).
[52] F. Echeverria, Phys. Rev. D, 40, 3194 (1989).
[53] N. Andersson, K.D. Kokkotas, IJMPD, 10, 381, arXiv:gr-qc/0010102v1 (2001).
[54] M. Davis, R. Ruffini, W. H. Press and R. H. Price, Phys. Rev. Lett., 27, 1466
(1971).

175[55] É. É. Flanagan and S. A. Hughes, Phys. Rev. D., 57, 4566 (1998).
[56] A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D, 75, 124018 (2007).
[57] V. Kalogera and G. Baym, Ap. J. Lett, 470 L61 (1996).
[58] L. Ferrarese and H. Ford, Space Sci. Rev., 116, 523 (2005).
[59] M. C. Miller and E. J. M. Colbert, IJMPD, 13, 1 (2004).
[60] J. D. E. Creighton, Phys. Rev. D, 60, 022001 (1999).
[61] B. Allen, ”GRASP documentation”, http://www.lsc-group.phys.uwm.edu(1997).
[62] R. Adhikari, “Sensitivity and noise analysis of 4 km laser interferometer gravitational wave antennae”, Ph.D. Thesis, Massachusetts Institute of Technology(2004).
[63] ”LIGO Algorithm Library”, LIGO Project, California Institute of Technology,
Pasadena, CA, http://www.lscgroup.phys.uwm.edu/daswg/projects/lal.html.
[64] Y. Tsunesada and the TAMA Collaboration, Class. Quant. Grav., 21, S703
(2004).
[65] G. L. Turin, IRE. Trans. Inform. Theory, 6, 311 (1960).
[66] K. S. Thorne and R. Blandford, Applications of Classical Physics, (unpublished) 2008.
[67] V. Kumar, A. Mahalanobis, and R. Juday, Correlation Pattern Recognition,
Cambridge University Press, New York (2005).
[68] B. J. Owen, Phys. Rev. D 53, 6749 (1996).
[69] http://www.cs.wisc.edu/condor.
[70] The LSC, ”Tuning Matched Filter Searches for Compact Binary Coalescence”,
LIGO-T070109-01-Z (2007).

176[71] R. Powell, http://www.atlasoftheuniverse.com.
[72] R. K. Kopparapu, C. R. Hanna, V. Kalogera, R. O’Shaughnessy, G. González,
P. R. Brady, and S. Fairhurst, Ap. J., 675, 1459 (2008).
[73] E. S. Phinney, Ap. J., 380, L17 (1991).
[74] R. O’Shaughnessy, C. Kim, T. Fragos, V. Kalogera, and K. Belczynski, Ap. J.,
633, 1076 (2005).
[75] S. T. Portegies Zwart, J. Makino, S. L. W. McMillan, and P. Hut, A&A, 348,
117 (1999).
[76] M. A. Gürkan, J. M. Fregeau, and F. A. Rasio, Ap. J., 640, L39 (2006).
[77] J. M. Fregeau, S. L. Larson, M. C. Miller, R. O’Shaughnessy, and F. A. Rasio,
Ap. J., 646, L135 (2006).
[78] S. Chatterji, ”The search for gravitational wave bursts in data from the second
LIGO science run”, PhD Thesis, Massachusetts Institute of Technology, Dept.
of Physics (2005).
[79] P. R. Brady, J. D. E. Creighton, and A. G. Wiseman, Class. Quantum Grav.
21, S1775 (2004).
[80] S. Fairhurst and P. R. Brady, arXiv:0707.2410v1 (2007).
[81] F. Pretorius, Phys. Rev. Lett., 95, 121101 (2005).
[82] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev.
Lett., 96, 111101 (2006).
[83] J. G. Baker, J. Centrella, D. Choi, M. Koppotz, and J. van Meter, Phys. Rev.
Lett., 96, 111102 (2006).
[84] F. Herrmann, I. Hinder, D. Shoemaker, and P. Laguna Class. Quant. Grav.,
24, S33 (2007).

