Open Issues in the Search for Gravitational Wave
Transients
byLindy L. Blackburn
A.B. Physics with Honors and Mathematics, University of Chicago(2003)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2010
c Lindy L. Blackburn, MMX. All rights reserved.
�
The author hereby grants to MIT permission to reproduce anddistribute publicly paper and electronic copies of this thesis documentin whole or in part.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 31, 2010
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Erotokritos Katsavounidis
Associate Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Krishna Rajagopal
Professor of Physics
Associate Department Head for Education

2

Open Issues in the Search for Gravitational Wave Transients
byLindy L. Blackburn
Submitted to the Department of Physics
on August 31, 2010, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy in Physics

Abstract
The LIGO-Virgo network of kilometer-scale laser interferometric gravitational-wave
detectors reached a major milestone with the successful operation of LIGO’s fifth (S5)and Virgo’s first (VSR1) science runs during 2005–2007. This thesis presents several
issues related to gravitational-wave transient detection from the perspective of thejoint all-sky, un-triggered burst search over S5/VSR1 data.
Existing searches for gravitational-wave bursts must deal with the presence of
non-Gaussian noise transients which populate the data over the majority of sensitive
signal space. These events may be confused with true signals, and are the current
limiting factor in search sensitivity and detection confidence for any real event. The
first part of this thesis focuses on the development of tools to identify, monitor andcharacterize these instrumental disturbances in LIGO and Virgo data. An automatedprocedure is developed and applied to the S5/VSR1 search in order to safely remove
noise transients from the analysis without sacrificing sensitivity by making use of thewealth of auxiliary information recorded by the detectors.
The second part of this thesis focuses on the interpretation of outlier events in thecontext of a non-Gaussian, non-stationary background. An extensive follow-up procedure for candidate gravitational-wave events is developed and applied to a single burst
outlier from the S5/VSR1 search, later revealed to be a blind simulation injected into
the instruments. While the follow-up procedure correctly finds no reason to reject
the candidate as a possible gravitational wave, it highlights the diﬃculty in making
a confident detection for signals with similar waveform morphology to common instrumental disturbances. The follow-up also deals with the problem of objectively
defining the significance of a single outlier event in the context of many semi-disjoint
individual searches. To address this, a likelihood-ratio based unified ranking is developed and tested against the original procedures of the S5/VSR1 burst search. The
new ranking shows a factor of four improvement in the statistical significance of theoutlier event, and a 12% reduction using fixed thresholds and 38% reduction using aloudest event statistic for a rate upper limit on a mock signal population.
Thesis Supervisor: Erotokritos Katsavounidis
Title: Associate Professor of Physics
3

4

Acknowledgments
I’d like to begin by thanking my grandparents—I can’t imagine growing up in a moreloving and welcoming family. My entire approach to life I owe to my father—thankyou for teaching me to think for myself. Thank you Mom for your irreplaceable loveand for teaching me to appreciate good food. To Lawrence Lee for being such anincredible family friend. The coolest things were always at your place, and thank you
for showing us how to live and love life. I thought maybe if I became a physicist I could
be more like you. Jim Riley taught a truly amazing 5–6th grade class at Hawthorne
Elementary. Thank you for showing us about the wonder in the world, and the trips
to Balboa Park, Mission Bay, the symphony, circuits, magic, combinatorics, puzzles,
art, ambigrams, square dancing, ultimate aerobie. . . And finally a late thank you to
Mr. Feynman and Mr. Sagan for leaving your written legacy. No doubt you havetransformed millions of people, including me.
I’ve met genuinely great people during my graduate student career at LIGO. I’dlike to start with my advisor Erik Katsavounidis whose better judgment I benefited
from innumerable times. I’d also like to thank my excellent committee members Scott
Hughes and Nergis Mavalvala. Scott, thanks for teaching me G.R., but also, thanks
for happily accommodating my rushed defense just before your wedding! When I
started at MIT-LIGO, I shared an oﬃce with Shourov Chatterji. Thanks for being
such a great example (aside from graduating in a timely manner!), but really foreverything: attention to detail, an unrelenting drive to figure things out and do themright, amazing patience, Matlab wizardry. I’m not at all surprised to still see influenceof your work and style all over the collaboration.
To Peter Shawhan—thanks for always being the first to understand and appreciate
my garbled presentations to the burst group. And to Laura Cadonati, thank you foryears of support and guidance. It was a pleasure every time I had a chance to workwith you. Jolien Creighton—I know you mostly through emails not addressed to me,
but there is always great stuﬀ there and a wealth of good ideas, a few of which madetheir way into this work. Peter Saulson and Alan Weinstein—I always looked forward5

to listening to all the valuable things either of you had to say. Josh Smith, Nicolas
Leroy—it was great working with you both. Brennan Hughey, John Zweizig, Igor
Yakushin—much of this thesis would have been impossible without great help fromall of you, thank you! To my lab-mates Jared Markowitz and Leo Stein, it was fun
brainstorming through problems and random stuﬀ, best of luck to you both.
Thank you David Deveau for bringing me back to the piano in graduate school.
To you and Marcus Thompson, I’ll never forget all the great times in AMP and CMS.
Thank you for making my life at MIT that much richer. To Rosa, and Paymon—yourdecade of friendship is invaluable to me. And to Amanda, thank you for your loveand patience in seeing this through.
The authors gratefully acknowledge the support of the United States National
Science Foundation for the construction and operation of the LIGO Laboratory, the
Science and Technology Facilities Council of the United Kingdom, the Max-PlanckSociety, and the State of Niedersachsen/Germany for support of the construction andoperation of the GEO600 detector, and the Italian Istituto Nazionale di Fisica Nucleare and the French Centre National de la Recherche Scientifique for the construction
and operation of the Virgo detector. The authors also gratefully acknowledge thesupport of the research by these agencies and by the Australian Research Council,
the Council of Scientific and Industrial Research of India, the Istituto Nazionale di
Fisica Nucleare of Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria d’Economia Hisenda i Innovació of the Govern de les Illes Balears, the Foundation
for Fundamental Research on Matter supported by the Netherlands Organisation for
Scientific Research, the Polish Ministry of Science and Higher Education, the FOCUS Programme of Foundation for Polish Science, the Royal Society, the Scottish
Funding Council, the Scottish Universities Physics Alliance, The National Aeronautics and Space Administration, the Carnegie Trust, the Leverhulme Trust, the David
and Lucile Packard Foundation, the Research Corporation, and the Alfred P. Sloan
Foundation.
Portions of this work do not reflect the scientific opinion of the LSC and were notreviewed by the collaboration. This document has LIGO number LIGO-P1000093.
6

Contents
1 Introduction

15

2 Gravitational radiation

19

2.1

Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2

Weak field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2.1

Transverse traceless gauge . . . . . . . . . . . . . . . . . . . .

24

2.2.2

Eﬀect of gravitational waves on matter . . . . . . . . . . . . .

25

2.2.3

Stress-energy of gravitational waves . . . . . . . . . . . . . . .

26

2.2.4

Generation of gravitational waves . . . . . . . . . . . . . . . .

28

Sources of gravitational radiation . . . . . . . . . . . . . . . . . . . .

30

2.3.1

Compact binary coalescence . . . . . . . . . . . . . . . . . . .

30

2.3.2

Gravitational core collapse . . . . . . . . . . . . . . . . . . . .

35

2.3.3

Rotating neutron stars . . . . . . . . . . . . . . . . . . . . . .

37

2.3.4

Stochastic background . . . . . . . . . . . . . . . . . . . . . .

40

2.3

3 Detecting gravitational waves

45

3.1

Resonant mass detectors . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Interferometric detectors . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.3

Laser Interferometer Gravitational-wave Observatory (LIGO) . . . . .

50

3.3.1

Fabry-Perot cavities . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.2

Length sensing and control . . . . . . . . . . . . . . . . . . . .

53

3.3.3

Noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.4

Advanced LIGO . . . . . . . . . . . . . . . . . . . . . . . . . .

56

7

4 Gravitational wave bursts

59

4.1

Burst parametrization . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.2

Detecting bursts in stationary noise . . . . . . . . . . . . . . . . . . .

61

4.2.1

Power spectral density . . . . . . . . . . . . . . . . . . . . . .

61

4.2.2

Matched filter signal-to-noise ratio . . . . . . . . . . . . . . .

62

4.2.3

Time-frequency decomposition . . . . . . . . . . . . . . . . . .

66

Multiple detector observation . . . . . . . . . . . . . . . . . . . . . .

70

4.3.1

Coincidence in observed parameters . . . . . . . . . . . . . . .

70

4.3.2

Coherent observation . . . . . . . . . . . . . . . . . . . . . . .

71

4.3.3

Time-shift method of background estimation . . . . . . . . . .

73

4.3

5 Non-Gaussian noise transients

75

5.1

Transient population in LIGO data . . . . . . . . . . . . . . . . . . .

76

5.2

Examples of transients . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.3

kleineWelle: wavelet-based identification . . . . . . . . . . . . . . . .

82

5.3.1

Wavelet decomposition . . . . . . . . . . . . . . . . . . . . . .

83

5.3.2

Data conditioning . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.3.3

Event generation . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.3.4

Parameter estimation . . . . . . . . . . . . . . . . . . . . . . .

89

5.3.5

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .

90

6 Detector characterization, data quality, and vetoes

93

6.1

On-line detector characterization . . . . . . . . . . . . . . . . . . . .

93

6.2

Data quality flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

6.3

Event-by-event vetoes for burst background . . . . . . . . . . . . . .

96

6.3.1

Veto eﬀectiveness metrics

99

6.3.2

Veto safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.3

Selection of veto conditions . . . . . . . . . . . . . . . . . . . 101. . . . . . . . . . . . . . . . . . . .

7 Candidate event follow-up
7.1

107

Detection checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8

7.2

Blind injections during S5 . . . . . . . . . . . . . . . . . . . . . . . . 110

7.3

Follow-up of S5/VSR1 burst outlier . . . . . . . . . . . . . . . . . . . 111
7.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3.2

The event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3.3

State of the instruments . . . . . . . . . . . . . . . . . . . . . 115

7.3.4

All-sky search and background estimates . . . . . . . . . . . . 122

7.3.5

Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . 127

7.3.6

Astrophysical analysis . . . . . . . . . . . . . . . . . . . . . . 130

7.3.7

Other events in S5 . . . . . . . . . . . . . . . . . . . . . . . . 135

7.3.8

Conclusion of the follow-up procedure . . . . . . . . . . . . . . 137

8 Unified ranking for gravitational-wave events

143

8.1

Trial factor issues in S5/VSR1 . . . . . . . . . . . . . . . . . . . . . . 145

8.2

Inverse false-alarm rate ranking . . . . . . . . . . . . . . . . . . . . . 148

8.3

Likelihood-ratio ranking . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.3.1

Signal population . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.3.2

Background population . . . . . . . . . . . . . . . . . . . . . . 153

8.3.3

Analytic fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.3.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9 Conclusions

165

A Statistical significance of Poisson and χ2 processes

169

A.1 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.2 Gaussian approximation . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.3 Calculating Poisson significance . . . . . . . . . . . . . . . . . . . . . 171
A.3.1 Gamma function approximations . . . . . . . . . . . . . . . . 171
A.3.2 Extreme tail of the Poisson CDF . . . . . . . . . . . . . . . . 173
A.4 χ2 distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B KleineWelle channel configuration

175

B.1 S5 channels and frequency range . . . . . . . . . . . . . . . . . . . . . 175
9B.2 S6 channels and frequency range . . . . . . . . . . . . . . . . . . . . . 178
C Auxiliary channel veto statistics

183

C.0.1 Veto performance on H1 transients . . . . . . . . . . . . . . . 183
C.0.2 Veto performance on H2 transients . . . . . . . . . . . . . . . 185
C.0.3 Veto performance on L1 transients . . . . . . . . . . . . . . . 187
D Burst detection checklist

189

10

List of Figures
2-1 The eﬀect of a gravitational wave on a ring of particles . . . . . . . .

27

2-2 Binary black hole merger waveform . . . . . . . . . . . . . . . . . . .

34

2-3 Gravitational waveform from a stellar collapse . . . . . . . . . . . . .

37

2-4 Sources and limits of a stochastic gravitational wave background . . .

43

3-1 Current resonant bar detector strain sensitivities . . . . . . . . . . . .

47

3-2 A two mirror resonant cavity . . . . . . . . . . . . . . . . . . . . . . .

51

3-3 LIGO optical layout . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3-4 LIGO S5 sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5-1 Transient population rate vs SNR during S5 . . . . . . . . . . . . . .

77

5-2 Transient population frequency and Q during S5 . . . . . . . . . . . .

79

5-3 Transient rate vs day for S5 . . . . . . . . . . . . . . . . . . . . . . .

80

5-4 An event caused by magnetic disturbance at LHO . . . . . . . . . . .

81

5-5 Discrete wavelet decomposition tree using conjugate mirror filters . .

84

5-6 Example of LPEF+DWT applied to a test signal in noise . . . . . . .

86

5-7 Definition of distance in the dyadic wavelet tiling . . . . . . . . . . .

88

6-1 S5 online kleineWelle trends . . . . . . . . . . . . . . . . . . . . . . .

94

6-2 Application of event-by-event veto window . . . . . . . . . . . . . . .

96

6-3 Hierarchical veto tuning procedure . . . . . . . . . . . . . . . . . . . 103
6-4 Auxiliary channel veto performance on burst background . . . . . . . 105
7-1 Filtered time series of LIGO data at the time of the blind injection . 113
7-2 Spectrogram of LIGO calibrated strain data . . . . . . . . . . . . . . 114
11

7-3 H1H2 consistency of the blind burst injection . . . . . . . . . . . . . 115
7-4 Q Pipeline correlated vs coherent energy . . . . . . . . . . . . . . . . 116
7-5 Q Pipeline single-interferometer triggers in ±8 s of the blind injection

118

7-6 Q Pipeline rate vs. time . . . . . . . . . . . . . . . . . . . . . . . . . 119
7-7 Q Pipeline SNR vs. time . . . . . . . . . . . . . . . . . . . . . . . . . 120
7-8 Q Pipeline frequency vs. time . . . . . . . . . . . . . . . . . . . . . . 122
7-9 Results from the kleineWelle+CorrPower online analysis . . . . . . . 123
7-10 Coherent Waveburst S5 H1H2L1 background distribution . . . . . . . 124
7-11 Coherent Waveburst S5 H1H2L1 background properties . . . . . . . . 125
7-12 Results from the Q/Ω Pipeline analysis . . . . . . . . . . . . . . . . . 127
7-13 Coherent Waveburst sky statistic for H1H2L1 and H1H2L1V1 networks 128
7-14 Coherent Waveburst best-fit strain waveform . . . . . . . . . . . . . . 129
7-15 Coherent best fit waveform in data . . . . . . . . . . . . . . . . . . . 130
7-16 Ω Pipeline Bayesian statistic sky map . . . . . . . . . . . . . . . . . . 131
7-17 Hilbert-Huang transform of candidate event . . . . . . . . . . . . . . 132
7-18 Equal mass black hole merger waveform . . . . . . . . . . . . . . . . 133
7-19 Time series of equal mass black hole merger simulation in LIGO noise 133
7-20 Spectrogram of equal mass black hole merger simulation in LIGO noise 134
7-21 L10 density up to 100 Mpc . . . . . . . . . . . . . . . . . . . . . . . . 136
7-22 Sine-Gaussian f =100 Hz Q=9 hardware injection . . . . . . . . . . . 137
7-23 Gaussian 3 ms hardware injection . . . . . . . . . . . . . . . . . . . . 137
7-24 Comparison of injected and reconstructed waveforms . . . . . . . . . 140
7-25 Comparison of injected and reconstructed sky location . . . . . . . . 141
8-1 Coherent Waveburst H1H2L1V1 likelihood-ratio maps . . . . . . . . . 160
8-2 Coherent Waveburst H1H2L1 likelihood-ratio maps . . . . . . . . . . 161
8-3 Q/Ω Pipeline H1H2L1 likelihood-ratio maps . . . . . . . . . . . . . . 162
8-4 Q/Ω Pipeline H1H2V1 likelihood-ratio maps . . . . . . . . . . . . . . 163
8-5 Q/Ω Pipeline L1V1 likelihood-ratio maps . . . . . . . . . . . . . . . . 164

12

List of Tables
2.1

Predicted rates for binary coalescence . . . . . . . . . . . . . . . . . .

31

6.1

Data quality categories for S5 transient searches . . . . . . . . . . . .

97

7.1

Basic information on burst event 090722 . . . . . . . . . . . . . . . . 112

7.2

State of the LIGO, GEO600, and Virgo instruments . . . . . . . . . . 116

7.3

Auxiliary channel triggers detected by kleineWelle . . . . . . . . . . . 121

7.4

Coherent Waveburst single-interferometer parameters for candidate event128

7.5

Coherent Waveburst parameters for reconstructed gravitational wave

7.6

Electromagnetic events from the GCN catalog around 2007-09-22 . . 131

8.1

Configurations and live-times for S5/VSR1 burst searches . . . . . . . 147

8.2

Comparison of hand-chosen and likelihood-ratio thresholds . . . . . . 157

13

130

14

Chapter 1
Introduction
One of the most fascinating predictions of Einstein’s general theory of relativity is
the existence of gravitational radiation—ripples in the geometry of space-time which
propagate as waves at the speed of light. Due to the weakness of gravitational interaction, it takes an immense amount of matter accelerating at relativistic speeds
to make an appreciable amount of gravitational radiation. While any terrestrial attempt to generate gravitational waves would result in waves much too weak to be
detectable, suﬃciently extreme conditions do occur in the universe to produce gravitational waves at the Earth which approach the limit of current detector sensitivities,
such as the coalescence of two neutron stars or black holes.
The experimental quest to detect astrophysical sources of gravitational radiation
began with the work of Joseph Weber in the 1960s at the University of Maryland [1].
Weber’s idea was to use large suspended resonant metal bars whose modes would
be excited by a passing burst of gravitational radiation. Today, the most promisingdetectors are broadband laser interferometers which began development in the late1960s with the independent work of Weiss [2], Moss, Miller and Forward [3]. The
largest and most sensitive interferometric detectors currently in operation include thetwo US Laser Interferometer Gravitational-wave Observatory (LIGO) 4 km-baseline
instruments at Hanford, WA and Livingston, LA [4], and the 3 km French-ItalianVirgo detector near Pisa, Italy [5].
Although gravitational waves have yet to be directly observed, strong evidence of15their existence comes from the measured orbital decay of the binary pulsar system
PSR 1316+16, which was discovered by Russell Hulse and Joseph Taylor in 1974 [6].
Over decades of observation, the measured energy loss of the double neutron starorbit has proved very consistent with the amount of energy and angular momentum
radiated away in gravitational waves predicted by general relativity [7]. The LIGO
and Virgo instruments are designed to capture, among other things, the signal produced during the final coalescence and merger of such binary systems. Gravitational
waves can provide a unique undisturbed signature of the inner dynamics of systems
which are either dark in or opaque to photons and neutrinos, providing an entirely
new window through which to observe the universe.
The material in this thesis deals primarily with the search for gravitational-wave
transients in data from LIGO’s fifth science run (S5) which took place between November 2005 and October 2007. During this time, LIGO collected one full year of datawith all three LIGO instruments (including the currently decommissioned 2 km instrument at Hanford) operating at design sensitivity. The latter half of S5 overlappedwith Virgo’s first science run (VSR1), which provided the opportunity for a coordinated joint search.
An introduction to gravitational waves arising from Einstein’s theory of generalrelativity is presented in chapter 2, accompanied by an overview of astrophysics
sources relevant to ground-based detection. A brief description of ground-based detectors, including resonant-mass detectors and interferometric detectors, is presented
in chapter 3. Additional detail is given to the LIGO instruments during S5, alongwith current and future upgrades. The basic techniques for the detection of burst
sources (short-duration gravitational-wave transients) in detector noise are presented
in chapter 4. This includes burst parametrization, the concept of matched-filter
signal-to-noise, time-frequency decompositions and coherent network analysis, andthe time-shift method of background estimation.
In chapter 5, we describe the character of the noise transients which populate
the gravitational-wave data from the LIGO instruments. Accidental coincidencesof noise transients across multiple detectors dominate the background in existing16

gravitational-wave transient searches. These noise events are the limiting factor
in search sensitivity as well as the ability to detect a real event with confidence.
An eﬃcient wavelet-based method called kleineWelle is developed to identify andparametrize time-series transients in real-time across the hundreds of test points(channels) recorded across the sites. In chapter 6, a hierarchical method is describedwhich reduces the large amount of auxiliary (non-gravitational) information into thatwhich is most eﬀective at identifying and removing periods of instrumental disturbances. These techniques are ultimately used to reduce background in the S5/VSR1
all-sky un-triggered burst search [8].
The S5/VSR1 burst search resulted in a single outlier event which had an associated false-alarm rate from background of once per 43 years. As a potential detection
candidate, the outlier prompted an extensive follow-up, described in detail in chapter 7. The outlier event was ultimately revealed as a blind injection of a simulated
ad-hoc burst gravitational waveform. As such, it provided a valuable test of the detection readiness of our search procedures, as well as valuable experience in candidate
event follow-up and interpretation for the search teams.
The outlier event follow-up also highlighted the need for an objective procedure
to evaluate the global significance of an event (unlikelihood to have arisen from background) in the context of the many semi-disjoint sub-searches which together make up
a burst search covering multiple networks, frequencies, and analysis methods (together
we define these choices as the search configuration). This is achieved most cleanly
through a global ranking of all events. When evaluated for a single population ofsimulated signals, the likelihood ratio provides a natural ranking which is directlycomparable across sub-searches. This unified ranking is applied to events from theS5/VSR1 burst search in chapter 8 using a homogeneous population of ad-hoc burst
signals and time-shift sample background, and various metrics of performance areused to compare the new ranking with the hand-tuned fixed thresholds used in theoriginal search.

17

18

Chapter 2
Gravitational radiation
2.1

Relativity

The principle of relativity is the idea that the laws of physics do not depend on
the inertial reference frame of any particular observer, and so no experiment can beconstructed to measure ones velocity with respect to a fundamental notion of spaceat rest. The additional empirical requirement that light is observed to travel at a
fixed speed regardless of reference frame gives us special relativity as described by
Einstein in 1905. In special relativity, spacetime is a four dimensional manifold witha Minkowski metric (in Cartesian coordinates t, x, y, z with natural units c = 1),ηµν = 


−1 0 0 0
0 1 0 0 ,
0 0 1 0 
0 0 0 1

(2.1)

which defines the squared distance between two neighboring points, or spacetime
interval:
ds2 = ηµν dxµ dxν
= −dt2 + dx2 + dy 2 + dz 2 .
19

(2.2)
(2.3)

In this and following expressions, we adopt tensor notation used by MTW [9] andmany others where Greek indices (µ, ν, . . . ) represent space-time coordinates while
Latin indices (i, j, . . . ) represent spatial coordinates only. We will also use shorthandfor partial derivative operators ∂µ = ∂/∂x µ . Repeated indices in an expression indicate a sum over all possible values. Indices which appear on each side of an equationmeans the equation holds for all possible values. ds2 is invariant under the specialgroup of Lorentz transformations which relate coordinates among diﬀerent inertial
observers. For a clock moving along a particular spacetime path, d�x2 = 0 in the
instantaneous moving inertial frame of the clock, so that the that the proper time√dτ = −ds2 elapsed for the world-line of a particle is also an invariant quantity andcan be used to parametrize its path.
To include the eﬀects of gravity into relativistic theory, Einstein made the conceptual leap to describe spacetime as fundamentally curved. In general relativity, gravity
is not an invisible and instantaneous force between two masses, but a manifestation of
the behavior of locally straight lines (geodesics) in a curved space. This is suggestedby the observation that all objects undergo the same acceleration in a gravitational
field regardless of mass or composition. At the same time, the local curvature itself
is governed by mass-energy density.
The geometry of curved space is characterized by a symmetric metric gµν which
replaces the flat Minkowski metric ηµν of special relativity. The metric cannot be
arbitrary–the geometry must reduce to the flat spacetime of special relativity at a
suﬃciently small scale. In a small enough laboratory at free-fall, we should not beable to observe the eﬀects of gravity at finite precision.
The essential nature of curvature is contained in the Riemann tensor,
Rρ σµν = ∂µ Γρ σν − ∂ν Γρ σµ + Γρ λµ Γλσν − Γρ λν Γλσµ ,(2.4)

written here in terms of the aﬃne connection coeﬃcients which relate partial deriva20

tives to covariant derivatives suitable for curved space,1Γρ µν = g ρλ (∂µ gλν + ∂ν gλµ − ∂λ gµν ) .
2

(2.5)

The Riemann curvature tensor records the way vectors change slightly in curved
space when parallel transported about infinitesimal loops of arbitrary orientation.
Parallel transport means the vector undergoes no first order rotation in its locally flatneighborhood as it moves along a path. The metric with raised indices is the inverseof the original metric, so that g µλ gλν = δ µ ν where repeated indices indicate a sum overcoordinates. The metric can also be used to raise or lower indices: Vµ = gµν V ν , whichallows us to take inner products: Aµ Bµ = gµν Aµ B µ and contractions: Aµ µ = gµν Aµν .
In a local inertial frame for some point in spacetime, the metric gµν reduces tothe Minkowski metric at that point and its first derivatives vanish, setting the aﬃneconnection to zero when evaluated at the point as well. This shows us that spacetime
curvature and the Riemann tensor are fundamentally characterized by second derivatives of the metric. In general relativity there are 20 second derivatives of the metric
which cannot be removed by choice of coordinates, and there are 20 independent
components of the Riemann tensor as well.
There is one independent contraction of the Riemann tensor which gives the Ricci
tensorRµν = Rλ µλν(2.6)

R = Rλ λ .

(2.7)

and Ricci scalar

We can now write down the Einstein equations which relate the dynamics ofspacetime curvature to the local stress-energy tensor T µν ,18πG
Rµν − g µν R = 4 T µν .
2
c

(2.8)

The trace-reversed form of the Ricci tensor on the left hand side of the equation is
known as the Einstein tensor, Gµν = Rµν − 12 g µν R, which is the unique second-rank21

tensor built from linear contributions of the Riemann tensor that allows Equation 2.8
to satisfy conservation of energy-momentum (∇µ T µν = 0). The constant factor on
the right hand side gives the correct Newtonian limit (subsequent equations will use
natural units where c = G = 1 unless otherwise stated). The equations provide
ten independent constraints on the initial conditions and dynamical evolution of themetric. If an initial metric is chosen which satisfies Einstein’s equations, their uniqueevolution (up to a choice of coordinates) will continue to satisfy the equations for all
time, thus determining all geometric properties of the spacetime.

2.2

Weak field limit

In the limit of weak curvature, the metric can be considered approximately Minkowskian
plus a small perturbation,gµν � ηµν + hµν ,�hµν � � 1.

(2.9)

In this limit, we ignore all contributions to curvature that are more than first orderin hµν . The connection coeﬃcients become1Γρ µν = η ρλ (∂µ hλν + ∂ν hλµ − ∂λ hµν )2(2.10)and the Riemann tensor becomesRρ σµν = ∂µ Γρσν − ∂µ Γρσµ(2.11)

as theΓ 2 terms drop out. The Riemann tensor depends entirely on second derivativesof hµν giving rise to the interpretation that hµν is a a small curvature field resting on
top of flat background spacetime.
In addition to the usual Lorentz transformations, we are free to choose coordinates
22which represent an infinitesimal gauge transformation,�xµ = x µ + ξ µ(2.12)hµ� ν � = hµν − ∂µ ξν − ∂ν ξµ ,(2.13)

for some small vector ξ µ = η µν ξν . Under this transformation the metric maintains
the form of Equation 2.9 to first order. There is also no change in the Riemanntensor as Rρ σµν is already small, and the new coordinates represent the same physicalcurvature.

Particularly useful is the harmonic or Lorentz gauge �xµ = 0 → g µν Γρ µν = 0. We

have introduced the D’Alembertian wave operator � = ∂ µ ∂µ . In the weak field limit,
this requirement becomes
µν

∂µ h

=0

(2.14)

1= hµν − η µν hλ λ ,2(2.15)

for the trace-reversed metric perturbation
µν

h

µν

which can be satisfied by absorbing the divergence of h

into the available coordinate

perturbation,µν�ξ ν = ∂µ h .

(2.16)In the Lorentz gauge, the linearized Einstein’s equations take the simple form
µν

�h

=−

16πG µνT .
c4(2.17)We have reintroduced the factor of G/c4 in the linearlized Einstein equation to emphasize the smallness of the perturbations.
23

2.2.1

Transverse traceless gauge

For vacuum (T µν = 0), the linearized Einstein equations give a three-dimensionalµν

wave equation for metric perturbation h ,
�

�
∂2
µν
2
− 2 + ∇ h = 0.
∂t(2.18)The equation has general solutions that can be represented as linear combinations ofplane waves,µνh= C µν exp [ikσ xσ ] .

(2.19)

Here C µν is a constant complex symmetric tensor orthogonal to the wavevector(C µν kν = 0). The wavevector itself must be null (k σ kσ = 0) to satisfy 2.18 sothat a single mode represents a wave traveling in some direction at the speed of light.

The orthogonality requirement that C µν kν = 0 comes from the harmonic gauge
condition (Equation 2.14). However the gauge is not unique, and any additional
coordinate perturbation which satisfies the wave equation will allow us to remain inµν

the harmonic gauge as it does not change ∂µ h . By using a transformation of theformξ µ = B µ exp [ikσ xσ ](2.20)

we can, for the correct choice of B µ [10, 6.50], further restrict C µν to satisfy two new
conditions,
C µµ = 0(2.21)U µ Cµν = 0,(2.22)

for some arbitrary timelike vector U µ . If we choose U µ = (1, 0, 0, 0) and orient kσ torepresent a wave traveling in the z direction, the new conditions mean that hµν takes
24the simple form,


0 0
0


0 Cxx CxyhTTµν = 
0 Cxy −Cxx

0 0
0


0


0
 exp [−iω(t − z)] .

0

0

(2.23)

This is known as the transverse-traceless (TT) gauge because the metric perturbationis trace-free (due to 2.21) and only has components transverse to the direction of
propagation (due to the orthogonality requirement and 2.22). Because hTTµν is alreadytrace-free, we can drop the bar from hµν . Also note that Cxx = −Cyy (trace-free) andthat Cxy = Cyx (symmetry of Cµν ). Since the wavevector is null we have ω = ktt =−kzz to give the final form of 2.23. Although we have defined here the TT gauge fora single mode, the TT gauge exists for any radiative wave.
With this in mind, we write the general form of a gravitational plane wave travelingin the z direction,


0
0
0


0 h+ (t − z)h× (t − z)hTTµν = 
0 h× (t − z) −h+ (t − z)

0
0
0

0





0
.

0

0

(2.24)

h+ and h× are independent real-valued functions representing the two polarizationsof a gravitational wave.

2.2.2

Eﬀect of gravitational waves on matter

Free particles will follow geodesics in curved space meaning their paths will satisfy
the geodesic equation,d2 xρ
dxµ dxν
ρ
= 0.
+
Γ
µν
dτ 2
dτ dτ

(2.25)A particle initially at rest with dxµ /dτ = (1, 0, 0, 0) will experience no change incoordinates in the presence of a gravitational wave asΓ ρ 00 = 0 in the TT gauge.
25

Therefore coordinates in the TT gauge follow geodesics of free particles at rest.
While we have chosen coordinates which do not change for a particle at rest in
the presence of a gravitational wave, there is a change in the measurable distance
between nearby particles. This is seen by considering the light travel time between
two coordinates separated by δx: δt2 = (1 + h+ )δx2 (this is just the rearranged spacetime interval for a null geodesic δs2 = 0 in the perturbed metric with δy = δz = 0).
The general eﬀect of a gravitational wave on a region of nearby particles can bequantified in terms of the geodesic deviation which shows how a vector S µ pointing
from a reference geodesic to nearby geodesics changes in a curved spacetime. In thiscase the geodesics are the world lines of our free test particles at rest with fourvelocities U µ = (1, 0, 0, 0).
The equation of geodesic deviation is
D2 µ
S = Rρ σµν U σ U µ S ν ,2dτ(2.26)

which gives us for the vector S µ between free particles [10, 6.62],∂2 µ 1 ν ∂2 µ
S = S
h ν.
∂t2
2 ∂t2

(2.27)

Deviation occurs only in spatial directions perpendicular to the direction of wave
propagation. For a wave traveling in the z direction we have,
�
�
�
1
1
x
S = 1 + h+ S (0) + 1 + h× S y (0)
2
2
�
�
�
�
1
1
y
y
S = 1 − h+ S (0) + 1 + h× S x (0).
2
2
x

2.2.3

�

(2.28)(2.29)

Stress-energy of gravitational waves

The concept of local gravitational energy is inconsistent with general relativity. General relativity assumes no preferred coordinate system, and the equivalence principle
tells us that it would be impossible to measure any kind of gravitational energy
26

tFigure 2-1: The eﬀect of a gravitational wave on a single ring of particles. The
gravitational wave propagates into the page, and causes deviations of the apparent
position S µ of particles at rest from the perspective of an observer in the center as
space is stretched and contracted (Equation 2.28). The ring is shown at various stages
in its phase evolution. The top row shows the eﬀect of a monochromatic wave at fixedintervals in phase with only h+ polarization (h× = 0), and the bottom row shows pureh× polarization (h+ = 0). Both waves are linearly polarized, and can be swapped
under a ±π/4 rotation about the direction of propogation. As in electromagnetism,
circularly polarized states can be formed by a superposition of equal amplitude h+ andh× components with a π/2 phase oﬀset. A general gravitational wave with arbitraryh+ and h× components is invariant under a rotation of π along the direction of
propagation which is a reflection of the spin-2 nature of the field.

in a small enough laboratory undergoing free fall. However it is possible to define
concepts of stress-energy of fluctuations on a comparatively smooth background curvature. Thus we require that the eﬀective wavelength of fluctuations is much shorter
than any background curvature: λ � R. For the same reason, the energy cannot be
localized to within a wavelength of a particular fluctuation, but we must average overmany wavelengths to get a meaningful measure. Even the energy radiated from astrong non-linear isolated source can be calculated this way as long as space becomes
asymptotically flat at some suﬃciently far away shell of integration surrounding thesource.
Actual calculation of the energy in fluctuations on flat background requires a more
careful approach than simple linearized theory presented here as the energy density27

is quadratic in the gravitational field amplitude. The shortwave formalism [11, 12]
handles more general gravitational waves by splitting the metric into backgroundcurvature and fluctuations, and keeping contributions up to second order in hµν in the
Einstein curvature tensor. The stress-energy contribution from the fluctuations are
identified by their second-order contribution to the background curvature in vacuum.
In the transverse-traceless gauge it takes the form [9, 35.70],1TT�∂µ hTTjk ∂ν hjk �,
32π

(GW)
Tµν
=

(2.30)

where quantities between angle brackets �. . .� are to be averaged over many wavelengths. We see from Equation 2.30 that the energy density varies quadratically withboth amplitude and frequency of a monochromatic wave of the form 2.23.

2.2.4

Generation of gravitational waves

Gravitational radiation is sourced by accelerating mass+energy m much in the same
way that electromagnetic radiation is a result of accelerating charge q. Similarly, it
is useful to decompose gravitational radiation into multiple moments. The monopole
moment is simply the total mass of the system, which does not change and therefore
cannot produce any radiation. Next would be mass (“electric”) and mass current(“magnetic”) dipole radiation, which would be caused by changes in the mass distributions,��ρ(x) x d3 x

(2.31)

ρ(x) x × v(x) d3 x.

(2.32)

Changes in these distributions would violate conservation of linear and angular momentum, so there can be no gravitational dipole radiation as well. For slowly moving
sources, we expect the dominant form of radiation to come from changes in the mass
quadrupole moment,
Iij =

�

ρ(x)xi xj d3 x,28(2.33)

which is not generally conserved.
Due to the complexity of the Einstein equations, it is diﬃcult to solve a generalform for the radiation from an arbitrary source. A useful approximation comes fromsolving the linearized Einstein equations (Equation 2.17) far from an isolated source
which is moving at non-relativistic speeds. In this case we can apply the known generalsolution of the wave equation with a source and outgoing boundary conditions,µνh̄ (t, x) = 4

�3 � [T(t� , x� )]ret,|x − x� |µν

dx(2.34)

where the source −16πT µν is evaluated at the retarded time. In the long-wavelength(low velocity) far-field approximation, this becomes4h̄ (t, x) −→
r→∞ r
µν

�

d3 x� T µν (t − r, x� ).

(2.35)

Here we see that the amplitude of a gravitational wave diminishes linearly with the
distance to an isolated source.
Using the flat-space conservation law ∂µ T µν = 0 and non-relativistic approximation T tt = ρ, it is possible to solve for the spatial components [9, 36.20],
2 TT
-̈
hTT
ij −→ I ij (t − r).
r→∞ r(2.36)

This simple result is known as the quadrupole formula for gravitational wave generation. In order to project onto transverse traceless coordinates, we have used thetransverse traceless projection of the reduced quadrupole moment I- ij ,1
I- ij = Iij − δij I k k =
3

�

�

1
ρ(x) xi xj − δij r2
3
1
I- TT
ij = Pia I ab Pbj − Pij Pab I ab2

Pab = δab − n̂a n̂b

�

d3 x

(projection operator onto n)(2.37)(2.38)(2.39)The gravitational wave amplitude depends on the second time derivative of the reduced quadrupole moment. By plugging into Equation 2.30 and integrating around29

all solid angles, we get for this limit an expression for the total luminosity of a sourcein gravitational waves,

1 ... ...
LGW = � I- ij I- ij �.
5

2.3

Sources of gravitational radiation

2.3.1

Compact binary coalescence(2.40)As two gravitationally bound compact objects such as neutron stars (NS) or black
holes (BH) orbit each other, they gradually lose orbital energy and angular momentum
to gravitational radiation and proceed toward an inevitable collision. During the final
highly relativistic stages of this process, an immense amount of gravitational radiation
corresponding to as much as 5−10% of the total system mass is released in a very
short amount of time making these sources the most distant objects potentially visible
to gravitational wave detectors.
The existence of NS/NS binary systems has been confirmed through several galactic observations where one of the objects is a radio pulsar. In this case, orbital parameters can be deduced from the periodic Doppler shifting of the radio signals. The
first and most famous pulsar in a binary system found to date is PSR 1913+16 which
was discovered in 1974 by Russell Hulse and Joseph Taylor [6]. Careful measurementof the orbital decay over many years [13] showed remarkable agreement with the prediction from general relativity of energy loss due to gravitational radiation. Hulse
and Taylor received the 1993 Nobel prize in physics for their discovery. While the
Hulse-Taylor binary will take ∼300 million years to coalesce, the number of observed
binary pulsar systems can be used to infer rates of compact object coalescence inMilky Way type galaxies. These predicted rates, along with rates predicted fromstellar population synthesis are presented in Table 2.1.
Compact object mergers where at least one member is a neutron star may be
the mechanism which drives gamma-ray bursts (GRBs) [15]. The short (<2 seconds)
burst of hard gamma-rays are thought to arise from shocks created by the relativistic
30

Compact binary coalescence rates per L10 per Myr.
Source Rlow Rlikely Rhigh RLIGO I [yr−1 ] RAdvLIGO [yr−1 ]
NS/NS
0.6
60
600
0.02
40
NS/BH 0.03
2
60
0.004
10
BH/BH 0.006
0.2
20
0.007
20
Table 2.1: The compact binary coalescence rates (lowest possible rate, likely rate,and highest possible rate given observations) per 1010 solar blue light luminosities
per million years as estimated from the number of observed binary pulsar systems, aswell as from stellar population synthesis models. One Milky Way equivalent galaxy
corresponds to approximately 1.7 L10 . Contributions from elliptical galaxies andold stellar populations are not explicitly included. Also shown are predicted likely
detection rates for initial and advanced LIGO detectors. The low and high estimatescan diﬀer from this likely estimate by as much as three order of magnitude. Quoted
rates are taken from [14].
jets produced. That this process happens so quickly and that short GRBs have been
associated with old elliptical galaxies [16] with no young massive stars support themerger theory. If this is the case, there is the possibility that a nearby short-GRB
could be associated with a characteristic gravitational-wave signature in ground-based
detectors, confirming the model and providing unique observation in gravitational
waves of the tidal breakup of the neutron star [17, 18].
The adiabatic inspiral phase of a binary coalescence lasts while the objects arein stable orbits which decay only as a result of gravitational wave emission. As theobjects lose orbital energy, the period of their orbit shrinks and gravitational wave
amplitude increases due to the increased velocities. This gives a characteristic chirp
waveform. The gravitational wave emission is well approximated by post-Newtonian(PN) methods which analytically solve the field equations for the two point masses
with arbitrary spin in quasi-circular orbit to some order in (v/c)2 or equivalently
M/R of the binary by the virial relationship of the gravitationally bound system.
At lowest order, the PN formalism gives the quadrupole formula for gravitational
wave emission. For binary systems, the PN approximation has been calculated tomuch higher orders [19] to get the accuracy required at small separations and highvelocities.
When the two objects pass their innermost stable circular orbit (ISCO) at r � 6M ,31

fISCO � 4.2(M⊙ /M ) kHz, they inevitably fall into each other regardless of gravitational radiation emitted. This marks the end of the adiabatic inspiral and the beginning of the merger. The highly relativistic conditions provide a unique opportunity
to observe strong field gravitational eﬀects [20]. For the same reason, however, postNewtonian approximations break down and no complete analytic solution of a generalmerger exists. Fortunately numerical relativity, after decades of work, has recently
been able to model complete black-hole binary coalescence [21].
The final phase in binary coalescence is characterized by the ringdown of theresulting black hole as it settles into its stable Kerr state. The dynamics are studied
using black hole perturbation theory, and can be described as excitations of thequasinormal modes of the black hole which decay exponentially due to the emissionof gravitational waves at the corresponding frequencies [22, 23]. Eventually the lowest
mode dominates and we expect all coalescence waveforms to end with an exponentiallydamped sinusoid at the l = m = 2 quasinormal mode fundamental frequency and Q,
��
fQNM � (2πM )−1 1 − 0.63(1 − a)0.3
Q � 2(1 − a)−0.45 .