177[85] M. Campanelli, C. O. Lousto, and Y. Zlochower, Phys. Rev. D, 74, 084023(2006).
[86] M. Campanelli, C. O. Lousto, Y. Zlochower, B. Krishnan, and D. Merritt Phys.
Rev. D, 75, 064030 (2007).
[87] M. Campanelli, C. O. Lousto, Y. Zlochower, and D. Merritt Ap. J., 659, L5(2007).
[88] J. G. Baker, W. D. Boggs, J. Centrella, B. J. Kelly, S. T. McWilliams, M. C.
Miller, and J. R. van Meter, Ap. J, 668, 1140 (2007).
[89] A. Buonanno, Y, Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. Mc Williams,
and J. R. van Meter, Phys. Rev. D, 76, 104049 (2007).
[90] P. Ajith et al., Class. Quant. Grav., 24, S689 (2007).
[91] B. Abbott et al., “Search of S3 LIGO data for gravitational wave signals
from spinning black hole and neutron star binary inspirals”, arXiv:0712.2050v2(2008).
[92] T. Baumgarte, P. Brady, J. D. E. Creighton, L. Lehner, F. Pretorius, and R.
DeVoe, arXiv:gr-qc/0612100v1 (2006)[93] S. Babak, H. Grote, M. Hewitson, H. Lück and K. A. Strain, Phys. Rev. D,
72, 022002 (2005).
[94] L. Goggin and A. Weinstein, ”Study of the Optical Parameters of the 40m
LIGO Prototype”, LIGO-T000122-00-R (2000).

178

Appendix A
Ringdown Search ConfigurationFile; ringdown pipeline configuration script.
;; Id;; this is the configuration file for the inspiral DAG generation
program; lalapps inspiral hipe that creates a condor DAG to run the ringdown; analysis pipeline.

[condor]; setup of condor universe and location of executables
universe = standard
datafind = /opt/lscsoft/glue/bin/LSCdataFind
tmpltbank = /bin/false
inspiral = /archive/home/lgoggin/bin/lalapps ringinca = /bin/false
thinca = /archive/home/lgoggin/bin/lalapps rinca
trigtotmplt = /bin/false

179sire = /bin/false
cohbank = /bin/false
chia = /bin/false
inspinj = /bin/false
frjoin = /archive/home/lgoggin/bin/lalapps frjoin
coire = /home/false[pipeline]; tagging information for the configure script
version = Id
cvs-tag = N ame; user-tag here can be overidden on the command line oflalapps inspiral pipe
user-tag =; data choice (playground only|exclude playground|all data)
playground-data-mask = all data

[input]; the segments file should be the output from segwizard with DQ
flags applied; if no segment file if specified, assumed no data from that IFO.
h1-segments = H1triplesegs.txt
h2-segments = H2triplesegs.txt
l1-segments = L1triplesegs.txt
g1-segments =ligo-channel = LSC-DARM ERR
geo-channel =geo-bank =geo-bank =ligo-type = RDS R L3

180
geo-type =; injection file (if blank then no injections)injection-file = HL-INJECTIONS 8-793130413-2548800.xml
num-slides =[calibration]; location of the calibration cache and the cache files
path = /archive/home/lgoggin/projects/ringdown/s4/calibration
L1 = l1 calibration.cache
H1 = h1 calibration.cache
H2 = h2 calibration.cache[datafind]; type of data to use
type = RDS R L3
url-type = file
match = localhost/archive[data]; data conditioning parameters common to all ifos
pad-data = 8;segment-length = 1048576;number-of-segments = 16
;sample-rate = 4096
sample-rate = 8192
block-duration = 2176
segment-duration = 256[ligo-data]; data conditioning parameters for ligo data

181
highpass-frequency = 40
cutoff-frequency = 45
dynamic-range-factor = 1.0e+20[geo-data]; data conditioning parameters for geo data[tmpltbank]; not used in ringdown pipeline[tmpltbank-1]; not used in ringdown pipeline[tmpltbank-2]; not used in ringdown pipeline[inspiral]; analysis parameters -- added to all ring jobs
bank-max-mismatch = 0.03
bank-min-frequency = 50
bank-max-frequency = 2000
bank-min-quality = 2.0
bank-max-quality = 20.0
bank-template-phase = 0
maximize-duration = 1
debug-level = 33
;approximant = ringdown;segment-overlap = 64

[no-veto-inspiral]

182; not used in ringdown pipeline[veto-inspiral]; not used in ringdown pipeline[h1-inspiral]; h1 specific inspiral paramters
threshold = 5.5[h2-inspiral]; h2 specific inspiral parameters
threshold = 5.5[l1-inspiral]; l1 specific inspiral parameters
threshold = 5.5[g1-inspiral][inspinj]; not used in ringdown pipeline[inca]; not used in ringdown pipeline[thinca]; common coincidence parameters -- added to all thinca jobs
debug-level = 33
multi-ifo-coinc =maximization-interval = 1