(2.41)(2.42)A 10 M⊙ black hole has a fQNM � 3.2 kHz.
Searches for binary compact object coalescence are typically done with matchedfilter techniques. This procedure uses a known waveform weighted against the stationary spectral noise as a linear filter to check for the presence of a signal. For asingle waveform, the matched filter maximizes the signal-to-noise ratio in the caseof a match. A neutron star binary system will chirp outside of the LIGO sensitive
band of around 100–150 Hz well before reaching its ISCO, so PN inspiral waveforms
are suﬃcient for a template search. Similarly, a massive black hole system with total
mass of several hundred M⊙ will have most of its inspiral and early merger hidden by
the low frequency noise in ground based detectors. These can be found eﬃciently by atemplate of ringdown waveforms. Such searches have been performed on recent datafrom the LIGO and Virgo instruments [24, 25, 26]. The inspiral searches in LIGO
32

S5 data can detect sources out to 30–100 Mpc depending on mass, and set 90% rate
upper limits for binary coalescence with total mass 2–35 M⊙ of 1.4 × 10−2 (NS/NS),

10
7.3 × 10−4 (BH/BH), and 3.6 × 10−3 (BH/NS) yr−1 L−1
10 where L10 represents 10

solar blue light luminosities. Published results for searches for intermediate mass
black hole ringdowns in LIGO S4 data set a limit of 1.6 × 10−3 yr−1 L−1
10 for binary
mergers with total mass 85–390 M⊙ , which can be seen out to ∼85 Mpc in S4 data.
For a large range of total mass between 20–500 M⊙ , a significant amount of thedetectable signal for current ground based interferometers is present in the binary
merger phase [27]. The best sensitivity will be achieved by coherently integrating
signal energy across the entire binary evolution. However this is only possible withmatched filtering if the waveform is known. In 2005, the first successful equal mass,
non-spinning complete black hole merger simulation was achieved [28]. Since then,many groups have run successful simulations using a wide variety of codes and formalisms [21]. The results have been consistent with each other, as well as with PN
approximations for the inspiral and perturbative ringdown. The numeral simulations
have explored a large portion of the binary black hole parameter space, including
precessing orbits with misaligned spin [29]. However, running simulations is computationally expensive, and it is currently not possible to build up a template bank thatsuﬃciently samples the parameter space for a real search.
To be able to make a set of templates suitable for matched filter analysis, hybrid
semi-analytic waveforms have been developed. In one method, phenomenological
waveforms [30] are created by stitching PN inspirals to numerical mergers. A certain
simple parametrization in the frequency domain is then assumed to representat thegeneral hybrid waveform. The parametrization has several coeﬃcients which can besmoothly varied to fit any particular binary configuration. The latest parametrization
have been able to include the eﬀects of unequal mass and aligned and anti-aligned
spins [31]. The eﬀective one-body (EOB) formalism [32] has also been able to successfully model a complete binary coalescence. In the EOB approach, the dynamics ofthe binary system are mapped to those of a single test particle in a eﬀective external
metric which changes as a result of radiation damping. This gives a resummation of33

the PN series approximation, with the advantage of providing a smooth transition tomerger and ringdown of a single object. The lower order terms are determined bytheir correspondence with the known PN expansion, while higher order PN and otheradjustable terms are fit to numerical relativity simulations. The EOB-NR technique
has been used to model unequal mass black hole mergers with aligned and anti-aligned
spins as well [33]. The general case of misaligned spins which cause precession andchanges in the orbital plane has yet to be modeled.

EOB Re(h22) R/M
NR Re(h22) R/M

0.2

0.4

0.1

0.2

0.0

0.0

-0.1

-0.2

-0.2

-0.4
0

500

1000

1500

2000

2300 2350 2400 2450

Δφh (rad)
ΔA / A

0.01

0.1
0

0

-0.1
-0.2

-0.01
0

-0.3
500

1000

1500

2000

t/M

2300 2350 2400 2450

t/M

Figure 2-2: Numerical simulation of a binary black hole merger using SpEC [34]with EOB-NR waveform superimposed. The initial configuration is equal mass black
holes with spins ∼0.44 opposite the orbital angular momentum. The top plots show
the waveforms, where the right plot is a zoom-in on the merger and ringdown. The
EOB-NR waveform is an eﬀective one-body PN expansion with 6 adjustable terms
calibrated by the numerical simulation. The bottom plots shows the phase and amplitude consistency between the waveforms. This figure is reproduced from [33]34

2.3.2

Gravitational core collapse

Gravitational core collapse of massive stars is another highly anticipated source ofgravitational radiation. Core collapse is the natural conclusion of main sequencestellar evolution for stars �9 M⊙ . These central areas of these stars are hot and denseenough to form iron cores as the end product of nuclear fusion. As the degenerateiron core accumulates mass and contracts under gravity, the environment becomesso hot and dense that photodissociation of nuclei and electron capture by protons
can occur. These two processes remove thermal and electron degeneracy pressure–the only means of support, and thus trigger a gravitational collapse where the coreshrinks to a radius of ∼20 km in ∼1 second [35].
The equation of state rapidly stiﬀens as the core becomes dense enough for neutron
degeneracy pressure to become significant. For stars with an initial main sequencemass of �25 M⊙ [36], the resulting bounce of the inner core and outward shock wave
when it hits the falling outer layers, along with neutrino pressure from the hot, convective protoneutron star, carry enough of the released gravitational energy to blow
oﬀ the surrounding envelope of the star in a Type II supernova (SN). A hot, rapidlyrotating neutron star remnant is left. For heavier stars, the shock plus neutrinos do
not transfer enough energy to unbind all the outer material. A weak, Type Ib supernova is expected, and a short lived proto neutron star eventually collapses to a blackhole due to matter infall. Stars with initial mass �40 M⊙ and low metallicity collapse
directly to black holes due to their larger cores at time of collapse. Long-duration
gamma-ray bursts, which are the most commonly observed, have been connected tothe collapse of very massive stars in Type Ic supernovae [37], and are thought to becaused by relativistic jets created by the accretion disk of stellar matter surrounding
a newly formed black hole.
The most notable observational evidence to support the massive stellar core collapse and subsequent explosion model of Type II supernovae is the coincident observation of a burst of 20 neutrinos from the Kamiokande II and IMB water Cherenkov
detectors approximately three hours before visible observation of SN1987a [38]. Mod35

ern neutrino detectors should see around 2000 neutrinos from a galactic Type II supernova. Planetary nebula have also been linked to the expelled stellar envelopes ofhistorical supernovae. The Crab nebula has been identified with a bright supernova
recorded by Chinese and Arab astronomers in 1054. At the center of the nebula is
the Crab pulsar - the rapidly rotating neutron star remnant which was the first of its
type to be found in 1968.

While perfectly spherical collapse would not emit any gravitational radiation,
the collapse of a rotating core will instead flatten along the axis of rotation due tocentrifugal forces. The axisymmetric but aspherical collapse, bounce, and subsequentcore oscillations are a reliable source of linearly polarized gravitational radiation.
Convection and anisotropic neutrino emission are additional significant sources ofcollapse asymmetry in existing simulations [39]. Triaxial rotational instabilities candevelop in the proto neutron star itself such as the standard bar mode instabilities,
or radiation reaction driven r-modes. These draw energy from the core’s rotation andcan be a powerful source of gravitational radiation. In the collapse of a very massive
star, accretion onto a newly formed black hole can excite the quasinormal modes ofthe black hole, which ring down through gravitational radiation.

Numerical simulation of stellar collapse is a very active field as the exact processof collapse has yet to be definitively answered. The field has provided a number ofsophisticated 2+1D axisymmetric and fully 3D models, with the 2D models able toallow for a larger parameter space and more computationally demanding microphysics
at the expense of being able to investigate non-axisymmetric behavior which tends todevelop after the initial collapse [40]. Many models produce radiation strong enoughfor a galactic supernova, with a rate of one per 50 years, to be detected by first
generation broadband interferometric detectors. Advanced detectors will extend therange by a factor of ten into the local group, but it will likely be third generation
detectors which will be able to reliably detect supernovae from the Virgo cluster at a
rate of 1 per year.
36h+D (cm)

100
0

−100
−200

h+ D equatorh+ D pole−300
h× D (cm)

100
0

−100
−200
−300

h×D equator
h×D pole

0

10

20

30

40

50

60

70

t - tbounce (ms)Figure 2-3: Gravitational waveform from a 20 M⊙ rotating stellar collapse. At agalactic distance of D = 10 kpc, 100 cm on the vertical axis corresponds to a strainof 3.24×10−21 . The two polarizations are shown in separate plots, while colors distinguish equatorial observers (black) and those looking down the pole (red). In this 3D
simulation, the initial collapse remains axisymmetric emitting a typical linearly polarized bounce waveform perpendicular to the rotation axis, while non-axisymmetricinstabilities contribute to circularly polarized radiation along the rotation axis beginning ∼30 ms post bounce. Approximately 7.5 × 10−8 M⊙ is released in gravitationalradiation with about equal contributions from bounce and late-time 3D dynamics.
This figure is reproduced from Ott et al [40]2.3.3

Rotating neutron stars

Any kind of oﬀ-axis asymmetry in a rapidly rotating neutron star will radiate gravitationally. The asymmetry could be due to a misaligned stiﬀ crust, electromagnetic or
accretion induced anisotropies, precession, or the excitation of some form of rotational
instability. Gravitational radiation, regardless of mechanism, will draw rotational energy and angular momentum out of the star, slowing it down. In many cases thisprocess is quite slow and results in weak radiation over a timescale that is much longer
37

than a typical search. For this reason, such objects are often referred to as continuous
wave sources.
Weak deviations from mass axisymmetry along the rotational z axis can be parametrized
by the ellipticity for a triaxial ellipsoid, written here in terms of the moment of inertia
tensor Iij = δij I k k − Iij (see Equation 2.33), along principal axes,�=

Ixx − Iyy.
Izz

(2.43)

Neutron star crusts are thought to be able to support an ellipticity no larger than
∼10−6 . The characteristic strain observed at the Earth at rotational frequency ν ish0 =16π 2 G �Izz ν 2.
c4
r(2.44)The gravitational wave frequency will be at 2ν since, as typical of rotating systems,
the quadrupole moment goes through two cycles for every rotation. While the gravitational wave emission is essentially monochromatic, the observed signal at the Earthis Doppler shifted by non-negligible amounts by the Earth’s motion in the galactic
frame.
Loss of angular momentum through gravitational radiation is a leading candidate
to explain the observed sharp cutoﬀ at ∼720 Hz in the rotation rate distribution of
millisecond accreting X-ray pulsars [41]. The neutron stars are spun up by accretionfrom a companion star, yet the maximum observed frequency is well below theirbreakup frequency of ∼1500 Hz for typical equations of state. Bildsten originally
postulated [42] that accreting X-ray pulsars would reach an equilibrium between spinup due to accretion and spin down due to gravitational radiation, making it diﬃcult
such systems to reach frequencies greater than 600–800 Hz. If angular momentum loss
through gravitational waves dominates over electromagnetic eﬀects, their strength isset to balance the torques due to mass transfer. In this optimistic limit, a few ofthe brightest sources corresponding to the largest accretion rates may be detectable
by Advanced LIGO [43], especially if the instrument is tuned for the frequencies ofinterest.
38

This type of upper limit for gravitational radiation can be established for any
known pulsar where the frequency ν and spin down rate ν̇ are observed through radioor X-ray timing. Assuming the spin down is due completely to gravitational radiation,
the strain at the Earth would be
hsd =

�

5 GIzz |ν̇|
2 c2 r 2 ν

�1/2

.

(2.45)The spin down limit on gravitational radiation depends on the moment of inertia Izz
as this is what determines the rate of angular momentum loss given a measured spin
down rate. In many cases, one can assume canonical value of Izz = 1038 kg m2 whichcorresponds to a 1.4 M⊙ neutron star with uniform density and radius 10 km. To
date, the only known pulsar for which a direct observational limit on gravitationalradiation is smaller than the limit imposed by spin down is the Crab pulsar. The
Crab pulsar has an unusually high spin down limit of hsd = 1.4 × 10−24 due to its
close proximity and high spin down rate, and data from LIGO’s fifth science run has
been used to beat this limit on gravitational radiation by a factor of 7.2 for the moststraightforward model of emission [44]. This is not surprising given the large expected
electromagnetic torques slowing down the young supernova remnant.

In addition to rotating mass asymmetry, neutron stars have a rich spectrum of
non-radial quasi-normal mode oscillations which can radiate gravitationally [45]. The
modes can be easily excited by the core bounce and fall-back during a gravitational
collapse, but also by other sources like irregular accretion and crust quakes which
could accompany a pulsar glitch or soft gamma repeater (SGR) flare. Unlike thetriaxial rotational instabilities, these pulsations do not require rotation to occur, but
the rotation profile will aﬀect their frequencies and mixing in non-trivial ways [46]. In
neutron star astroseismology, one uses information carried by the gravitational waves
from normal mode oscillations to model fundamental properties of the neutron star.
39

2.3.4

Stochastic backgroundThe stochastic gravitational wave background is a random field of gravitational perturbations throughout all space. The cause of the perturbations may be an ensembleof unresolved astrophysical point sources, or they may be a result of fundamental
processes in the early universe. Gravitational waves from the early universe are particularly interesting because they decouple from other particles at extremely highenergies. The universe remained opaque to electromagnetic radiation until the epochof recombination around 3.8 × 105 years after the Big Bang, and the cosmic microwave background (CMB) radiation we detect today gives us a fairly undisturbed
snapshot of the structure of the universe at that time. We expect a similar neutrino
background from ∼2 seconds after the Big Bang, although their low energy today of∼1.95 K makes direct detection very remote. A gravitational wave background fromas early as the Planck era (∼10−42 s) is also expected to be very diﬃcult to detect
although many processes in the very early universe are expected to leave signatures
which could be observed by future instruments. Gravitational waves produced in theearly universe will have maximum wavelength on the order of the horizon size at timeof production, so that waves entering as early as ∼10−25 seconds would be redshifted
into the LIGO sensitive band today [47].
Because the spacetime perturbations carry energy, they contribute to the totalenergy density of the universe. This contribution is generally represented in terms ofthe gravitational wave energy density per unit logarithmic frequency,ΩGW (f ) =1 dρGW.
ρc d ln f(2.46)The energy density is written as a fraction of the critical density needed to makethe universe flat in standard cosmology given the measured Hubble expansion rate
H0 � 72 km s−1 Mpc−1 [48, 49],ρc =3c2 H02.
8πG(2.47)

The flatness of the universe has been verified to ∼2% using CMB measurements.
40However, it is often assumed to be exactly flat as a consequence of inflation.
The inflationary stochastic gravitational wave background arises from quantum
vacuum spacetime perturbations which were stretched and amplified to classical scales
during inflation. Exponential inflation contributes a flat spectrum toΩ GW (f ) today over a very large range of frequencies [50]. The current best limits on gravitational waves from inflation come from the smallness (10−6 ) of large scale temperature
anisotropies measured in the CMB. Long wavelength gravitational waves create observed temperature anisotropies through gravitational redshift, known as the SachsWolfe eﬀect. The low modes (2 ≤ l ≤ 30) in the CMB set a limit ofΩ GW (f ) < 10−13

at 10−16 Hz [47], which for a flatΩ GW (f ) spectrum is well below the noise in othermeasurements of the gravitational wave spectrum. Future measurements of CMB
polarization may be able to decrease this limit or detect gravitational waves frominflation by measuring their unique contribution to polarization B modes [51].
The gravitational wave spectrum is much more poorly constrained at higher frequencies. The total energy in gravitational radiation,
ρGWΩTotal
=
GW =
ρc

�

d(ln f )ΩGW (f ),(2.48)

cannot exceed 1.5 × 10−5 for gravitational waves which were present at the time of
Big Bang nucleosysynthesis [52]. A larger amount of gravitational radiation would
increase the relative radiation density of the universe, which speeds up expansion andimplies an earlier freeze-out for neutron-proton thermal equilibrium. The resulting
overabundance of neutrons due to a higher freeze-out temperature is inconsistent
with the observed fraction of 4 He [50]. The limit only constrains gravitational waves
present during Big Bang nucleosynthesis minutes after the Big Bang. The wavelength
is limited by the horizon scale at that time, corresponding to frequencies today which
are greater than 10−10 Hz. A similar bound on the total radiation density during theepoch of recombination can be made by calculating its eﬀect on matter and CMB
power spectra. The total energy in gravitational radiation when the CMB decoupled
is constrained to be less than 1.3 × 10−5 [53] and includes gravitational waves which41

are now greater than 10−15 Hz. This limit is projected to improve as the CMB power
spectrum becomes better constrained.
Precision pulsar timing of millisecond pulsars has established a bound ofΩ GW (f ) <3.7 × 10−8 at 4 × 10−9 Hz [54]. A stochastic gravitational wave background contributes to timing residuals of the extremely regular pulses. Measurements of correlations between the two LIGO sites over the course of two years have also beenused to set a direct upper limit on the stochastic gravitational wave background ofΩGW (f ) < 6.9×10−6 at ∼100 Hz [55]. Advanced LIGO will be sensitive to anΩ GW (f )which is smaller by a factor of 100 assuming a factor of 10 increase in strain sensitivity. This will test cosmic-string [56] and pre-Big Bang [57] models which predictsignificant high frequency stochastic gravitational wave backgrounds.
The energy density in gravitational waves is determined by the average amplitudeof their fluctuations, and thus we can relate the gravitational wave energy spectrum toan observable power spectrum of strain fluctuations (signal power per unit frequency),
SGW (f ) =3H02 −3
f ΩGW (f ).
10π 2(2.49)The stochastic gravitational wave background contributes directly to the strain power
spectrum of a gravitational wave detector such as LIGO. For the LISA space based
interferometric detector, the galactic population of unresolved white dwarf binary
systems is expected to be a limiting noise source at low frequencies. Better sensitivity
can be achieved for two uncorrelated detectors by looking for an excess in correlated
noise, which is caused by gravitational wavelengths which are longer than the distance
between instruments. The exact parametrization of the degree of correlation is called
the overlap reduction function, γ(f ), which is a function of frequency as well as
distance and relative orientation between the detectors [58]. An optimal filter todetect a particular spectrum of gravitational radiation will also weigh contributions
by frequency according to the individual detector sensitivities and their overlap. The
sensitivity of such a measurement scales with the square root of observation time.

42

−4

10

CMB & Matter
Spectra−6
10
PlanckLIGO S4BBN

LIGO S5

PulsarLimit−8

Cosmic Strings

ΩGW

10

AdvLIGO
−10

10

LISA
CMB Large
Angle

−12

10

Pre−Big−Bang

Inflation

−14

10

−18

−16

−14

−12

−10

10 10 10 10 10

−8

−6

−4

−2

0

2

4

6

8

10

10 10 10 10 10 10 10 10 10 10
Frequency (Hz)Figure 2-4: Cosmological source models as well as limits of and future sensitivities toa stochastic gravitational wave background. The limits based on BBN constraints aswell as from CMB and matter spectra are integral limits on the total energy in gravitational radiation over the frequencies shown, in units of the critical density of theuniverse. The models shown for standard inflation, cosmic strings, and pre-Big Bang
scenarios, as well as remaining limits from large angle CMB anisotropy, millisecond
pulsar timing, and current and future ground and space based interferometric detectors are limits on gravitational wave energy density per unit logarithmic frequency:ΩGW (f ) = dΩTotal
GW /d ln f . This figure is reproduced from [55].

43

44

Chapter 3
Detecting gravitational waves
3.1

Resonant mass detectors

Passing gravitational waves transfer a small amount of energy to a rigid body as
atomic forces fight the stretching and squeezing of space. In a resonant mass (bar)detector, gravitational waves at the fundamental frequency of the bar will cause thegreatest amplitude of excitations, which are converted into signals using sensitive electromechanical transducers attached to the bar. The first such devices were designed
and tested in the 1960s by Joseph Weber [1, 59]. Weber used two 1.5 ton cylindricalaluminum bars with a strain sensitivity of 10−16 at 1660 Hz to search for coincidences from astrophysical gravitational waves. Today’s bar detectors are ∼3 meterswith masses ∼1,000 kg and have fundamental longitudinal mode resonant frequencies
about 1,000 Hz. When cooled to temperatures as low as 0.1 K, they reach sensitivitiesof ∼10−21 , approaching their quantum limit for zero-point energy fluctuations.
The three primary noise sources for resonant mass detectors are sensor noise, thermal noise, and recently quantum noise. Sensor noise refers to noise in reading theposition of the bar endpoints. The sensor noise is the limiting noise source away fromresonance where the amplitude of oscillations becomes small. Modern detectors makeuse of resonant sensors matched to the bar frequency which allow gravitational waves
near the resonant frequency to first transfer energy to the bar whose oscillations thendrive the resonant sensor. The bandwidth of such detectors is limited by the time it45

takes to drive the resonant sensor to peak amplitude. Random thermal fluctuationsin the bar also cause noise which can be large compared to an expected gravitationalwave induced perturbation. To minimize thermal noise, resonant bars are cooled
to below liquid helium temperatures. This also maximizes the quality factor, Q, ofthe bars mechanical resonance which keeps the thermal fluctuations restricted to avery narrow bandwidth about resonance. If sensor and thermal noise are made smallenough, a bar detector reaches a fundamental quantum uncertainty in any measurement of the position of the bar endpoints. This limits strain sensitivity to ∼10−21 fora typical bar configuration, just at the edge of being astrophysically interesting.
Current cryogenic gravitational wave bar detectors include the ALLEGRO detector in Baton Rogue, Louisiana [60], the AURIGA detector in Lengaro, Italy [61], the
EXPLORER detector in Geneva, Switzerland [62], and the NAUTILUS detector in
Rome, Italy [62]. The four detectors took part in a coincident run of data-taking
from 2005-2007 under the second International Gravitational Event Collaboration(IGEC-2) [63, 64]. The joint analysis, sensitive to short <1 s signals with frequency
content above the detector noise, found no gravitational wave candidate events. Interferometric detectors such as LIGO and Virgo have since eclipsed the bar detectors
in sensitivity and bandwidth. The ALLEGRO detector ceased operation in 2007. The
remaining detectors will provide important coverage during the substantial down-timeof interferometric detector upgrades over the next several years, and they will also provide provide enhanced detection confidence for the case of a loud gravitational-wave
event at bar frequencies.
A cylindrical resonant bar is sensitive to strain along its main axis which can
excite the bar’s fundamental longitudinal mode. The geometric projection of a gravitational wave from arbitrary sky location onto components of strain measured by thedetector is represented by the detector antenna response. The two gravitational wave
polarizations, h+ (t) and h× (t) contribute a measured strain of,h(t) = F+ h+ (t) + F× h× (t),
46

(3.1)

−18

10

EXPLORER
NAUTILUS

ALLEGROS

1/2(1/ √ Hz )
h

−19

10

−20

10

AURIGA
−21

10

860

880

900
920
Frequency (Hz)

940

960

Figure 3-1: Typical strain noise sensitivities of current cryogenic resonant bar detectors for data taken during 2005-2007. This figure is reproduced from [64]where F+ and F× are the antenna factors for plus and cross polarization. For a bar
detector, these antenna factors are,F+ = sin2 θ cos 2ψ

(3.2)

F× = sin2 θ sin 2ψ,(3.3)

where θ represents the angle of the source from the bar axis, and ψ represents theorientation of the gravitational wave polarization axes with respect to an orientation
defined by the bar axis-source plane. Resonant bars are sensitive to a large area ofthe sky perpendicular to their axis. Spherical resonant detectors have been proposedthat would provide a uniform coverage of the sky with no blind spots. Because
they have no preferred oscillation axis, they can also resolve both gravitational wave
polarizations, and provide some directional information for a single source. Two
prototype detectors include the MiniGRAIL sphere [65] in the Netherlands, and the
Schenberg detector[66] in Brazil.
47

3.2

Interferometric detectors

Interferometry provides very precise measurement of relative length changes.

A

Michelson interferometer with a single beam splitter and two arms at 90 degrees
is particularly well suited to measure the diﬀerential strain produced by a passing
gravitational wave. Over bar detectors, interferometers have the fundamental advantage of being able to be made much larger. As they do not depend on any resonancebetween the instrument and signal, they are also broadband detectors sensitive to amuch larger range of frequencies than resonant bars. Development of laser interferometric gravitational wave detectors began in the late 1960s with the independent
work of Weiss [2], Moss, Miller, and Forward [3]. Only in the last several years withthe successful construction of first generation km-scale interferometers have they met
and surpassed bar detector sensitivity.
In an idealized Michelson interferometer, a single beam of light is sent througha 50:50 beam splitter at a 45◦ angle. Half the light intensity is transmitted down
the same direction as the incident beam, and half is reflected at 90◦ . Each beam isreflected back at some distance away by end mirrors and they recombine at the same
beam splitter to form a reflected beam back toward the original source direction, andan anti-symmetric output opposite the first reflected beam. A monochromatic laser
input will have electric field,
Ein = E0 ei(wt−kx) .

(3.4)

The 50:50 beam splitter may be characterized by amplitude transmission coeﬃcient√√i/ 2 and reflection coeﬃcient 1/ 2, while the end mirrors have a reflection coeﬃcientof −1. The field exiting the antisymmetric port is a superposition of the portion of thebeam first transmitted then reflected by the beam splitter, and the portion reflected
first then transmitted by the beam splitter on the way back. The amplitude of the
field exiting the antisymmetric port is,|Eantisym | = E0 cos k(Lx − Ly ).
48

(3.5)

A photodetector placed at the antisymmetric port will measure the intensity of exiting
light which is related to the average squared electric field, and is therefore related tothe input power by,
Pantisym = Pin cos2 k(Lx − Ly ).

(3.6)

Measurement at the antisymmetric port reflects the accumulated phase diﬀerence
between the light traveling in each arm of the interferometer.
A linearly polarized gravitational wave passing through the plane of the interferometer and aligned to the interferometer arms is optimally oriented to produce achange in the diﬀerential arm length. Like bar detectors, the single strain signal measured by an interferometer from a source with arbitrary direction and polarization is
subject to a geometric projection factor,
h(t) = F+ h+ (t) + F× h× (t).

(3.7)

A detector with arms along x̂ and ŷ has antenna factors,
1
F+ = (1 + cos2 θ) cos 2φ cos 2ψ − cos θ sin 2φ sin 2ψ
2
1
F× = (1 + cos2 θ) cos 2φ sin 2ψ + cos θ sin 2φ cos 2ψ,
2

(3.8)
(3.9)

where (θ, φ) represents the standard spherical coordinate system inclination andazimuthal angles from zenith and the y = 0 plane respectively, and ψ is a polarization
angle which represents the angle from the coordinate axes defined by (−φ̂, θ̂) at aparticular point in the sky to the chosen gravitational wave (x, y) coordinate axes.
This is a choice of convention as the gravitational wave can be written with any choiceof polarization axis.
The geometric antenna pattern is only valid in the long wavelength limit where
the gravitational wavelength is much longer than the total light travel distance in
the interferometer so that the instrument is at any particular time measuring anessentially static gravitational field. For shorter wavelengths, or equivalently highgravitational wave frequencies, the light circulating the interferometer arms is both49

redshifted and blueshifted before being recombined, thus canceling the signal. Thisis further complicated by the resonant cavities used in modern interferometers wherelight is trapped for a variable number of trips in each arm before returning to thebeam splitter. Longer arm lengths do result in proportionally greater path lengthchanges at fixed strain amplitude, so the ideal size of an interferometer will balance
the two eﬀects in addition to practical considerations.
First generation large scale interferometric gravitational wave detectors have comeonline within the last decade. These include the two 4 km and one 2 km (currently
decommissioned) LIGO detectors in Hanford, Washington and Livingston, Louisiana[4], the 3 km Virgo detector in Cascina, Italy [5], the 600 meter GEO600 detector
near Hannover, Germany [67], and the 300 meter TAMA300 detector outside Tokyo,
Japan [68]. The LIGO and GEO600 detectors operate within the LIGO ScientificCollaboration, which has entered into close data sharing and joint analysis agreementswith the Virgo Collaboration in order to make best use of the scientific data. While no
gravitational wave events have yet to be observed (consistent with predicted detection
rates at current sensitivities), the successful technological development and operationof first generation detectors is an important step toward the realization of advanced
detectors sensitive enough to make regular detections.

3.3

Laser Interferometer Gravitational-wave Observatory (LIGO)

The Laser Interferometer Gravitational-wave Observatory (LIGO) consists of threekilometer-scale gravitational wave interferometers located at two sites. An observatory at Hanford, Washington houses a 4 km baseline (H1) and 2 km (H2) detector
which share a common vacuum enclosure and seismic isolation. Another nearly identical observatory three thousand miles away at Livingston, Louisiana has a single 4 km
detector. Between November 2005 and September 2007 the detectors engaged in theirfifth science run (S5) where they operated at their design sensitivity and collected in50

EcircEin

Etrans
L

Erefl
+

−
r1 t1

−

+
r2 t2

Figure 3-2: A simple two-mirror Fabry-Perot resonant cavity of length L. The reflection and transmission coeﬃcients for each mirror must satisfy energy conservation:
r2 + t2 = 1 − losses. Light reflecting from the inner (coated) mirror surface suﬀers
a sign change. There are both right (shown) and left circulating fields. Typicallythe mirrors have high reflectivity. On resonance, light escaping the cavity interferes
destructively with the promptly reflected input light from the first mirror causing
laser light two build up between the two mirrors.
total a year of science data with all three detectors operating in coincidence. The
LIGO instruments H1 and L1 have since undergone minor upgrades for a currently
running sixth science run. The upgrades are part of a transition to Advanced LIGO
which will require several years of commissioning beginning in late 2010.
The LIGO instruments are modeled after Michelson interferometers with suspended end mirrors that act as freely falling gravitational test masses over the frequency range of interest (40–7000 Hz). Through S5, the instruments operated on adark fringe with no light exiting the antisymmetric port without the presence of agravitational wave (or some other disturbance). To increase the eﬀective arm length
as well as the total amount of circulating light sampling the gravitational wave, theinstruments make wide use of resonant cavities which we will review here briefly due
to their importance.

3.3.1

Fabry-Perot cavities

Each arm of a LIGO interferometer is a Fabry-Perot resonant cavity, which traps
light between two parallel mirrors. A Fabry-Perot cavity is a kind of interferometerin itself as light in the cavity interferes with other beams of light which have madevarious numbers of paths between the optics.
51

The steady state fields for a Fabry-Perot cavity for a constant input laser source
are given by the superposition of light from all possible paths between both mirrors. A
field E0 incident on a mirror is split into a reflected field of amplitude ±rE0 dependingon which side of the mirror reflection takes place and transmitted field of amplitude
tE0 . The right circulating field in a two mirror cavity of length L (Figure 3-2) withreflection and transmission coeﬃcients r1,2 and t1,2 is found by adding up all roundtrips of the incident light,
∞
�
�
�n
Ecirc
t1
.
r1 r2 e−2ikL =
= t1
−2ikL
Ein
1
−
r
r
e
1
2
n=0

(3.10)

The cavity is on resonance when 2kL = 2πn for some n. For highly reflective mirrors,
the denominator becomes very small, and the amount of light trapped can be verylarge. The transmitted field is the portion of this field that escapes through thesecond mirror,
Etrans
t1 t2 e−ikL.
= t2 e−ikL Ecirc =Ein
1 − r1 r2 e−2ikL(3.11)A cavity with highly reflective mirrors acts as a filter for the light, only passing
through modes which are resonant in the cavity. The reflected field is the sum of thepromptly reflected light with the light escaping the cavity back toward the source,
Erefl
r2 t21 e−2ikL.
= r1 − r2 t1 e−2ikL Ecirc = r1 −Ein
1 − r1 r2 e−2ikL(3.12)

Oﬀ resonance, most of the light is promptly reflected if the first mirror has highreflectivity. The reflected light can also be written in the form,
Erefl
r1 − r2 (1 − l1 )e−2ikL=
Ein
1 − r1 r2 e−2ikL(3.13)

where l1 = 1 − r12 − t21 represents small absorption and scattering losses due to the firstmirror. When both mirrors are identical and losses are negligible, a special case arises
for resonance where all the light is transmitted by the cavity and none is reflected
back.

52

For the LIGO arm cavities, the end mirrors are made to be as reflective as possible.
The input mirrors have small nonzero transmission. In this case, the coeﬃcient forreflected light from the cavity becomes very close to −1. Near resonance, the reflected
light can be expanded to first order about L + δL,
Erefl
=
Ein

�

Erefl
Ein

�

res

�

�
r1 r 2
1−
2ikδL .
1 − r1 r 2(3.14)The phase shift of the reflected light is larger by a factor of r1 r2 /(1 − r1 r2 ) fromthe phase shift due to a single round trip. Therefore, when applied to a Michelsoninterferometer, the Fabry-Perot cavities greatly increase the eﬀective arm length. The
full gain in phase sensitivity happens as long as δL varies on a time scale longer than
than the eﬀective light storage time. Above the cavity pole frequency of (c/4πL)(1 −r1 r2 ), phase sensitivity falls as f −1 . For LIGO 4 km instruments, the reduction insensitivity begins around 90 Hz.
The LIGO detectors make use of another resonant cavity to increase the eﬀective
laser power going to the beam splitter. High laser power is desirable in order toreduce photon counting statistical noise which is the limiting noise factor at highfrequency. In this case a partially transmitting power recycling mirror is placed before
the Michelson interferometer. Since the interferometer operates on a dark fringe, it
acts as a second mirror for the resonant cavity reflecting incident light back toward
the input port. Power builds up for this cavity in the same way as for a standard
two mirror Fabry-Perot cavity. By making use of power recycling, the incident laser
power on the beam splitter is increased by a factor of ∼100.

3.3.2

Length sensing and control

The main drawback of resonant cavities is that they must be kept on resonance. The
tolerance for cavity length changes can be very small, around ∼10−10 m for LIGO.
This requires active control of mirror position which is done by way of small magnets
bound to the optics. Variations on Pound-Drever-Hall (PDH) reflection locking areused to construct error signals used to drive these magnets and keep the instrument at
53

ETM

ETM
ITM
BS
PRM
MC
FI
IO

End test mass
Input test mass
50/50 beamsplitter
Power recycling mirror
Mode cleaner
Faraday isolator
Input optics

AS Anti-symmetric port
PO Pick-off port
REF Reflection port
RF length detector
RF alignment detector
Quadrant detector

FI
REF

250 W

IO

5W

10 W

Laser

PRM

PO

BS

AS

15 kW

ITM

MC

4 km

Figure 3-3: Optical layout of a LIGO interferometer. This figure is reproduced from[69].
the ideal operating point. First the 1064 nm (f � 280 THz) laser light is modulated
at radio frequencies fm ∼ 25 MHz by an electro-optical modulator. This has the eﬀect
to first order of introducing two new small sideband fields at frequencies f ± fm . If
these sideband fields are not resonant in an optical cavity, they will be promptly
reflected. The reflected sidebands will mix with the reflected carrier light forming
optical beats at the modulation frequency. The amplitude of this signal is easily readby demodulating at the original modulation radio frequency and is proportional to thephase diﬀerence between the reflected carrier and sideband signals. Since this phase
diﬀerence is itself proportional to small deviations from resonance, the demodulated
signal can be used in a feedback loop to keep the cavity locked.
In LIGO two sets of sidebands are introduced. The first set from modulation
at 62.5 MHz are not resonant in the power recycling cavity (PRC) and are directlyreflected at the recycling mirror (RM). The second set from modulation at 25 MHz
are resonant in the PRC and enter the Michelson where they can be used to control
the individual arm cavity lengths. No sidebands are resonant in the arm cavities
themselves. The final degree of freedom is the diﬀerential length of the Michelsoninterferometer. Since the interferometer operates on a dark fringe (minimum), theamplitude of light at the antisymmetric (AS) port varies quadratically with diﬀerential
54arm length and thus does not make a good control signal. Instead, a 30 cm Schnupp
asymmetry [70] is introduced into the Michelson arm lengths. The asymmetry is
an integral number of wavelengths of the carrier light to maintain the dark fringe
condition, but it is chosen so that some of the sideband power is channeled to the AS
port. As in the PDH scheme, this gives a linear signal appropriate to measure andcontrol the diﬀerential degree of freedom.

3.3.3

Noise sources

Three primary noise sources limit the design sensitivity of the initial LIGO detectors. At low frequency, seismic noise couples to random motion of the mirrors. At
high frequencies, photon counting statistics limit the measurement accuracy as shotnoise. At middle frequencies where the detector is most sensitive, thermal noise whichcauses random fluctuations in the mirror surfaces dominate. Other important noise
sources which must be minimized are laser frequency and amplitude noise as the laser
is not perfectly stable, various environmental noise such as acoustic or magnetic disturbances which can couple to mirror motion, and electronics noise. Mirror shapeand alignment issues can also reduce sensitivity and must be carefully controlled.
The LIGO mirrors are isolated from ground vibration by four-layer passive seismic
isolation stacks which provide f −8 suppression of seismic noise above ∼10 Hz. In
addition, an active seismic pre-isolator was deployed at the Livingston site due tothe high ground motion there. The mirrors themselves are hung by steel wire which
provides another f −2 level of isolation above the pendulum frequency of ∼0.75 Hz. At
low frequencies, the amount of ground motion that couples to mirror motion increases
dramatically, creating an eﬀective seismic noise wall at ∼45 Hz.
Thermal noise enters the instrument through excitations of the suspension modes
and fluctuations on the mirror surfaces. It can be diminished by choosing materials
with very low mechanical loss and high Q, thus concentrating the noise at well defined
frequencies which ideally can be placed outside of the detector band. The eﬀects ofthermal fluctuations at the mirror surface can be averaged out by using a large beam
size. The Japanese LCGT project [68] aims to beat thermal noise by cooling the55

optics to cryogenic temperatures.
At high frequency, shot noise limits the detector sensitivities. Even if the mirrors
were perfectly undisturbed with zero displacement noise, the random nature of individual photons arriving at the photodetector means that rapid changes in intensity
become unresolvable. As for any Poisson process, shot noise has a flat power spectrum which means there are no correlations between one measurement and the next.
However the response of an interferometer with Fabry-Perot cavity arms decreases as
f −1 above frequencies near the inverse light storage time. This must be multipliedby the shot noise spectrum to get the frequency dependent sensitivity to diﬀerential
motion. The obvious way to reduce shot noise is to increase laser power. High power
can lead to increased radiation pressure noise, another quantum noise source which
will be important for Advanced LIGO. It also means more heat deposited on the
optics which requires better thermal compensation to avoid distortion.

3.3.4

Advanced LIGO

The Advanced LIGO sensitivity goal calls for a factor of ∼10 increase in sensitivity
over initial LIGO across all frequencies. While the advanced interferometers will use
the same site and vacuum enclosure as the current instruments, a large number of
upgrades will be necessary to reach the new target. Some changes have already been
implemented for Enhanced LIGO [71] and the current S6 run. These include anincrease in laser power from 10 to 35 W and implementation of a DC readout scheme
for the diﬀerential error signal.
Advanced LIGO will upgrade the laser to 200 W and add a signal recycling mirror
to the antisymmetric port, creating a resonant cavity for gravitational wave induced
sidebands near a chosen frequency. This allows tuning of the instrument to reduce
shot noise at frequencies of interest. The test masses will be significantly larger andheavier to allow for bigger beam size and a reduction of radiation pressure noise.
The use of fused silica rather than steel suspensions will greatly reduce suspension
thermal noise, and seismic isolation will be improved by more complicated passive
and active isolation systems. The factor of ten increase in strain sensitivity translates
56

−19

Equivalent strain noise (Hz−1/2)

10

−20

10

−21

10

−22

10

−23

10

2

10

3

10
Frequency (Hz)Figure 3-4: Strain noise sensitivities for the three LIGO interferometers during the
S5 data run 2005-2007. H1 (red) and L1 (green) have a 4 km baseline and are at ornear their initial LIGO design sensitivity (dashed). H2 (blue) has a 2 km baseline
reducing sensitivity by a factor of two at low frequency. The instruments reach a peakstrain sensitivity around ∼150 Hz, but are sensitive to a very wide bandwith less somenarrow spectral lines due largely to 60 Hz harmonics and suspension resonances. At
the most sensitive frequencies, the 4 km instruments have reach a root-mean-square(rms) strain noise of 3 × 10−22 when restriced to a 100 Hz band. This figure is
reproduced from [69].

57into a factor of 1,000 in eﬀective search volume, and regular detection of gravitational
waves with the advanced detectors is expected.

58

Chapter 4
Gravitational wave bursts
Short bursts of gravitational radiation are expected from violent astrophysical phenomena such as the core collapse of massive stars and the compact binary merger
scenario discussed in section 2.3. A search for the signatures of gravitational-wave
bursts in detector data must maintain sensitivity to a large variety signals representing a range in progenitor initial conditions, as well as signals from sources withdynamics that may be poorly modeled or completely un-modeled. Thus, the genericsignals targeted by burst analyses are only loosely constrained by their properties intime, frequency, shape, and by detector noise and network antenna response.

4.1

Burst parametrization

Burst signals typically have some natural frequency due to orbital, rotational, or
acoustic timescales, and so a parametrization in time-frequency is a useful characterization. The burst signal, as detected by a single instrument, has representations intime and frequency which are related by the Fourier transform,

−1

� +∞

[h̃(f )] =h̃(f )e+2πif t df−∞
� +∞h(t)e−2πif t dt.
h̃(f ) = F [h(t)] =h(t) = F

−∞

59

(4.1)
(4.2)

We assume the burst signals have finite extent in time and frequency, with detector
noise placing a practical limit on frequency extent.
The total signal energy is conventionally defined as the sum squared signal, andis related in time and frequency representations by Parseval’s theorem,
2

�h� =

� +∞
−∞

2

|h(t)| dt =

� +∞
−∞

|h̃(f )|2 df.