183parameter-test = ds sq
h1-time-accuracy = 2h2-time-accuracy = 2
l1-time-accuracy = 2;h1-freq-accuracy = 20
;h2-freq-accuracy = 20
;l1-freq-accuracy = 20
;h1-quality-accuracy = 3;h2-quality-accuracy = 3;l1-quality-accuracy = 3
h1-ds sq-accuracy = 0.05
h2-ds sq-accuracy = 0.05
l1-ds sq-accuracy = 0.05
do-veto =h1-veto-file = combinedVetoesH1-23.list
h2-veto-file = combinedVetoesH2-23.list
l1-veto-file = combinedVetoesL1-23.list[thinca-1]; not used in ringdown pipeline[thinca-2]; not used in ringdown pipeline[thinca-slide]; time slide parametersh1-slide = 0h2-slide = 10
l1-slide = 5

184[trigtotmplt]; not used in ringdown pipeline[sire]; not used in ringdown pipeline[sire-inj]; not used in ringdown pipeline[coire]; not used in ringdown pipeline[coire-inj]; not used in ringdown pipeline[cohbank]; not used in ringdown pipeline[coh-trig]; not used in ringdown pipeline[chia]; not used in ringdown pipeline

185

Appendix B
Segmentation of the Data
Here we give a concrete example of how the data is segmented (segmentation was
initially discussed in chapter 5).
In section 5.3 we stated that the minimum length of an analysis segment was 2176
s. In table 5.1 we listed the first five H1 science segments. The first science segment
was 1020 s in duration and the second was 240 s in duration; both of these are too
short and have to be discarded. Therefore, the first analysis segment comes fromscience segment number 3, beginning at GPS time 793166413 and ending at GPS
time 793170673, with a total duration of 4260 s, and from this we create the first
analysis segments.
As mentioned in section 5.4.1 an extra 8 s is read in before the start of eachanalysis segment for data conditioning, and then removed again before the filtering
stage. However, for the first analysis segment in a science segment, there is no previous
analysis segment to “borrow” the 8 s of data from and so the first 8 s of the science
segment are effectively lost. Thus, the first analysis segment begins at 793166413+8 =793166421 and ends 2176 s later at 793168597. An extra 8 s is also read in at the endof the segment, and because the analysis segment is more than 2176 s long, this canbe taken from the next analysis segment.
For filtering purposes, the analysis segment is divided into sixteen 256 s blocks,
which overlap the previous block by 64 s and the next block by 64 s. Then, for eachblock only triggers from the middle 128 s are recorded. As a consequence, no triggers
come from the first or last 64 s of the analysis segment. Once again, for this particular

186
analysis segment there is no segment preceding it and thus, any triggers from thistime are lost. This is not so with the last 64 s of this first analysis segment; these
are analysed in the succeeding analysis segment. Therefore, for this analysis segment,
the first trigger will occur after 793166421 + 64 = 793166485 and the last trigger willappear before 793166485 + 16 ∗ 128 = 793168533, 64 s before the end time of theanalysis segment.
The second segment begins 128 s before the first segment ended, at 793168597 −
128 = 793168469 and ends 2176 s later, at 793170645. Now this time the extra 8 s at
the start of the segment can be read in from the previous segment and therefore, nodata is lost. Once more the segment is divided into sixteen 256 s blocks each of which
overlaps the previous and subsequent blocks by 64 s, and the middle 128 s of each isanalysed. As we started the second analysis segment 128 s before the first segment
ended, the first trigger from the second segment can occur immediately after the lastpossible trigger in the first segment. Therefore, no data is lost here. The last trigger
in the second segment can occur at 793168469 + 2048 = 793170581.
Now the second segment ended 28 s before the end of the science segment; thatleaves 28 + 64 = 92 s of unanalysed data. We analyse this by creating a segmentthat starts (2176 + 8) s before the end of the science segment, at 793168489, but only
analyse data after the last possible trigger in the second segment, (i.e., at 793170581).
As was the case at the start of the first analysis segment, the last 8 + 64 = 72 s are
discarded, and so in this particular example, we get 20 s of data from the last analysis
segment in the science segment. The last analysis segment ends at 793170665.

187

Figure B.1: An illustration of how science segment number 3 (yellow box) is divided
into analysis segments (red, blue, and green dotted lines). The solid boxes denote thetimes that are analysed in each analysis segment.