(4.3)|h(t)|2 is the density of signal energy in time, while |h̃(f )|2 = h̃∗ (f )h̃(f ) is the signal

energy per unit frequency, or energy spectral density. In this formula, h̃∗ is thecomplex conjugate of h̃. In order to have a representation of gravitational-wave
amplitude independent of detector orientation and response, we also define a quantity
hrss as the summed contributions from h+ and h× components in the gravitationalwave plane,h2rss = �h+ �2 + �h× �2 .

(4.4)

The two amplitude values �h� and hrss are equal only for the case of a linearlypolarized wave which is optimally oriented (h× = 0 and F+ = 1).
Central time and frequency of the signal are given by the first moments of thesquared signal,
� +∞1
tc = �t|h(t)| � =t|h(t)|2 dt�h�2 −∞
� +∞22
f |h̃(f )|2 df.
fc = �f |h̃(f )| � =
2
�h� 0
2

(4.5)
(4.6)

The factor of 2 for the frequency integral is to compensate for the integration overpositive frequencies only as h̃(−f ) = h̃∗ (f ) for real signals h(t). Signal duration andbandwidth are given by the second central moments,
� +∞
1
σt =
(t − tc )2 |h(t)|2 dt�h�2 −∞
� +∞2(f − fc )2 |h̃(f )|2 df.
σf =
2
�h� 0
60

(4.7)
(4.8)

Finally, the quality factor, Q, of the signal is,
Q=

fc
.
σf

(4.9)

The quality factor is a measure of the sharpness of the energy distribution about
the central frequency. The nomenclature comes from the quality of low dispersion
resonators. Q is useful as a parameter because it is a property of the shape of the
waveform, and is invariant under arbitrary scaling in time.
The central time and frequency measurements and their spread are just the mean
and variance of the respective values if the normalized squared signal is interpreted
as a probability distribution. Like the Heisenberg uncertainty limit on position andmomentum wave-functions in quantum mechanics, σt and σf satisfy a time-frequency
volume bound,σt σf ≥ 1/4π.

(4.10)

Minimal uncertainty waveforms have σt σf = 1/4π and have the form of a Gaussian
windowed exponential wave packet. In this case, high Q wave packets will have longer
duration σt . From 4.9 and 4.10 one can count Q/2π oscillations within ±σt about
the central time for a Gaussian wave packet regardless of frequency.

4.2

Detecting bursts in stationary noise

4.2.1

Power spectral density

Ideal detectors will have stationary noise n(t) that can be characterized as a stochastic
process with a given auto-correlation between points in time,rn (τ ) = �n(t)n(t − τ )�.
61(4.11)

The Fourier transform of the auto-correlation gives the two-sided power spectral density, a frequency domain representation of the signal power per unit frequency,
Gn (f ) =� +∞rn (τ )e−2πifτ dτ

(4.12)−∞The power spectral density is the expected energy spectral density per unit time.
Signal energy is introduced in Equation 4.3, but because the noise is infinitely long,it has infinite signal energy unless restricted to a finite amount of time. An alternate
definition of the power spectral density takes the expected energy spectral density
per unit time for a finite interval of duration T ,1
�|ñ(f )|2 �.
T →∞ TGn (f ) = lim(4.13)For real-valued signals, one often encounters the one-sided power spectral density,

Sn (f ) =
2Gn (f ) f ≥ 00,(4.14)

f <0

which is nonzero for non-negative frequencies f ≥ 0 and contains contributions at
each non-negative frequency from both the equal magnitude left (positive frequency)
and right (negative frequency) complex oscillators which make up the real signal.

4.2.2

Matched filter signal-to-noise ratio

The noise power spectrum tells us about the contribution from noise to mean-squared
amplitude fluctuations over some frequency range. A gravitational wave signal must
be above these random fluctuations in order be detectable. In a general detection
scheme, the signal-to-noise ratio refers to the amplitude of the measurement in thepresence of a signal divided by the standard deviation on the measurement due tonoise fluctuations, assuming the noise leads to Gaussian fluctuations. To characterize
the strength of a signal relative to a particular noise spectrum, we use the signal-to62

noise ratio for a linear matched filter,
2

ρ =

� ∞
0

4|h̃(f )|2df.
Sn (f )

(4.15)The signal-to-noise ratio ρ scales with any measure of signal amplitude including theroot-sum-squared amplitude at the detector �h� defined in Equation 4.3. If the noise
spectrum is nearly constant over the frequency range of the signal, ρ takes the simple
form,
ρ2 �

2�h�2
.
Sn (fc )(4.16)The matched filter maximizes the signal-to-noise ratio for the case of a linear realfilter b(t) in data which is the sum of some isolated signal h(t) plus stationary noise:x(t) = h(t) + n(t). The output y(t) is the filter convolved with the detector data,y(t) = b(t) ∗ [h(t) + n(t)](4.17)ỹ(f ) = b̃(f )[h̃(f ) + ñ(f )].

(4.18)

The asterisk ∗ denotes convolution in the time domain,[f ∗ g](t) =

� +∞
−∞

f (τ )g(t − τ ) dτ,(4.19)

which is equivalent to multiplication in the frequency domain,
F [f (t) ∗ g(t)] = F [f (t)]F [g(t)].

1/2(4.20)−1/2

If we multiply the frequency domain filter b̃(f ) by Gn Gn= 1, we can seethat maximizing the signal-to-noise ratio for b(t) ∗ x(t) is the same as maximizing
1/2

the signal-to-noise for the filter b̃� (f ) = b̃(f )Gn (f ) applied to the whitened data
−1/2x̃� = Gn(f )x̃(f ). The signal-to-noise ratio is the magnitude of the filter output−1/2

for filter b� applied to a pure whitened signal Gn
63(f )h̃(f ) at the time of the signal

−1/2

divided by the standard deviation of the filter output for whitened noise Gn
� +∞ �
b (−t)h� (t) dt
−∞
ρ= �
.
� +∞
�
2
|b̃ (f )| df−∞(f )ñ(f ),(4.21)The denominator is constant if we fix a normalization for b� . In that case, the numerator is maximized by choosing b� (−t) ∝ h� (t). In the vector analogy, this maximizes

the dot product of vectors b� (−t) and h� (t) by choosing them to be in the same direc-

tion. In the frequency domain this gives b̃�∗ (f ) ∝ h̃� (f ), which defines the matchedfilter,b̃(f ) ∝h̃∗ (f ).
Gn (f )

(4.22)

We can directly compute the observed signal-to-noise ratio for the matched filter by
expanding 4.21 in the case of a match,
��

� +∞

h̃∗ (f )h̃(f )G−1
n (f ) df−∞
ρ = ��=+∞ ∗−1 (f ) df
(f
)
h̃(f
)G
h̃
n
−∞

+∞
−∞

h̃(f )∗ h̃(f )Gn (f )

�1/2,

(4.23)

which is the same as Equation 4.15.

By thresholding on the amplitude of the filtered output, one can decide if there isa signal present in the noise. For a standard normalized (such that the denominatorin 4.21 is 1) matched filter with a real valued template, contributions to amplitude
measurements from stationary noise alone are Gaussian distributed about zero withunity variance. The best estimate of the true signal-to-noise ratio of a signal present
in noise is then just the amplitude measurement y(tsignal ) itself. The statistical significance of a measurement ρ0 is defined as the negative log probability of observing
a higher apparent signal-to-noise from noise alone,�1
Z = − ln √
2π

� ∞
ρ0

−ρ2 /2e�dρ .

(4.24)In the case of a gravitational wave from a compact binary inspiral, the orbital
dynamics and gravitational radiation may be known, but the phase oﬀset of the64

observed signal at the detector depends on the orientation of the source. Since theinspiral evolution assumes adiabatic orbital decay from a dominant quadrupole mode,
we can make use of the stationary phase approximation to search over arbitrary phaseoﬀset [72, 73]. This requires a complex filter of the form,b(−t) = A(t)eiΦ(t) = A(t) cosΦ( t) + iA(t) sinΦ( t),(4.25)

where the known amplitude envelope A(t) is assumed to vary slowly relative to the
known frequency dΦ(t)/dt. The real and imaginary projections represent two standard matched filters for the waveform with π/2 phase oﬀset. The magnitude of thecomplex projection is the same as the magnitude of a standard matched filter using
the correct phase oﬀset, and the phase oﬀset can be read oﬀ from the phase of thecomplex projection. The expected signal energy from the projected noise, however, istwice the signal energy of the real-valued case as the equal and uncorrelated contributions from real and imaginary components must be added. Therefore the ratio offiltered signal amplitude to root-mean-square noise fluctuation is reduced by a factor√of 1/ 2 due to the use of a complex filter. However, it may be misleading to characterize the signal with a signal-to-noise ratio reduced by the same amount because
the noise fluctuations are no longer Gaussian distributed. Rather the squared noise
fluctuations arise from the sum squared of two Gaussian random variables and arethus exponentially distributed. Instead we can again use statistical significance for ameaningful comparison with other measurements,
Z = − ln� ∞E

χ22 (E) dE = − ln� ∞E

1 −E/2e,
2(4.26)

where E represents the observed signal energy |y(t)|2 . The expected observed energy
for a signal present in noise is related to the intrinsic matched filter signal-to-noise,�E� = �|y(t)|2 � = ρ2 + 2,

(4.27)

where ρ2 is the contribution from the signal projection and 2 comes from the energy
65contribution from both phases of noise. One can then threshold on detected energy
E to decide if a signal is present.
The statistics regarding matched filters presented so far only deal with a single
template at a single point in time. For gravitational-wave searches, we are interested
in searching over a range of times, and most likely a range of many other parameters as
well. This requires the use of a bank of matched filter templates which covers the spaceof signals we want to detect. For continuously varying parameters (time, frequency,
mass, mass ratio, etc.), templates are generally chosen to suﬃciently sample thespace in order to maintain a maximum fractional energy loss due to mismatch for any
particular signal. This ensures near optimal sensitivity to signals at a fixed energy
threshold. One of the advantages of the complex filter used in the stationary phaseapproach is that is makes template spacing in time much easier. Computational cost
places a practical upper limit on the number of templates used. In addition, thenaive false alarm probability from noise fluctuations scales directly with the numberof templates. The increase in false alarms from a dense template bank ignores thelarge overlap between nearby templates, and in practice some sort of ad-hoc downselection is used to isolate loud matches.

4.2.3

Time-frequency decomposition

The space of signals targeted by a burst search is generally too large and poorly
constrained to practically cover with a bank of matched filter templates. Instead of
searching coherently for a single waveform, it is possible to integrate signal energy
over a number of (generally orthogonal) templates in order to search for excess total
signal power over what is expected from stationary noise. Any signal which contains
appreciable signal energy within the chosen time-frequency bounds can be detected
this way.
To isolate a signal in time-frequency, one option is to use the windowed Fourier
66

transform defined for a time series x(t) by the integral,X(τ,φ ) =
=

� +∞

−∞
� +∞

x(t)w(t − τ )e−i2πφt dt(4.28)x̃(f + φ)w̃∗ (f )e+i2πfτ df.

(4.29)−∞The transform projects a time series onto a windowed complex exponential centeredat time τ and at frequency φ. Represented as an integral in time, we recognize theoperation as a Fourier transform of the windowed time series x(t)w(t − τ ). Alternatively the transform can be represented as an integral over all frequencies using
Parseval’s relation. This representation is useful if the shape of the window function
does not vary with time τ but does vary with φ, as is the case for a scale-invariant
wavelet transform. It is important to note the diﬀerence in time-frequency variables
for the data and filter (t, f ) versus those for the transform X(τ,φ ).
The degree of time-frequency isolation provided by the projection depends on the
narrowness of the windows w(t) and w̃∗ (f ). The operation captures the signal energy
which overlaps a tile in time-frequency whose geometry is determined by the window
extent. In 1944, Gabor investigated the use of Gaussian window functions for w(t)for the windowed Fourier transform. Gabor’s filters have minimal time-frequency
volume σt σf (as in 4.10) and thus best isolate signal power in time-frequency. The Q
transform [74] uses approximately Gaussian windowed exponential filters of varying
widths to sample the signal power in a space parametrized by τ , φ, and Q = φ/σf .
At fixed Q, the transform is scale invariant can be described as projections onto asingle function stretched and compressed in time by various amounts.
The windowed Fourier transform and derivatives are highly redundant as they are
by definition evaluated continuously over a bank of templates parametrized by at leastτ and φ. When a signal is thought to contain power over a number of time-frequency
elements, it is desirable to have orthogonal basis elements so that the signal content
can be captured in a small set of coeﬃcients. The Q transform attempts to do thisby a hierarchical method of down selection in which significantly overlapping tiles are
discarded. However because there is no clear delineation between Gaussian windowed67

complex exponential templates, some information can be lost in the down selection
process. The Excess Power pipeline [75] and TFClusters pipeline [76] also make useof the windowed Fourier transform to capture signal energy in time-frequency. Bothpipelines directly achieve an orthogonal basis suitable for clustering by using simple
disjoint rectangular windows in either time or frequency. The disadvantage is thatfor a rectangular window in time, frequency isolation (which takes the form of a sincfunction) is poor and vice versa.
The wavelet transform [77, 78] provides an alternate time-frequency decompositionof a signal onto an orthogonal wavelet basis. The continuous wavelet transform for atime series x(t) is defined by the integral,��
1 ∗ t−τW (τ, s) =dt (τ, s) ∈ R ⊗ R+
x(t) √ ψ
s
s
−∞
� +∞
√
x̃(f ) sψ̃ ∗ (sf )e+i2πfτ df.
=
� +∞(4.30)(4.31)−∞The time series is projected onto scaled versions of a single mother wavelet ψ(t) whichis assumed to be time localized and zero-average. The scale factor s determines theamount of stretching or shrinking of the wavelet which otherwise does not change
in shape. As is the case for the Q transform at constant Q, the windowed Fouriertransform with window size that scales inversely with frequency φ can be considereda special case of the wavelet transform, with the main diﬀerence being that the phaseof the oscillatory component is absolute in time for the Fourier transform. A numberof real-valued wavelets have been discovered which have the unique property thatwhen evaluated on a dyadic grid,1
ψ(j,n) (t) = √ ψ2j�

t − 2j n
2j�(j, n) ∈ Z2 ,(4.32)

they form an orthonormal basis for a real valued signal. The dyadic (discrete) wavelet
transform provides a multiresolution decomposition of the signal with small spacingδτ = 2j n between elements in time and large spacing δf = 2f between elements
in frequency at small scale j (large f ), and correspondingly poor time resolution
68and better frequency resolution at large scale. Alternatively one may continue todecompose the coeﬃcients at a given scale in a wavelet packet decomposition which
results in a rectangular array with constant time and frequency resolution at all
frequencies similar to the fixed-windowed Fourier transform. This decomposition
is used by the Coherent Waveburst pipeline [79]. While the wavelets have larger
time-frequency volume than Gaussian wave-packets, many still provide very goodtime-frequency localization while retaining all the benefits of an orthogonal basis.
Each basis element can be characterized by the parameters defined in section 4.1.
The linear transformations for orthogonal templates are energy-preserving so that the
total signal energy can be reconstructed from adding up the signal energy (squared
coeﬃcients) in the transformed space. Quantities such as central time, frequency
and their second moments can also be reconstructed easily. Alternatively one may
simply keep the values of the few top coeﬃcients which give a good approximation
to the signal in order to retain information about more complicated time-frequency
structure.
Time-frequency decomposition is an eﬀective way to isolate a localized signal fromnoise. Typically data is whitened and normalized prior to any transformation so thatthe noise power spectrum is flat and individual noise samples are uncorrelated. A
projection of noise onto a normalized real template will then be a Gaussian randomvariable. Such is the case for the dyadic wavelet decomposition. The squared magnitude of a windowed Fourier coeﬃcient is exponentially distributed as the sum of two
squared Gaussian random variables corresponding to the real and imaginary phasecomponents. Total noise energy from a cluster of independent coeﬃcients then follows
a χ2 distribution where the number of degrees of freedom is equal to the number ofcoeﬃcients (×2 for complex filters). In general the noise energy from a time-frequency
area A will be χ2 distributed with 2A degrees of freedom [75]. This result follows
most quickly from considering the number of independent samples necessary to represent a signal of duration∆ t and bandwidth∆ f according to the Nyquist criterion.
One can them keep only coeﬃcients or clusters of coeﬃcients whose signal energy isstatistically significant.
69

4.3

Multiple detector observation

One of the generally accepted requirements for a gravitational-wave detection is thatthe event be observed simultaneously in multiple detectors. For this reason, the
LIGO detectors consist of two sites separated by a large distance (3,000 km) in orderto minimize environmental coupling. Coincident observation greatly enhances ourconfidence in an event by dramatically reducing the false alarm probability. It also
serves as the principle means for evaluating the background of a search throughartificial relative time-shifts applied to data from two detectors.

4.3.1

Coincidence in observed parameters

The most basic criterion for coincidence is that the events from multiple detectors
happen at the same time. For two gravitational-wave detectors, the coincidence time
window is determined by the light travel time between detectors (∼10 ms for thetwo LIGO detectors) plus any uncertainty from noise in arrival time estimation forthe search algorithm. Misaligned detectors sensitive to diﬀerent gravitational-wave
polarizations may also contribute to observed timing diﬀerences. The coincidence
window∆ t is generally chosen to be large enough to cover all these eﬀects. If thesingle detector accidental event arrival times can be modeled as a Poisson processwith rate λ and we use a coincidence window ±∆t that is small compared to the time
between events, the coincident rate between two uncorrelated detectors is,
λ12 = 2λ1 λ2 ∆t.

(4.33)The rate decreases rapidly as we add additional detectors so long as λ∆t � 1.
In addition to time, other parameters such as frequency and amplitude estimatedby a search algorithm can be used as coincidence criteria to further reject coincident
noise events. When the inconsistency parameters are approximately independent,
the naive likelihood ratio is particularly eﬀective at distinguishing signal from background [80]. In this Bayesian approach, distributions for n inconsistency parameters70x = (δtc /σt , δfc /σf , etc.) are formed for a large number of simulated signals aswell as sample background. The likelihood ratio for a coincident event is the productof these distributions evaluated at x for simulations (hypothesis H1 ) divided by the
distributions for pure noise (hypothesis H0 ),n
P (x|H1 ) � P (xi |H1 )Λ(x) =
�P (x|H0 ) i=1 P (xi |H0 )4.3.2(4.34)

Coherent observation

Coherent techniques make use of the known response of a network of detectors to fit
a gravitational waveform to data which can then be checked for consistency with the
hypothesis that the data consists of the best fit waveform plus Gaussian noise. A
single measurement of gravitational-wave strain (h+ , h× ) across D detectors can be
represented as a matrix equation,


x1





F1+ F1×





n1



  
      +
 ×
 x2  F2 F2  h+
n 
 =
  +  2
 ..   ..
 .. .. 
 .   .
 . 
.  h×
  

 
+
×
FD FD
xD
nD

(4.35)or in compact notation,
x = Fh + n.

(4.36)

Here x are the measurements from detector data appropriately time-shifted for theassumed source direction of incident gravitational wave h. F(Ω̂) represents the directional antenna factors of each detector in the gravitational-wave frame, and n are
the contributions to x from noise. While the measurements x could in principle be
the measured strain for each detector, we are generally interested in situations where
the noise contributions n are Gaussian random variables with standard deviations
σi . Real detector data is colored with potentially narrow features. To maintain a
Gaussian distribution from noise, the data point must represent some spectral feature, such as a single narrow-band wavelet coeﬃcient, of the data and gravitational
71

wave. We cannot simply whiten the gravitational wave h along with the data for a
network analysis as 4.35 requires it to be the same for all detectors
The likelihood ratio is defined in Equation 4.34 as the probability density of observing the data x across all detectors assuming the signal h is present (H1 ) divided
by the probability density of observing the data assuming h is not present (H0 ).
Assuming the detector noise is uncorrelated, this gives��
1(xi − Fi h)2√
P (x|H1 ) =
,
exp −
2
2σ
2πσ
i
i
i=1
�
�
D
�
x2i
1
√
P (x|H0 ) =exp − 2 .
2σi
2πσi
i=1
D
�

(4.37)(4.38)For N independent measurements (such as a cluster of wavelet coeﬃcients), we canwrite the log likelihood ratio,

L = lnN�
P (x[k]|H1 )P (x[k]|H0 )
k=1

=

N �
D �
�
xi [k]Fi h
k=1 i=1σi [k]2(Fi h)2−2σi [k]2�(4.39)The best-fit waveform (h+ , h× ) and sky location Ω̂ is that which maximizes L [81].
This can be solved analytically for a fixed sky location, but generally requires samplingover a grid of locations in order to find the global maximum. At typical frequencies,
thousands of sky locations must be sampled for each event which causes coherent
methods have high computational cost. For this reason, a coherent stage is usuallytriggered by the computationally cheaper search for coincident excess power.
One problem with fitting to the maximum value of L is that it ignores any priorinformation we may have about the gravitational wave h. This becomes most apparent when dealing with coincident noise transients from a small number of detectors.
In that case, it is not diﬃcult to find arbitrary (h+ , h× ) which manage to cancelout a large portion of the noise. Various types of regularization [82, 83] and otherconstraints [84, 85] have been used to steer the search toward physically plausible
waveforms. A fully Bayesian approach [86] solves the problem by marginalizing overgeneric but physically meaningful signal and noise transient priors.
72

Coherent analyses provide not only excellent network sensitivity to gravitationalwave signals, but also a reconstructed waveform and sky location. The reconstructed
waveform can be subtracted from the data in each detector leaving residuals in thecase of accidental noise coincidences. This can provide a very powerful test for removing transient background [87]. The probability sky maps based on the likelihood
ratio are also useful for coincident searches with electromagnetic (EM) transients aswell as for the prospects of a gravitationally-triggered EM follow up.

4.3.3

Time-shift method of background estimation

The time-shift method of background estimation is critical for the ability of a networkof detectors to be able to detect gravitational waves with confidence. The procedure
relies on the assumption that if an entire analysis is run with data from the various
detectors artificially time-shifted with respect to one another, the resulting events
will be drawn from the same underlying background distribution as for un-shifted
data. Real gravitational waves, however, should be either invisible or negligibly weakcompared to background in the time-shifted analysis. These assumptions requirethat true signal rate is low which is valid for current gravitational-wave searches andis easily verified at the end of the analysis. The time shifts cannot be longer than
the timescale of non-stationarity at each instrument, and to maintain independence
they must be spaced at least as far apart as a typical coincidence window. For
burst searches it is possible to achieve thousands of independent realizations of the
background distribution this way, and thus estimate the underlying distribution to ahigh degree of accuracy.

73

74

Chapter 5
Non-Gaussian noise transients
The matched filter, excess power, and coherent maximum likelihood signal analysis
techniques presented in the previous chapter are all built on the assumption that thedetector noise can be reduced to stationary white Gaussian noise. This means thatthe noise can be completely characterized by its power spectral density, and that any
changes in the underlying power spectrum occur slowly enough so that the spectrum
can still be estimated from the data itself to a high degree of accuracy. The backgrounds present in analyses of real data for gravitational-wave transients, however,are typically dominated by noise transients non-Gaussian in character. These short
instrumental disturbances are caused by a variety of sources including coupling ofenvironmental disturbances to mirror motion saturation of various control signals orother instrumental artifacts.
The presence of noise transients does not necessarily make the Gaussian noise
assumption a poor characterization. A real gravitational-wave signal will most likely
rest on top of nearly stationary noise, so the maximum likelihood approach can,for example, still provide valid sky location and waveform reconstruction. However
the amplitudes of transient disturbances in the gravitational-wave signal from noise
fluctuations can be quite large which corresponds to a vanishingly small probability ofarising from Gaussian noise. Any set of noise events ranked by statistical significanceunder the Gaussian assumption will be dominated by such non-Gaussian transientsat suﬃciently large amplitudes.
75

Coincidence requirements across multiple detectors bring the false alarm probability of a transient search down to manageable levels of O(1) per analysis livetime,and coherent waveform consistency checks are particularly good at rejecting events
which stand out well above the Gaussian noise. Still, accidental coincidences of noisetransients which manage to pass coherence tests are the limiting factor for the sensitivity of current transient searches, and their presence makes it very diﬃcult to assign
a real event a background rate suﬃciently low enough to claim a detection. This andthe following chapters deal with characterizing the noise transient populations in the
instruments, and then using information about transients from the hundreds of auxiliary environmental, diagnostic, and control channels being monitored simultaneouslywith the gravitational-wave channel in order to identify disturbances in the detectors and thus remove potential transient background without using data from thegravitational-wave channel itself.

5.1

Transient population in LIGO data

As we are interested in identifying the noise transients which might be confusedwith gravitational-wave bursts, the methods to find both types of events are very
similar. The time-frequency excess power techniques discussed in subsection 4.2.3
are well suited to the problem as they maintain sensitivity over a very large classof waveforms with time-frequency structure. The parametrization provided by timefrequency methods are also very useful for characterizing the transient population.
The most telling (and problematic) property of noise transients is that their distribution in amplitude falls oﬀ much more slowly than what is expected from Gaussian
fluctuations. This is shown in Figure 5-1 which shows the rate of single interferometer(not coincident) transients in LIGO S5 gravitational-wave data above a certain threshold in signal-to-noise ratio. The transients occur during science data after removing
times representing known problems with the instruments (control channel saturations, calibration line dropouts, large power main glitches, drops in circulating lightin the arm cavities). They are identified by the Q Pipeline [74] which matches each76

Q Pipeline transient amplitude distribution
−2

H1
H2
L1
White noise

10

−3

10

−4

rate [Hz]

10

−5

10

−6

10

−7

10

1

10

2

10

3

10
SNR threshold

4

10

Figure 5-1: Rate of single-interferometer noise transients during the entire S5 run as
a function of signal-to-noise ratio threshold as detected by the Q Pipeline [74]. The
gravitational-wave data from the LIGO instruments during S5 contains populations ofnoise transients well in excess of those expected from stationary white noise (dashedcurve) which for the Q Pipeline are exponentially distributed in SNR2 . The searchcovers the LIGO sensitive band of 64-2048 Hz with 4 < Q < 64. Since these signals do
not survive coincidence requirements across multiple sites beyond what is expectedfrom random coincidences, we know they are overwhelmingly not of astrophysicalorigin.

77

transient to a best fit sine-Gaussian, parametrizing it in amplitude, time, frequency,
and Q, and they cover the LIGO sensitive band of 64-2048 Hz with 4 < Q < 64.
Figure 5-2 shows the population of noise transients in terms of their distribution in
frequency and Q. The number of transients is represented by a fractional excess above
what is expected in ideal stationary noise. The number is slightly diﬀerent for diﬀerent
templates as longer low frequency or high Q templates are spaced further apart andhave correspondingly lower false alarm rates in Gaussian noise. The expected numberof events with signal-to-noise ratio greater than 8 is vanishingly small for Gaussian
noise, so the relative excess for a real population of non-Gaussian noise transients canbe quite large. All instruments show a considerable excess of low-frequency low-Q
transients. These are moderately loud signals (Figure 5-1) around 60-150 Hz whichlast for just a few cycles. Not surprisingly they also dominate the population of
accidental coincidences across multiple instruments.
The average rates for the same population of transients (restricted to the LIGO
sensitive band 64-2048 Hz) is shown in Figure 5-3 as a function of day in S5. The dramatic variation in rates arises from environmental changes such as increased seismic
activity due to storms or earthquakes as well as instrumental changes. The variation in daily rates highlights the fact that not only is the baseline stochastic noise
non-Gaussian and non-stationary, but the transient population is also highly nonstationary. For this reason, care must be taken when interpreting aggregate transient
statistics over a long period of time.

5.2

Examples of transients

The most straightforward type of transient in the gravitational-wave data is caused bya direct coupling of environmental noise into mirror motion. In LIGO’s second sciencerun (S2), a single outstanding event remained at the end of the burst analysis [88].
During follow-up investigations, the source was identified as acoustic coupling froman overhead airplane at the Hanford site. These can couple directly to the mirrors(which do not sit in vacuum) or through the induced ground motion. This relatively
78

7
6
5
4
3

512
256
128

2
1

64

0
4

8

16
Q

32

10
9
1024

8
7
6
5
4
3

Frequency [Hz]

1024

8

H2:LSC−STRAIN clustered triggers SNR > 8
log10(trigger excess relative to white noise)9Frequency [Hz]log10(trigger excess relative to white noise)

H1:LSC−STRAIN clustered triggers SNR > 8
10

512
256
128

2
1

64

0

64

4

8

16
Q

32

64

9
1024

8
7
6
5
4
3

Frequency [Hz]log10(trigger excess relative to white noise)

L1:LSC−STRAIN clustered triggers SNR > 8
10

512
256
128

2
1

64

0
4

8

16
Q

32

64

Figure 5-2: Population of noise transients in the LIGO instruments for S5 as identified
by the Q Pipeline [74]. The counts are represented in terms of a fractional excess ofcounts over what is empirically expected at the same SNR threshold from Gaussian
noise. All instruments exhibit an excess of low frequency, low Q transients (below∼150 Hz), and there is a population of higher Q power transients from nonstationarities in the 60 Hz line as well as mirror suspension resonances. The short-cycle low
frequency transients overlap with the most sensitive frequency band of the LIGO instruments, and their accidental coincidences dominate background rates for transient
searches.

79

Figure 5-3: Rate of transients detected in the gravitational-wave data for S5 with
SNR > 8. The transients are detected by the Q Pipeline [74] and averaged over eachday. Therefore any intra-day variation in rates (which can be very large) are notrepresented here. Rates can vary by orders of magnitude and are aﬀected by changes
in the environment (e.g. storms) as well as changes in the instrument (for example
due to commissioning). The Q Pipeline also makes use of a trigger down-selection
process which artificially limits the rate to 1.0 Hz. The transients shown come fromthe same population as in 5-2 except here they are restricted to 64-2048 Hz.
loud and semi-coherent H1H2 signal was in random coincidence with a weak transientat Livingston. Since S2, additional acoustic isolation of the optics has helped mitigate
this kind of event, though airplanes are still carefully monitored for their eﬀect ondata quality.
The environment can also couple to the mirrors through ground motion transferred through the suspension system, as well as ambient magnetic fields which create
forces on the mirror through small magnets attached to the mirrors for active control.
Through S5, environmental coupling was particularly problematic at the Hanford site
as it caused coincident signals at H1 and H2 with similar spectral properties. Coherent S5 burst analyses in particular relied strongly on waveform consistency between
the two co-aligned Hanford instruments in order to reject background. In S5, themost prominent source of these were magnetically induced events caused by glitches
in the power mains. This caused semi-coherent low-frequency signals in H1 and H2
as well as large disturbances in ambient magnetic fields measured all around the site.
Figure 5-4 shows an example of such an event.
Other sources of instrumental artifacts have been traced back to (among otherthings) scattered light between the two Hanford interferometers which share a com80

Figure 5-4: Semi-coherent signals in the gravitational-wave data at H1 and H2 during
the time of a site-wide magnetic disturbance from a glitch in the power mains. The
disturbance is easily picked up in environmental monitors measuring ambient magnetic fields making this particular source easy to remove from the analysis. The signals
are centered about 60 Hz (with the narrow line generally suppressed by whitening
filters). The magnetometers shown are from the end stations, and similar signals areseen at other locations.

81mon vacuum tube, saturation of digital control signals, transient misalignment of thebeams which cause a loss of circulating light in the resonant cavities, and glitches
in the digital mirror excitation or data acquisition system which happen as a resultof computer load or malfunction. Transients which can be clearly associated witha well understood environmental or instrumental disturbance are typically removed
from analyses through automated procedures. A great many noise transients, however, are not traced back to a specific source. Statistically they may be associatedwith extreme conditions in the instrument such as those created by elevated groundnoise. They may also be associated with specific instrumental disturbances if coincident with transients observed in auxiliary channels not sensitive to gravitational
waves. In such cases their removal must generally be balanced with a non-trivial costin analysis livetime.

5.3

kleineWelle: wavelet-based identification

KleineWelle [89] is a signal analysis pipeline which was developed to find and characterize non-Gaussian transients in an input timeseries. The method consists of thefollowing steps,• Decimate, high-pass, and whiten the data with a linear predictive error filter(LPEF)• Decompose the data onto a dyadic wavelet basis and identify amplitude outliers• Cluster wavelet coeﬃcient outliers based on time-frequency proximity• Rank clusters by the statistical significance of their excess total signal energy
and characterize cluster properties
The procedure is conceptually similar to other methods which search for excess powerin time-frequency introduced in subsection 4.2.3. The method has the advantages of
speed and ease of tuning, which are largely provided by the properties of the dyadic
wavelet transform and scale-invariant clustering. For this reason it is suitable for
rapid processing of large amounts of data.
82

5.3.1

Wavelet decomposition

The wavelet transform 4.30 for timeseries x(t) is defined by the integral,� +∞1W (τ, s) =x(t) √ ψ ∗
s
−∞

�

t−τ
s

�

dt,

(5.1)

where the mother wavelet, ψ, is a time-localized function of zero average. The transform is simply a projection of the series onto stretched and compressed representationsof a single function. The coeﬃcients W (τ, s) are evaluated continuously over times,
τ , and scales, s. Our ability to resolve in time and frequency is then determined by
the properties ψ assumes at each scale. At large scale, ψ is highly dilated yielding
improved frequency resolution at the expense of time resolution. At small scale, we
achieve good time resolution with large uncertainty in frequency.
For the case of discrete data, a computationally eﬃcient algorithm exists forcalculating wavelet coeﬃcients over scales and times that vary as powers of two:s ∈ {2j | j ∈ Z+ }, ∆t ∝ s (Equation 4.32). This is the dyadic wavelet transform,
which can be implemented for a limited family of wavelets using conjugate mirror
filters. The filters consist of a high pass filter, Ĥ, and low pass filter, L̂, which can beapplied in a cascade to obtain the wavelet coeﬃcients. Beginning with the originaltime series, A0 , of length N , two sequences of length N/2 are obtained by applicationof the high pass and low pass filters followed by down-sampling. The sequence ofdetail coeﬃcients, Dj , and approximation coeﬃcients, Aj , are defined at each level,j, of the decomposition by
Dj = Ĥ(Aj−1 ) and(5.2)

Aj = L̂(Aj−1 ).
The detail coeﬃcients for scale s, where s = 2j , calculated in this manner are thesame as the wavelet coeﬃcients obtained from Equation 4.30. If N is a power of two,so that N = 2m , the final approximation sequence will be Am and will contain one
point. The entire decomposition requires O(N ) computations.
83

A0
D1

A1
D2

A2
D3

A3

Figure 5-5: Discrete wavelet decomposition tree using conjugate mirror filters. The
original sequence A0 is decomposed using high pass (left) and low pass (right) conjugate mirror filters and then down-sampled resulting in detail and approximationcoeﬃcients at each successive scale. The decomposition ends for a finite series when
the approximation series is reduced to a single point. The detail coeﬃcients alongwith the final approximation series up to any given scale can be used to reconstruct
the original sequence A0.

The simplest dyadic wavelet is the Haar function [90]:ψ Haar (t) =











1 0 ≤ t < 1/2−1 1/2 ≤ t < 1(5.3)

0 otherwise.

The corresponding high pass and low pass filters are√
Ĥ Haar = [+1, −1]/ 2 and√L̂Haar = [+1, +1]/ 2,(5.4)

from which we see that the detail coeﬃcients are related to the diﬀerences of eachpair of points in the parent series, while the approximation coeﬃcients are related tothe averages of each pair. It is easy to check that for the Haar wavelet, the continuous
wavelet integral 5.1 and the detail coeﬃcients of the discrete dyadic decomposition
5.2 give the same values. It is also easy to check that the dyadic sampling of the Haar
wavelet provides an orthonormal basis. The Haar wavelet is often not the waveletof choice for signal processing applications because of its poor frequency localization.
84

However it has the advantages of being computationally very fast and compact making
it appropriate for current applications of the method.

5.3.2

Data conditioning

Instrumental noise is generally colored and may be populated with many narrow lines(Figure 3-4). KleineWelle makes use of linear predictive filtering [89] to whiten thedata prior to the discrete wavelet decomposition. In linear prediction, the nth sample
of a sequence is modeled by a linear combination of the previous L samples,x̂[n] =

L�

m=1

b[m]x[n − m].

(5.5)

A common choice of coeﬃcients b[m] involves minimizing the expected squared prediction error,
σe2 = E[|x[n] − x̂[n]|2 ].

(5.6)

Solving the least squares minimization for a limited amount of training data involves
estimating the auto-correlation coeﬃcients and solving the Yule Walker matrix equation. Application of this method in the context of gravitational-wave data conditioning is described in detail as a component of the Q Pipeline [74]. By subtracting theprediction sequence x̂[n] from the original data x[n], correlations are removed fromthe baseline stationary stochastic noise, reducing it to white noise. This process iscalled linear predictive error filtering (LPEF). Non-stationary transients which cannot
be predicted remain, but are shaped by the frequency response of the filter.
The LPEF filter length and training length are constrained by the properties ofthe data and parameters of the analysis. Narrow spectral lines in the noise represent
correlations over long time periods, and the filter length must be comparatively long inorder to suppress them. The training length must be long relative to the filter length inorder to allow reliable estimates of the auto-correlation of the data for the longest timedelay. It also must be long enough so that transient non-stationarities of moderate
strength contribute little to the auto-correlation estimates. A high pass prefilter must
85

−18

5

x 10

h 0

−5

0

200

400

600

800

1000

1200

1400

1600

1800

2000

time in s/16384

8

8192
4096

6

2048
1024

4

512
256

2

128
64
1
0
−1

Amplitude / Sigma

Approximate Frequency Resolution

Haar Wavelet Decomposition Coefficients

0
0.02

0.04

0.06
time (s)

0.08

0.1

0.12

Figure 5-6: Decomposition of a σ = 1 ms Gaussian injection with match filter signalto-noise ratio of 11 onto simulated LIGO S2 noise. Shown at top is the signal+noise
after high pass filtering at 50 Hz and LPEF whitening. Shown at bottom are the normalized Haar discrete wavelet decomposition coeﬃcient amplitudes for the whitened
data along with a timeseries of the original unfiltered signal.
be used to remove low frequency content which otherwise cause correlations on thetimescale of the filter length. Finally, an upper limit on filter length and training
length is set by computational considerations and the maximum timescale for which
the assumption of stationarity in the baseline noise is expected to hold.
In the kleineWelle pipeline, data is read sequentially in constant strides of length
N samples, or N/fs seconds where fs is the sampling frequency. The data is first decimated using a low-pass filter along with resampling such that the Nyquist frequency
is above the highest requested frequency of the analysis. The decimated series is thenhigh pass filtered using a 6th order Butterworth IIR high pass filter with cutoﬀfre86

quency at the lowest frequency of the analysis flow . For many channels representing
physical degrees of freedom in the instrument, this also removes the dominant seismic
noise contribution. IIR filter state is preserved across contiguous blocks of data inorder to avoid the initial transient response. In the case of a data discontinuity, atune-able length of initial data is additionally read, filtered, and discarded to remove
the filter transient. The first L < N biased auto-correlation coeﬃcients are calculated
from the high passed data stream and used for training the L linear predictor filter
coeﬃcients. The predicted sequence of data is then removed from the initial highpassed data leaving a whitened data stream. Both the auto-correlation coeﬃcientestimates and FIR filter application can be done eﬃciently in the frequency domain.
The linear predictor filter also keeps a history of the last N data points to avoid filtertransients for the case of continuous data. In the case of a discontinuity, the filtertransient will be at most L samples long and must be discarded as well.

5.3.3

Event generation

The data after high pass and linear predictor error filtering is a series of white noiseof length N . The stride is chosen such that N is a power of two, so that the entire
dyadic wavelet decomposition results in N − 1 detail coeﬃcients W(n,j) with N/2jcoeﬃcients at each scale where,j ∈ {1, 2, . . . , log2 N }n ∈ {0, 1, . . . , N/2j − 1}.

(5.7)

The final approximation coeﬃcient is zero as the data is zero-mean so that the original
sequence can be reconstructed from the N − 1 remaining detail coeﬃcients. At eachscale, the wavelet coeﬃcients can be assumed to be zero-mean Gaussian distributedwith some standard deviation σj . By setting a threshold on the absolute normalized
amplitude |W(j,n) |/σj across all wavelet scales, we build a list of outliers which arecalled black pixels.
A general transient signal isolated in time-frequency will result in a cluster of87scaletime

Figure 5-7: Definition of distance from a wavelet coeﬃcient (in white) to other coefficients in the time-scale decomposition. Tiles with a distance of 1 (light pink) and2 (dark pink) from the original tile in white are shown. A typical distance thresholdused for clustering outliers is 2.

large amplitude coeﬃcients in the index-scaling wavelet tiling. To capture the total
signal energy of such a transient, black pixels nearby in time-frequency are clustered
together to form a single event, which we call a trigger. Clustering is done based ona concept of index-scale distance,d = ∆j + ∆n

(5.8)

where∆ n is based on the time between adjacent coeﬃcients at the higher scale (larger
separation). This definition of distance is symmetric between two wavelet coeﬃcients.
Figure 5-7 illustrates the calculated distance between wavelet coeﬃcients in the dyadic
tiling. A cluster is made up of the largest set of wavelet coeﬃcients such that theminimum distance from any single wavelet coeﬃcient to all others does not exceed
some distance threshold dmax .
The discrete dyadic wavelet decomposition is both orthogonal and energy preserving, so the cluster of N independent wavelet coeﬃcients represents a signal with total
88

signal energy,
Ec =

N
�
i=1

|Ai |2(5.9)

where Ai are the amplitudes of all the normalized wavelet coeﬃcient amplitudes thatmake up the cluster. Since each normalized wavelet coeﬃcient is drawn from a zeromean unity-variance Gaussian random variable given pure stationary noise, the total
signal energy of a random cluster of N coeﬃcients will be χ2 distributed with N
degrees of freedom for stationary noise without the presence of any transient. The
statistical significance of a measure of total signal energy for a trigger is then definedas,Z(E,N ) = − ln� ∞EC

χ2N (E)dE.

(5.10)

which is a function of total signal energy EC and the number of pixels in the cluster
N . The significance represents the unlikelihood of measuring a cluster of coeﬃcientswith greater total signal energy in Gaussian noise. Appendix A provides calculational
details.