188

Appendix C
Loudest Ten Events in Double
Coincidence
Table C.1: The ten loudest candidate events from H1L1 doubles in triple time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 794949585.625 107.56 3.46
186.60 9.42 21.05
794949585.614 103.84 10.04
149.07 12.50
2 793829533.702
81.39 10.04
303.72 8.76 19.72
793829533.693
79.43 10.04
84.72 11.41
3 794291462.267
73.83 10.04
313.25 8.41 19.03
794291462.254
81.99 5.91
70.34 11.32
4 794989787.461
87.56 10.04
288.37 9.48 18.93
794989787.458
81.99 5.91
190.72 9.45
5 794317560.631
65.36 10.04
338.90 7.86 17.93
794317560.629
68.63 17.01
19.13 31.51
6 794943595.760 153.33 10.04
151.11 8.87 17.71
794943595.771 153.65 17.01
118.65 8.84
7 794006110.807 132.55 3.46
216.83 7.23 16.67
794006110.795 127.60 2.00
46.28 10.38
8 793894630.682
63.86 17.01
318.93 7.75 16.03
793894630.673
64.79 17.01
78.58 8.29
9 795118144.597
85.44 5.91
272.38 8.31 15.94
795118144.604
79.43 10.04
276.42 7.63
10 794783103.777
50.00 5.91
323.03 6.87 15.93
794783103.776
50.00 5.91
105.96 15.78

189

Table C.2: The ten loudest candidate events from H1H2 doubles in triple time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 793253792.205
63.20 2.00
1.55 1336.49 63.10
793253792.207
50.00 2.00
44.09
30.45
54.51 17.01
164.21
12.98 21.26
2 794654729.865
794654729.864
54.51 17.01
84.23
9.53
0.91
6.72 15.35
3 795398069.574 1803.63 5.91
795398069.570 1700.01 17.01
0.52
8.63
4 793728359.581
52.21 17.01
26.55
45.28 15.14
793728359.584
54.51 17.01
135.21
6.47
5 793726436.436 1559.35 17.01
0.87
7.78 14.81
793726436.440 1493.45 17.01
0.96
7.03
6 795400260.902 160.99 10.04
111.059.67 14.64
795400260.899 151.46 17.01
135.64
6.22
7 795273330.160
82.75 17.01
252.2911.09 14.51
795273330.158
83.95 17.01
214.01
6.16
8 795441060.564
70.63 17.01
183.2410.37 14.42
795441060.560
68.63 17.01
162.54
6.11
9 794115186.367 1628.16 17.01
1.12
6.62 14.35
794115186.370 1675.71 17.01
0.70
7.73
1.04
7.12 14.32
10 795437362.744 1589.49 10.04
795437362.747 1493.45 17.01
0.84
7.20

190

Table C.3: The ten loudest candidate events from H2L1 doubles in triple time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 794966223.382 1795.38 10.04
0.45 10.65 20.27
794966223.393 1907.47 17.01
0.50 9.62
0.60 9.48 17.32
2 794490884.837 1660.93 5.91
794490884.844 1700.01 17.01
0.40 7.83
0.36 9.17 16.38
3 793181575.946 1991.64 17.01
793181575.951 1885.01 10.04
0.47 7.20
4 794934058.049
67.65 17.01
142.63 8.31 16.18
794934058.036
64.79 17.01
147.70 7.88
5 795117184.353 1088.11 17.01
2.15 7.16 16.17
795117184.352 1042.12 17.01
2.08 9.01
6 794477109.831 902.42 17.01
3.53 7.03 16.10
794477109.828 902.42 17.01
2.52 9.07
7 794925830.874 915.50 17.01
3.50 6.85 15.91
794925830.862 915.50 17.01
2.63 9.77
8 793676381.075 1593.87 5.91
0.78 7.87 15.89
793676381.068 1551.24 10.04
0.48 8.01
9 794484924.385 1203.45 17.01
1.52 7.75 15.86
794484924.373 1152.59 17.01
1.21 8.12
1.76 7.18 15.71
10 794939243.941 1148.01 3.46
794939243.940 931.57 3.46
2.54 8.53

191

Table C.4: The ten loudest candidate events from H1L1 doubles in double time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 793258551.938
66.98 10.04
370.92 7.37 16.93
793258551.941
70.63 17.01
176.60 9.82
2 794506571.018
50.00 5.91
295.90 7.21 15.04
794506571.007
51.46 17.01
212.73 7.84
3 794368837.863 181.31 2.00
112.87 5.73 13.37
794368837.859 257.63 2.00
12.00 7.65
88.24 24.44 13.36
4 794195114.960 125.61 17.01
794195114.972 118.81 5.91
139.28 5.58
60.76 10.04
365.64 5.91 13.20
5 794361114.970
794361114.960
65.36 10.04
94.98 7.29
59.43 17.01
287.89 5.81 13.03
6 794958144.714
794958144.724
56.92 17.01
172.74 7.23
7 795477196.874
79.87 2.00
255.50 6.95 12.70
795477196.885
92.78 5.91
362.71 5.75
53.73 17.01
327.60 6.15 12.44
8 795601342.002
795601342.000
54.51 17.01
225.26 6.29
9 794368869.630 1885.01 10.04
0.93 5.82 12.36
794368869.629 1885.01 10.04
0.14 6.55
10 794362732.279 1130.17 10.04
2.95 5.75 12.35
794362732.280 1012.90 5.91
0.83 6.61