5.3.4

Parameter estimation

In addition to the total normalized energy and statistical significance of a trigger, thetime-scale wavelet decomposition provides a general time-frequency parametrizationof the signal. Each wavelet coeﬃcient represents a projection onto a basis element withnormalized amplitude Ai , scale j, index n, and central time t(j,n) = t0 +(n+1/2) 2j /fs .
For a cluster of N outliers, we use the energy weighted central values to parametrize
89the trigger,��start time ts = min ni 2ji /fs
i��
end time te = max (ni + 1) 2ji /fs
i
�
�
�
�
1 �
1
2
ji
2 /fs
|Ai | n +central time tc =
Ec i
2
1 �
|Ai |2 jcentral scale jc =Ec i�total normalized energy Ec =|Ai |2(5.11)(5.12)(5.13)(5.14)(5.15)

inumber of pixels N =

�

1

i

cluster significance Z = − ln

(5.16)� ∞EC

χ2N (E)dE.

(5.17)In addition we define the more interpretable approximate central frequency,
central frequency fc = fs /2jc .

(5.18)

Because the frequency isolation of the dyadic wavelet transform is poor, particularly
when using the Haar transform, we do not attempt to calculate frequency bounds or
bandwidth of the signal. The frequency resolution can be improved by making use of awavelet packet decomposition which no longer uses a multiresolution logarithmic tilingof the time-frequency plane, but a more traditional rectangular tiling or something
in between. Such an approach is used by the Waveburst pipeline for the detection ofgravitational-wave bursts [79].

5.3.5

Implementation

KleineWelle runs under the Data Monitor Tool (DMT) framework [91] which is a
C++ framework for rapid processing of instrumental data and publishing of results.
DMT is responsible for the majority of real-time feedback from instrumental datain the control room as well as the production of noise spectra, trends for transient
90

rates, sensitivity, and other values, and the production of various data-quality flags.
In online mode, the DMT environment provides a steady stream of live data fromthe instrument which is read from a shared memory partition. In oﬄine mode thisstream can be simulated from a series of frame data files on disk. The DMT libraries
also provide the necessary data structures and many of the signal processing routines
needed for standard signal analysis, as well as routines which can publish results
to disk or SQL databases. This keeps the kleineWelle specific code compact andrelatively easy to maintain.
LIGO’s fifth science run (S5) occurred between November 2005 and October 2007.
During this time, the three LIGO instruments collected one full year data when all
three detectors were operating in coincident science data taking mode. The individual
duty cycles were 78% for H1 and H2, and 66% at L1. During this time, kleineWelle
operated in a pseudo-online mode recording transients for a number of environmental
and interferometric channels at each site, including the gravitational-wave channel.
The kleineWelle jobs were launched on fresh data every five minutes under the Onasys
infrastructure [92]. In this environment, jobs were set up when the instrument was
fully locked. The Onasys daemon handled the submission of jobs to LDAS computing
clusters [93] at each LIGO site and monitored their progress. Online triggers wereused for prompt detector characterization as well as a low latency (∼1 day) burst
search for outstanding events.
In parallel to the online production, an oﬄine production of triggers was generatedusing the LDAS computing cluster resources at Caltech. The oﬄine production wasnecessary to cover holes in the online run due to failed services or other technicalissues. Oﬄine triggers were updated every few months with around 99% coverage ofscience data. In total, 201 inteferometric channels and 155 environmental channels
with frequencies of interest between 1 Hz and 4096 Hz (Appendix B) were processed
across the two sites and three interferometers. These oﬄine triggers were used forthe production of event-by-event vetoes for burst and inspiral searches, as well as anyother followup studies concerning auxiliary channel transients. A similar productionof kleineWelle triggers took place at Virgo using the Virgo interferometer V1 and site91

environmental data.
The LIGO instruments are currently in their sixth science run (S6) after a numberof upgrades lead up to the successful commissioning of Enhanced LIGO [71]. For thisrun, kleineWelle was moved to true online production under the DMT infrastructure, continuously reading and processing live data oﬀ shared memory in 32-second
intervals. In total, 339 interferometric channels and 158 environmental channels are
currently being processed across the two sites and interferometers H1 and L1 (H2 isnot being run for S6). At each site, the complete kleineWelle production runs on a single 2.6 GHz AMD Opteron core processing multiple channels in parallel. The memory
footprint is around 1 GB in addition to a shared 1 GB data input buﬀer. Under thecurrent configuration, processing runs at a factor of 2.6 faster than real-time resulting in a maximum latency of around 45 seconds between the time data is taken andtransients written to disk. Approximately 3 million transients are recorded per siteeach day. The transient rate at chosen thresholds for the gravitational-wave channel
alone is ∼0.2 Hz. The low latency for online auxiliary channel trigger production isnecessary in order to quickly identify disturbances in the instruments. There is a current eﬀort to detect gravitational-wave candidates within minutes of their occurrence
in order to trigger a targeted electromagnetic followup [94]. We want to make sure
not to trigger on events which can be easily traced back to non gravitational-wave
eﬀects.

92

Chapter 6
Detector characterization, data
quality, and vetoes
6.1

On-line detector characterization

An important application of online noise transient detection is continuous monitoringof the quality of gravitational wave data and state of the various instrumental subsystems. While stochastic rms noise leads to poorer resolvability and sensitivity to signalsof all types, noise transient rates contribute directly to the background of transient
searches and trace the overall stability of the interferometer. The characterization of
the transient behavior of the instrument is typically done in terms of the transient
rate versus time in the form of a rate trend. For S4, S5, and S6 kleineWelle rate
trends of gravitational wave and auxiliary channels were made available to operatorswith minutes latency to aid in tracking changes to the instruments.
Noise transients in the gravitational wave data itself understandably are subject tothe closest monitoring and follow-up by the LIGO glitch working group [95]. In addition to transient rate versus time, distributions in strength and frequency are checked
periodically. A noisy non-stationary line due to an approximately monochromaticnoise source might not be strong enough to cause concern in the noise spectrum, but
could lead to a significant excess of detected transients around the resonant frequency.
The regularity of trigger times can also point to interesting eﬀects. During S3, ex93

Figure 6-1: Rate of transients vs. time as identified by the kleineWelle pipeline.
The rates are calculated every few minutes and available through a web interface.
Shown here are correlations at L1 between single-detector noise transient rates in
the gravitational wave channel (DARM ERR: error signal for diﬀerential arm length
control) and alignment channel transients from the mirror wavefront sensors.

94

cess triggers were detected exactly at the 1/16 second boundary reflecting glitches inthe data acquisition system. During S5, a dramatic excess of triggers occasionally
showed up at the top of the hour. This was eventually traced back to scheduled
digital snapshots of the detector state whose computation load interfered with thedetector operation. Other regularities can show up in the time between successive
triggers. The LIGO instruments are extremely sensitive to seismic noise in the 0.1-1
Hz range which is caused mostly by weather and ocean waves. Transients are foundat points of extreme mirror motion leading to a preferred time between triggers at
multiples of half the inverse of the micro-seismic noise peak frequencies.

6.2

Data quality flags

Data quality (DQ) flags identify epochs in science data which may have a negative
impact on the analyses due to errors in data acquisition, poor sensitivity, excessive
contribution to the false event rate, or general un-trustworthiness of data. The intervals constructed target known problems with the instrument or environmental conditions. Individual data quality flags are evaluated by their eﬀectiveness at removing
single-interferometer and coincident noise transients, especially large amplitude ones,
from the gravitational-wave data streams. To make sure data quality flags remain
independent of the presence of a true gravitational wave, we check that they are nottriggered by hardware injections where simulated signals are physically injected into
the instrument by diﬀerential actuation of the mirrors.
Data quality flags from individual sources are selected for use by the burst analysis
based on their eﬀectiveness at removing non-Gaussian transients from the data while
minimally eﬀecting the live-time of the search. Each set of flags is tested over singleinterferometer transients found using kleineWelle as well as a sample of time-shifted
triple-coincident background events. Single-interferometer analysis provides the best
statistics and a clear picture of what happens at each instrument, while the timeshifted coincident analysis preferentially targets the sources of background that should
appear in the real search.
95

Figure 6-2: Simple veto logic used by auxiliary channel event-by-event vetoes for S5.
A channel-specific veto window is constructed around the central times of auxiliary
channel triggers above a certain threshold (represented as gray stars on the bottom
time-line). Gravitational-wave channel events are removed (red stars) if their central
time falls within the veto window. The veto procedure removes a small amount oflive-time from the analysis.
To organize the use of data quality flags in transient searches, individual sources
are categorized by the LIGO and Virgo detector characterization groups according
to their severity and our level of understanding [96, 8, 97, 98]. The categories thendetermine how the flags are used in an analysis. A description of these categories andexamples of the sources of disturbances in each for S5 can be found in Table 6.1.

6.3

Event-by-event vetoes for burst background

Event-by-event vetoes attempt to discard individual gravitational-wave (GW) channel
noise events by using information from the many environmental and interferometricauxiliary channels which measure non-GW degrees of freedom. Good vetoes are foundby looking for situations in which a short (∼ms) noise transient in an auxiliary channel
often coincides within a short interval (∼100 ms) with noise transients in the GW
channel. The work, then, is in identifying useful auxiliary channels which are wellcorrelated with noise transients in the GW data, choosing the relevant veto parameters
to use, and finally establishing that the veto procedure will not systematically throwout true gravitational waves.
96

Data quality categories for S5 burst analyses
Category
Description
1
Do not analyze. These flags define the set of data that is processed by search algorithms and exclude unlocked, uncalibrated,or non-science mode times, times aﬀected by data corruption
or missing data, photodiode saturations and calibration line
dropouts which cause extreme signal transients which can affect detector response and power spectral estimates, scattered
light contamination, and the often unstable 30 seconds prior toeach lock loss.
2
Unconditional post-processing cut. These flags are applied unconditionally to events surviving the analysis pipeline. Therefore, events during these times are never considered as
gravitational-wave candidates. Flagged times cover well understood disturbances in the instrument which have a reliable oneto-one correspondence with loud transients and include saturations in the alignment control system, severe impulsive glitches
in the power mains, uncertain calibration, and large glitches inthe thermal compensation system.
3
Poor quality data. Data during these times have poor sensitivity
or excess rates of loud noise transients and are removed from upper limit searches in order to set a better upper limit with morereliable background. Flagged times cover a full 120 seconds prior
to lock loss, noise in power mains, the possibility of cross instrument scattered light contamination from an unlocked Hanford
interferometer, very poor rms noise, severe seismic noise or windspeed, earthquakes and hurricanes.
4
Advisory flags. These flags have not shown a particularly strong
association with noise transients in the gravitational wave channel but cover times when one has special reason to investigate
data quality carefully in the case of a candidate event.
Table 6.1: The organization of data quality flags for LIGO S5 burst searches. The
flags are categorized by our level of understanding and the severity of their impact ontransient analysis. The categorization then determines how the flags are used. Detailin the data quality procedure for transient searches can be found in [96, 8, 97]. A
similar categorization is used for Virgo data quality [98].

97In this way, the event-by-event vetoes complement the data quality flags described
previously. While data quality flags generally identify a stretch of time when background rates are elevated or the data is in some way less trustworthy or problematic,
event-by-event vetoes are meant to specifically target individual transient disturbances in the instrument, so they cover a comparatively short time and are always
applied in post-processing. In their application to LIGO data through S5, vetoes havebeen applied in a very similar fashion to data quality flags where the veto choices areused to create a list of short time intervals covering excluded times. Events whichfall within the vetoed time intervals are removed in post-processing. More complicated veto methods (for example those which use additional information from thegravitational-wave channel) may not be easily representable as a direct list of exclusion segments. We choose to use the simple method of veto intervals and vetoapplication based on a single event time alone so that false dismissal probabilities canbe estimated in a straightforward manner.

The event-by-event vetoes used for the S5 burst search onward are divided into
two categories which follow the same naming convention used for data quality flags(Table 6.1). Category 2 vetoes are a conservative set of vetoes targeting known
electromagnetic and seismic disturbances at the LIGO and Virgo sites such as the
problematic site-wide magnetic disturbances caused by power main glitches at the
Hanford site (Figure 5-4). These are identified by requiring a coincident observationof an environmental disturbance across several channels at a particular site.

Category 3 vetoes make use of all available auxiliary channels shown not to respondto gravitational waves. While the physical source of a particular disturbance may
not be well understood or the coupling between the gravitational-wave channel andauxiliary channel poorly modeled, the presence of a transient in an auxiliary channel
not sensitive to gravitational waves can still be good reason to reject a transient in
the gravitational-wave channel if a strong correlation between noise transients form
the two channels can be made.
98

6.3.1

Veto eﬀectiveness metrics

A collection of non-overlapping veto time intervals (segments) is judged by its effectiveness at flagging gravitational-wave channel noise transients. This will dependon the live-time under consideration and the particular set of noise transients being
tested. Given a set of veto segments and noise transients we define the following
parameters:
Veto eﬃciency is the fraction of noise transients from the sample which are removed by our veto method. We use a simple veto logic where an event is vetoed if
its central time falls within a particular veto interval. Veto eﬃciency is dependent on
the initial sample of noise transients being tested.
Dead-time fraction is the fraction of live-time flagged by all the veto intervals.
Assuming that real gravitational-wave events are randomly distributed in time, deadtime fraction represents the probability of vetoing a true gravitational-wave eventby chance. For a realistic population of sources, detected gravitational-wave eventsare likely to scale with the sensitivity of the instrument which is non-stationary.
Still, dead-time fraction is a useful population-independent proxy for false dismissal
probability and is easy to calculate. If the veto segments are completely uncorrelatedwith the noise transients, measured veto eﬃciency should approach the dead-time
fraction. A veto eﬃciency greater than the dead-time fraction indicates a correlation
between the triggers and veto segments.
Veto significance reflects the level of statistical significance of a measurement ofexcess veto eﬃciency under the assumption of uncorrelated veto segments and noisetransients. Under this assumption, the number of events that fall within the flagged
dead-time is Poisson distributed with mean value µ equal to the number of noise
events times the fractional dead-time, or equivalently, the noise event rate times theduration of veto segments. We define the statistical significance of actually observing
N vetoed events as,
Z(N,µ) = − ln [PPoiss (x ≥ N |µ)].
Appendix A provides calculational details.
99

(6.1)

Eﬃciency/dead-time provides a measure of the performance of a set of veto segments. The ratio represents the relative excess rate of noise transients within theveto segments to those outside. All things equal, we want to remove the times corresponding to the highest noise transient rates first. Significance reflects our confidence
that the measured performance is not simply a statistical fluctuation. A veto withlow eﬃciency/dead-time of just a small excess over 1.0 may still be considered highly
significant if a large enough test sample is used. Similarly a high eﬃciency/dead-timemeasurement may be a result of low number statistical fluctuations if the test sample
is too small. Requiring a level of statistical significance corresponding to the numberof investigations being done helps protect against introducing random vetoes.

6.3.2

Veto safety

Veto safety deals with the possibility that a veto might systematically reject true
gravitational waves. Although the many auxiliary channels measuring test points
in the instrument and environment are not supposed to be sensitive to a passing
gravitational wave, we cannot assume that a strong gravitational wave would notleave a signature in channels other than those in the diﬀerential arm motion control
loop. The source of the coupling could be from mis-tuning of the interferometer,or something as mundane as electronic pickup. In order to explicitly check for an
auxiliary channel response, hardware signal injections simulating the passage of areal gravitational wave by actuating on the end mirrors using magnets are performedin each instrument at scheduled times throughout each run. The impulsive, burst-like
injections span the entire sensitive frequency range of the interferometer as well as avery large range of amplitudes including very loud events.
Each auxiliary channel is independently checked for veto safety by determining if it
has any measurable response to hardware simulations. First, transients are identified
in the auxiliary channel using kleineWelle as described in section 5.3). A set of vetosegments is formed by taking fixed ±100 ms time windows about the auxiliary channel
triggers at some threshold on trigger significance as in Figure 6-2. The dead-timeof the veto segments are evaluated for a local region of time surrounding transient100hardware injections. If there is no correlation between hardware injected signals andthe auxiliary channel, we expect the number of hardware injection times which fallwithin the veto intervals by chance to follow a Poisson distribution with mean value
equal to (dead-time fraction × number of injections).
The veto significance of the actual number of hardware injection times flagged
determines whether or not there is a measurable correlation between the injections
and auxiliary channel transients at the chosen threshold. In a long run such as S5,
there are thousands of injections and it only takes a small correlation to be significant.
To ensure veto safety, we require the threshold for auxiliary channel triggers to belarge enough such that at the chosen threshold and all higher thresholds there is no
significant correlation between the resulting veto segments and hardware injection
times to the 1% level.

6.3.3

Selection of veto conditions

Category 2 vetoes
Category 2 vetoes are applied unconditionally to all burst searches so they are chosenfrom the most conservative and best modeled instrumental disturbances. For S5, thisincluded seismic and magnetic transients with known coupling to the mirror motion.
Transients from each environmental channel were correlated with sample backgroundtransients in the gravitational-wave data, and veto windows and thresholds were tuned
in order to cover eﬃciently the observed overlap. Three classes of environmental
channels were adopted as vetoes. For LHO this included transients recorded in 24
magnetometers and voltmeters with a kleineWelle significance threshold of 200 andtime window of 100 ms, and 32 accelerometers and seismometers with a threshold on
the kleineWelle significance of 100 and a time window of 200 ms. For LLO these were
12 magnetometers and voltmeters with a kleineWelle threshold of 200 and a time
window of 100 ms.
The environmental disturbances which caused the most reliable response from theinterferometers were generally large enough to be detected site-wide and therefore
101were picked up by multiple channels of a given class. This allows us to reduce the deadtime and increase the reliability of the chosen vetoes by requiring coincidence acrossseveral channels in a given class before generating a veto for the gravitational-wave
channel. The coincidence step keeps genuine site-wide environmental disturbances
while greatly suppressing the contribution from noisy non-stationary channels. A
veto segment was only created from the window overlap of three or more channels
from a particular environmental class. This procedure removed in total about 0.1%
of coincident S5 live-time.
Category 3 vetoes
Category 3 vetoes attempt to eﬃciently clean the data as much as possible fromcontributions to transient background and attempt to make use of all the measured
transient behavior in the instrument. For S5 this includes kleineWelle triggers generated over approximately 200 interferometric channels and 150 environmental channels(subsection 5.3.5) which have been determined to be safe according to the procedure
outlined in subsection 6.3.2. In many cases the nature of the transient coupling
between channels is poorly known and the causal relationships between coincident
transient observations are not established. However statistical correlations between
noise transients in auxiliary channels and the gravitational wave channel can still be
used to construct vetoes which are eﬀective at systematically flagging background.
The vetoes constructed may also flag many times unrelated to actual disturbances in
the gravitational-wave data and for this reason they are not used as an unconditional
veto in a detection search. Their application does however improve the upper limit
by allowing for lower thresholds at the same false-alarm probability and the vetosegments themselves are useful for establishing confidence in the case of a detection
follow-up.
The task of constructing vetoes from the auxiliary channel transients is complicated by the desire to choose optimal veto frequency ranges, thresholds and windows,
and the fact that the veto channels themselves can be highly correlated with eachother so that applying one veto channel changes the incremental cost (in additional
102




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
















Figure 6-3: Procedure used for ordering a list of veto conditions from best to worst interms of channel, frequency range, window, and threshold. A set of vetoes can thenbe chosen for any choice of dead-time threshold by choosing the corresponding top
ranked vetoes. The particular method of refining the list order over several iterations
is chosen because it minimizes the number of steps which must be done in serial andtherefore is easy to implement on a computing cluster.