192

Table C.5: The ten loudest candidate events from H1H2 doubles in double time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 793589170.965
71.65 17.01
223.57 10.80 17.10
793589170.967
72.05 10.04
161.15 7.45
2 795114450.689
78.68 5.91
213.69 11.61 16.88
795114450.691
75.65 10.04
173.89 7.34
1.95 8.23 16.03
3 794692945.319 1130.17 10.04
794692945.320 1169.30 17.01
1.40 7.80
4 795385587.580 648.09 17.01
6.64 8.53 15.58
795385587.576 614.69 10.04
7.55 7.04
5 795382679.975 568.84 5.91
9.57 7.39 14.10
795382679.972 585.98 17.01
9.28 6.71
6 793694651.919 1991.64 17.01
0.74 6.07 13.99
793694651.916 1991.64 17.01
0.43 7.93
2.24 7.49 13.54
7 795384035.598 1072.56 17.01
795384035.599 1119.89 17.01
2.14 6.05
8 795382177.870 1136.12 17.01
2.21 6.59 13.51
795382177.868 1186.25 17.01
1.48 6.92
69.53 5.91
350.69 7.70 13.36
9 794725076.726
794725076.726
63.20 2.00
185.92 5.66
52.50 10.04
252.45 9.12 13.32
10 794158080.377
794158080.380
52.21 17.01
153.64 5.56

193

Table C.6: The ten loudest candidate events from H2L1 doubles in double time#t (s) f (Hz)Q def f (Mpc)ρρDS
1 794432410.606 1724.65 17.01
0.50 9.16 16.15
794432410.597 1651.77 17.01
0.18 6.99
2 793572623.619 1724.65 17.01
0.61 8.03 14.72
793572623.628 1675.71 17.01
0.49 6.68
3 794448103.696 1072.56 17.01
1.64 9.27 14.64
794448103.699 1055.52 5.91
0.77 6.22
0.99 8.63 14.26
4 793242141.579 1350.32 17.01
793242141.586 1369.89 17.01
1.07 6.03
0.87 9.20 14.20
5 795389328.279 1331.02 17.01
795389328.291 1369.89 17.01
1.67 6.00
3.20 5.86 13.93
6 793674572.094 972.01 5.91
793674572.087 858.96 5.91
2.66 8.11
7 795069044.029 1493.45 17.01
0.87 7.52 13.71
795069044.020 1589.49 10.04
1.13 6.20
0.70 8.31 13.64
8 795407874.988 1515.11 17.01
795407874.994 1472.11 17.01
1.52 5.72
9 793735902.987 1931.50 10.04
0.51 7.45 13.61
793735902.984 1879.52 5.91
0.23 6.16
10 794877527.436 667.02 17.01
6.15 8.54 13.58
794877527.435 657.49 17.01
10.08 5.69

194

Appendix D
Projects Undertaken at the 40 m
Interferometer
Prior to the construction of the LIGO 4 and 2 km interferometers the 40 m LIGO
prototype, an laser interferometer with 40 m long arms located on the Caltech campus
served as a test-bed for LIGO technologies. Currently it is used for testing Advanced
LIGO and future generation technologies.
I was part of the 40 m team during this upgrade period. My first project involved
modeling the interferometer optics [94]. For a given laser wavelength and length
constraints I used Matlab to trace the path of the beam through the interferometerand made recommendations for the radii of curvature and placement of various invacuum optics. These included the mode cleaner, a triangular configuration of mirrors
designed to isolate a single mode of light to send into the interferometer, and modematching telescopes, a pair of lenses to match the light from the mode cleaner to thebeam-splitter.
Once the new laser and optics were installed we found be useful to see the beam
on various optics throughout the interferometer. Cameras were installed and connected to an electronics rack linked to monitors in the control room. Using EPICS(Experimental Physics and Industrial Control System) I created an interface for theuser in the control room to select a particular camera and a monitor in which todisplay the image. A similar control system was employed to activate or deactivate
lamps illuminating the optics chambers.