103

dead-time) and benefit (in additional veto eﬃciency) of applying another. Applying
all vetoes which perform well by themselves often leads to an ineﬃcient use of deadtime as dead-time continues to accumulate while the same noise events are vetoedover and over.
For a particular set of GW channel noise events, we adopt a hierarchical approach
to choose the best subset of all possible veto conditions to use for a target dead-time.
This amounts to finding an ordering of veto conditions (veto channel, frequency range,threshold, and window) from best to worst such that the desired set of veto conditions
can be made by accumulating from the top veto conditions so long as the dead-time
does not exceed our limit, which is typically a few percent.
We begin with an approximately ordered list based on the performance of each vetocondition (frequency range, channel, window, and threshold) considered separately.
Incremental veto statistics are calculated for the entire list of conditions using theavailable ordering. This means that for a given veto condition, statistics are nolonger calculated over the entire S5 live-time, but only over the fraction of live-timethat remains after all veto conditions earlier in the list have been applied. The list
is then re-sorted according to the incremental performance metric and the process isrepeated until further iterations yield a negligible change in ordering.
The ratio of incremental veto eﬃciency to incremental dead-time is used as aperformance metric to sort veto conditions. This ratio gives the factor by which therate of noise events inside the veto segments exceeds the average rate. By adopting
veto conditions with the largest incremental eﬃciency/dead-time ratio, we maximize
total eﬃciency for a target dead-time. We also set a threshold of probability P <0.001 on veto significance (not to be confused with the significance of the triggers
themselves). This is particularly important for low-number statistics when large
eﬃciency/dead-time ratios can occasionally result from a perfectly random process.
Vetoes were optimized over several diﬀerent sets of gravitational-wave channel
noise events including low-threshold H1H2L1 coherent WaveBurst time-shifted events,
H1H2 coherent WaveBurst playground events (not used in the final analysis), as well
as Q Pipeline and KleineWelle single-interferometer triggers. For example, the eﬀect104

Hierarchical veto tuning on cWB events

60

0.12
0.10
0.08
0.06
0.04

removed by Cat2 environmental vetoes
removed by Cat3 data quality flags
removed by Cat3 hierarchical vetoes
surviving all data quality and vetoes

40

30

20

10

0.02
0.00
0.000

Data Quality and vetoes on cWB outliers

50

Time-shifted events

Fraction of events vetoed

0.14

0.005

0.010

0.015

0.020

0.025

Fraction of livetime removed

0
4.5

0.030

5.0

5.5

6.0

6.5

7.0

Coherent network amplitude

7.5

Figure 6-4: Left: Accumulated veto eﬃciency versus dead-time as vetoes are applied
cumulatively down the veto list to coherent WaveBurst time-shift H1H2L1 background
during the first calendar year of S5. The best vetoes are applied first, so we see
a general decrease in the eﬀectiveness of vetoes at higher dead-time. Vetoes fromenvironmental channels are artificially prioritized over interferometric channels, giving
rise to the knee in the plot around 0.8% deadtime where the environmental vetoes
are exhausted. Right: Histogram of coherent network amplitude, η, for coherent
WaveBurst time-shift background events representing 100 S5 year 1 live-times. The
diﬀerent shades show events removed by data quality cuts and vetoes at various stages
in the analysis.
of the S5 data quality flags and event-by-event vetoes on the sample of coherent WaveBurst time-shifted events is shown in Fig. 6-4. A final list of veto segments to apply to
the S5 upper limit analysis was generated from the union of these individually-tuned
lists.

105

106

Chapter 7
Candidate event follow-up
To minimize bias, existing searches for gravitational waves make use of a blind procedure to tune all parameters of the search using Monte Carlo and oﬀ-source data
before actually running once on on-source data. For burst searches, Monte Carlo
methods involve running over simulated ad-hoc signals added to the real data. The
signals simulate gravitational waves which sample in time, amplitude, frequency andmorphology the region of interest. Oﬀ-source data to estimate background is generated for a multi-site search by artificially time-shifting the data stream from one site
thereby destroying the coherence of real gravitational waves while ideally preserving
the noise properties of the on-source (unshifted) data (subsection 4.3.3).
To implement such a procedure eﬃciently, all aspects of the analysis must be automated so that the on-source analysis can proceed without any additional human input.
The choice of veto conditions (section 6.3), for example, is made using backgrounddominated single-interferometer transients or time-shift sample background only. The
decision about whether or not a particular event in the on-source analysis can be dismissed as an instrumental artifact is then set by predetermined rules. Existing burst
searches use information about the non-stationary noise power spectrum, transient
background rates and their distributions in strength and frequency, instrumental andenvironmental disturbances picked up in auxiliary channels, and consistency between
the waveforms seen in multiple detectors to separate signals from background. Constructing automated methods of folding this information into the search is a time107

consuming process but allows for an objective estimate of the false-alarm probability
associated with a given measurement critical for any detection statement. It also
allows for the unbiased measurements necessary for proper coverage in an upper limit
calculation. While the need for rapid online results has eroded at the concept of total
blindness for any subsequent analysis, the methodology of automated blind analysis
has not changed.
Events which stand out in the blind analysis are subject to a follow-up procedure[99] which provides a more careful subjective evaluation of the wealth of information
relating to the specific event. The follow-up procedure is designed to satisfy thefollowing functions:• provide a central resource for basic information about the event• serves as a careful secondary review of the specifics of the end-to-end analysis
that relate to the event• provide an opportunity to check for obvious reasons to dismiss a candidate event(e.g. clear environmental cause) or increase our interest in an event (e.g. opticalcounterpart)• outline specific event details which may be tangentially related to interpretation
as a gravitational wave candidate (e.g. single-interferometer signal properties)• provide a subjective but detailed evaluation of information which may enter
only crudely (e.g. in the form of hard cuts) in the blind analysis• provide a test-bed for new ideas which have not been able to make their way
into the quantitative detection statistic
Most importantly the follow-up procedure increases our confidence in the validity of
the analysis with respect to a specific event outlier by checking the sanity of the manydecisions which factor into the automated procedure. This is particularly relevant for
gravitational waves where no detection has yet been made.
108

7.1

Detection checklist

The follow-up procedure for S5 burst searches was implemented in the form of adetection checklist which was to be applied to an event from the blind analysis that
had any chance of being reported as a detection. The first time such a checklist was
implemented for LIGO burst analysis was during S2 [88] whose search yielded a single
H1H2L1 coincident event which narrowly survived all cuts used for the upper limit
analysis. These cuts were set to yield a background expectation of ∼5%, or 1.8 events
per year so that an event from background was unlikely but not too surprising. In
follow-up, the event was unambiguously linked to the acoustic coupling of an overhead
airplane into the mirrors at both Hanford interferometers, thus disqualifying it as agravitational wave candidate.
The second time the follow-up procedure was exercised was during the end of
LIGO’s fifth science run. The second calendar year of S5 was analyzed jointly with
Virgo’s first science run (VSR1) [8]. A single outlier was identified which passed all
upper limit cuts, except this time no reason was readily found to clearly reject theevent as non-astrophysical in origin. The result was an expanded and more thoroughfollow-up procedure from that applied to the S2 event. In final form, the detection
checklist for S5/VSR1 (Appendix D) consisted of an 80-point list of tests covering
the following categories:• zero-level sanity: reports in detector logs, check hardware injections• data integrity: frame file check-sum, undocumented injections, check against
raw frames• state of the instrument: obvious disturbances reflected in auxiliary channels,
verify coupling for any proposed veto, check by hand against known disturbances: dust, cosmic rays, power fluctuations, acoustic, etc.
• event properties: construct detailed spectrogram, reconstructed waveform anddirection, compare background from various methods, check signal consistency
across interferometers109• astrophysical interpretation: check for external EM or neutrino events, catalog sources consistent with reconstructed direction, compare waveform against
simulations

Ultimately it was decided that while the event passed all follow-up tests, the expectedbackground rate in the blind analysis for events of similar quality (between 1–10%
depending on the comprehensiveness of the search) was too high to report a detection.
Afterward it was revealed that the source of the event was a blind hardware injection
designed to test the end-to-end analysis pipeline.

7.2

Blind injections during S5

Beginning with S5, the burst and inspiral analysis teams were subject to a blind
injection challenge. In this test, a small random number (possibly zero) of simulated
events are coherently injected into the LIGO interferometers. The data analysis teams
know of the possibility of an injection but not whether or not an injection actuallyoccurres nor anything about the morphology of the waveform except that it should
be within the target parameters of the search. Only after the completion of the entire
analysis, including interpretation of any outliers, are the details of any blind injections
revealed. In this way, the blind injection serves as a test for the end-to-end procedures
involved in making a detection claim.
The event identified by the burst analysis was the only blind burst hardware
injection during S5. There was also a blind inspiral injection which was missed due
to data quality issues [100]. The blind injection provided a special opportunity to
exercise the follow-up procedure without the bias introduced from looking at a known
simulation. The following section summarizes the investigations of the event and was
prepared prior to the revelation of the event as a simulation. It has been edited only
briefly for clarity in order to maintain the unique perspective.
110

7.3

Follow-up of S5/VSR1 burst outlier

7.3.1

Introduction

Burst candidate event 070922 (September 9, 2007) was first observed in the burst
group KleineWelle-CorrPower S5 online analysis for H1L1, and reported at the next
weekly teleconference the burst group holds. The KleineWelle-CorrPower infrastructure consisted of single-detector kleineWelle (section 5.3) triggers generated online at
the sites, with a centralized daily coincidence and CorrPower [101] follow-up. The
H1L1 event stood out clearly above the expected 1-day background in CorrPower
distributions.
The September 22, 2007 event was quickly verified by other burst methods, Astroburst/BlockNormal [102] and Q Pipeline [74] which also generated online triggers.
KleineWelle and Astroburst recorded signals in H1 and L1, while Q Pipeline showed
an additional weak signal present in H2, visible in Q-scans. Astroburst was looking only for triple-coincident events throughout the run (none were found), while no
online coincidence was performed for Q Pipeline triggers. Coherent Event Display[103], which generally presents details of Coherent Waveburst events, showed the signals to be very consistent across the three detectors, and provided the first position
and waveform reconstruction of a possible gravitational wave.
The burst group resurrected a detection checklist, unused since the analysis of
the S2 data. A large number of items were added to investigate the event in detail,and the checklist would serve as a model for any future candidate events. Items wereassigned to relevant group members, and the checklist was gradually completed overthe course of a couple months. However it was decided that the event, appearing
essentially at the end of S5/VSR1, could only be interpreted properly in the context
of a complete S5/VSR1 analysis. Thus the S5 second year all-sky burst search, which
was finished over a year later, became a prerequisite for bringing the event to a close.
The September 22 event eventually became known in the burst group as the equinox
event due to its anticipation of the 2007 Autumn equinox.
111detector
H1
H2
L1

GPS timef874465554.7158 96.8 Hz
874465554.7119 110.9 Hz
874465554.7100 118.3 Hz

Q

SNR

4.7
22.6
4.7

11.8
5.4
11.3

Table 7.1: Basic information about the candidate event 090722. The time, frequency,
and Q represent the best-match sine-Gaussian as determined by the Q Pipeline [74,
104].

7.3.2

The event

Basic information
The equinox event occurred on Saturday September 22, 2007 03:05:40.71 UTC (Friday September 21 20:05 PDT, 22:05 CDT, 874465554.71 GPS). Around 100 Hz, theevent was a low-frequency, low-Q event similar to common transients seen in the
instruments. A basic table of Q Pipeline [74, 104] derived parameters follows. The
hrss , about 3 × 10−22 Hz−1/2 , is not estimated reliably by the Q Pipeline as it fits the
waveform in whitened space.

The measured signal
Figure 7-1 shows whitened plots of the uncalibrated signal from the gravitational-wave
channel LSC-DARM ERR for each instrument over 60–2048 Hz and a restricted 60–
140 Hz band. The signal, clearly visible in H1 and L1, is low Q with only a couple
cycles.
Figure 7-2 shows time-frequency spectrograms of the calibrated data. The spectrograms are created using the Q Pipeline basis at fixed Q. The space is oversampled
in time and frequency corresponding to low energy mismatch between tiles. Spectrograms made in this way are known as Q-scans [104]. Although the Q-scan shownfor H2 is generated for Q=4.2 for direct comparison to the H1 and L1 Q-scans, themost significant projection onto a minimal-uncertainty waveform in H2 has a Q in
the mid-20s.
112

Figure 7-1: Whitened plots of the uncalibrated signal LSC-DARM ERR (diﬀerential arm length error signal) for each instrument are shown over 60–2048 Hz and a
restricted 60–140 Hz band.
H1-H2 consistency
Figure 7-3 shows various combinations of calibrated H1 and H2 strain data which test
waveform consistency between the co-located, aligned detectors. The first plot is thecoherent sum (H1/SH1 )+(H2/SH2 ). This combination is designed to maximize signalto-noise for a true gravitational wave and shows a signal which is more significant thanpurely H1 alone. From the second plot of the H2−H1 null stream, we see that H2
is not completely consistent with H1, and there is a clear signal preceding the event
time at the same frequency present in the H2−H1 residual. However the secondary
H2 excess at the time of the event is removed by the subtraction. Strangely, thepreceding signal in H2 shows a larger SNR in H1−H2 than in H2 alone. The third
plot shows the incoherent H1/H2 ‘null’ stream [96] which represents what we would
expect to see in the null stream for randomly correlated signals of the same amplitude.
The incoherent null stream shows contributions from both the main event and thepreceding additional signal from H2, and we see that the actual H1−H2 null stream
113

Figure 7-2: Spectrograms of LIGO calibrated strain√ data for each instrument. Normalized energy Z is related to signal-to-noise ρ = 2Z. Note the diﬀerent color scales
in each plot, and the much weaker signal in H2. The spectrograms are generated overthe Q Pipeline tiling [74, 104].

is much weaker than expected from random signals.
We can also check where the H1H2 event falls in the space of correlated vs. coherent normalized energy. The coherent energy represents the coherent sum (first plot in7-3) which maximizes signal-to-noise assuming a common waveform. The correlated
energy is the signal energy present in the coherent sum minus that expected fromrandomly correlated H1 and H2 waveforms of the same strength. The correlated energy goes negative for anti-correlated signals. The left plot shows the equinox eventin the context of other time-shift and unshifted H1H2 triggers in the S5 second-year
search. It also shows where the simulated burst signals lie. The right plot shows theevent in the context of simulations of the same frequency, Q, and amplitude.
114

Figure 7-3: Consistency of the measured strain waveforms in the two co-locatedaligned Hanford detectors. The first is a spectrogram of the frequency-dependent
optimal combination (H1/SH1 )+(H2/SH2 ) designed to maximize signal-to-noise of
the combined signal. The second is the H2−H1 null stream which for calibrated
data should leave no gravitational-wave signal. The third spectrogram shows theincoherent H1/H2 ‘null’ stream which shows what we would expect to see in the nullstream for randomly correlated signals of the same amplitude.

7.3.3

State of the instruments

Detector state
All five LSC/Virgo detectors were in the middle of long stretches of science mode
(Table 7.2), with no major anomalies or data quality issues.
Hardware injections
Hardware burst injections were scheduled just six minutes after the event, beginningat 874465959 (03:12:25 UTC). These show up as yellow bars in the Q Pipeline plots
115

Figure 7-4: Q Pipeline correlated vs coherent energy in the context of expectedbackground (left) and simulated signals (right). The left plot shows the burst event(equinox event) as well as an earlier semi-blind inspiral injection. The equinox event
falls within the expected parameters for both all burst simulated signals (green bar)and specifically the closet match sine-Gaussian simulation (blue cloud). Black dotson the left plot show the population of sample time-shift background.

detector
H1
H2
L1
G1
V1

state

start time

relativestop time

relative

Science Mode 874438904
Science Mode 874441095
Science Mode 874452909
Science Mode 874453140
Science Mode 874449546-26650-24459-12645-12414-16008874501515
874478798
874488229
874479600
874547216+35961+13244+22675+14046+81662

Table 7.2: State of the LIGO, GEO600, and Virgo instruments during the time ofthe detected event. It is remarkable that all five instruments were operating in long
stretches of science mode.

116

shown in section 7.3.3. The burst hardware injections were identical strain waveforms
injected in coincidence across the three LIGO instruments. Weak persistent pulsar
hardware injections were also running at the time of the event. The persistent injections can occasionally cause transients if they suddenly turn on or turn oﬀ causing a
discontinuity, but do not appear related to this particular event.
Analysis of excitation channels by kleineWelle (excluding the blind injection channel) shows that the signal was not accidentally injected into any of the LIGO instruments else they would have shown up clearly in the excitation channels themselves.
Calibration
The event is seen both in methods which analyze uncalibrated gravitational-wave data
(kleineWelle and Astroburst), as well as methods which use calibrated h(t) strain data
(Q/Ω Pipeline and Coherent Waveburst). There are no artifacts in the calibration
channels or problems with calibration at the time of the event.
Single-detector noise transients
A plot (Figure 7-5) of the non-overlapping Q triggers within ±8 seconds of the eventshows weak broad-band transients in the H1 detector. No explanation of these transients has been found.
Q pipeline triggers for each detector are plotted in rate trends covering the hourof the event (Figure 7-6). The event occurs six minutes from the beginning on theplot, minutes before the hardware injections which are marked by the yellow bar. The
broad-band transients in H1 seen in the ±8 seconds surrounding the event show up
in the sixth point from the left as a substantial increase in these un-clustered trigger
rates. The H1 excess is not seen in clustered rates which only represent the loudest
trigger in each second. H2 and L1 rates are flat and quiet.
The strength distribution of the triggers is reflected in scatter plots (Figure 7-7)
with weak blue, moderate green, and loud red triggers. The yellow bars containing
many loud triggers mark a set of burst hardware injections. The same events areplotted in frequency-vs-time in Figure 7-8.
117

Figure 7-5: Plots showing Q Pipeline single-interferometer non-overlapping triggers
in time-frequency space in ±8 s of the event time. H1 shows a large number of weakertransients in the immediate neighborhood of the event.

In these plots, the equinox event shows up clearly as the first trigger in green for
H1 and L1. For H2, the event with an SNR of 5.4 cannot be resolved in this plot. We
can also see the series of small transients in H1 that accompany it. Clusters of triggers
show up later in the hour for H1, though the frequency of the strongest trigger is nolonger at 100 Hz, For L1, however, there are a couple more green triggers which show
up at 100 Hz throughout the hour, and many throughout the day. This population is
not particularly uncommon for that detector, though usually there are more strong
lower-frequency triggers as well. The event in H2 is the third blue dot from the left
in the horizontal line of triggers just below 128 Hz.
118

Figure 7-6: Q Pipeline trigger rate vs. time during the hour of the event which occurs
at 3:06 UTC. The rate is for non-overlapping (in time-frequency) triggers with SNR> 5 prior to 1 Hz down-selection. The event occurs during an increase in H1 trigger
rate. The yellow bar marks a series of hardware injections.
Data Quality and vetoes
There are no active data quality flags in any of the instruments at the time of theevent. Vetoes were checked by looking for nearby kleineWelle auxiliary channel triggers, as well as looking at full-frame Q-scans at the time of the event. While many
auxiliary channels are not particularly quiet, there is no indication of a coincident
transient observed in correlation with the gravitational-wave signals. There is no
signal observed in AS I for any of the interferometers.
Auxiliary kleineWelle triggers found within [874465553,874465556) and with kleineWellesignificance above 35 (SNR 15). Peak time is with respect to the equinox event peak
GPS time, 874465554.710 . The times of the events are ordered and are in seconds.
119Figure 7-7: Q Pipeline trigger SNR vs. time during the hour of the event which
occurs at 3:06 UTC. The event is seen as the first green dot in the H1 and L1 scatterplots. The yellow bar marks a series of hardware injections.

Frequency is in Hz.
At −200 ms there is also a 2.6 Hz excursion in H0:PEM-BSC3 ACCX, an accelerometer sitting on one of the optics tables, which was followed up by the burst
and glitch groups. However this channel was noted in the detector e-log as having excess noise inconsistent with other accelerometers for a month surrounding theevent, and Q-scans of random times (as opposed to times triggered by a gravitationalwave channel noise transient) often showed coincident disturbances in the channel.
None of the environmental channels at Hanford show a signal which would accountfor that measured in the gravitational-wave channels using the known environmental
couplings.
The DMT monitor PdNMon checks to see if the recorded signal was the same in120

IFO Channel
h0 coilmagx
h1 mcfl0 coilmagz
l0 coilmagx
h0 coilmagx
l1 mcfhifreq
h1 etmxexcdaq
h2 etmxexcdaq
l1 etmxcal
h1 mcfl1 etmxexcdaq
h1 asach1 reflih1 asach1 asach1 refldc
h1 mclh1 pobih1 prcctrl
h1 reflih0 coilmagx
h1 asach1 mclh1 reflih1 asach1 mclh1 michctrl
h1 pobq
h1 refldc
h1 refli
h1 prcctrl
h1 pobih1 reflq

Time [s] Dur. [s]-1.651 +- 0.001-1.577 +- 0.003-0.991 +- 0.003-0.991 +- 0.002-0.944 +- 0.004-0.831 +- 0.000-0.784 +- 0.003-0.784 +- 0.003
0.009 +- 0.004
0.071 +- 0.003
0.091 +- 0.009
0.202 +- 0.001
0.208 +- 0.002
0.616 +- 0.001
0.665 +- 0.001
0.657 +- 0.002
0.657 +- 0.002
0.673 +- 0.004
0.665 +- 0.004
0.665 +- 0.003
0.767 +- 0.004
0.759 +- 0.001
0.759 +- 0.004
0.755 +- 0.002
0.837 +- 0.009
0.837 +- 0.008
0.868 +- 0.014
0.852 +- 0.011
0.837 +- 0.009
0.868 +- 0.010
0.868 +- 0.014
0.868 +- 0.013
0.868 +- 0.013

Freq [Hz]
488.0
402.0
208.0
420.0
136.0
1686.0
645.0
565.0
395.0
274.0
380.0
568.0
284.0
658.0
411.0
221.0
359.0
263.0
179.0
219.0
373.0
652.0
220.0
443.0
376.0
131.0
125.0
161.0
131.0
186.0
125.0
135.0
78.0

Significance
80.68
76.40
52.22
56.10
49.35
37.27
36.21
35.95
37.69
115.87
38.39
54.55
40.04
90.96
189.0041.76
126.0036.63
38.33
132.23701.90195.4048.90
55.73
318.76
223.63
88.52
95.29
96.40
197.75
61.63
65.71
64.74

Table 7.3: List of auxiliary channel triggers near the time of the event with kleineWellesignificance above 35 (SNR � 15). The peak time is displayed relative to the event
peak time. None of the detected transients triggered a veto for the event under thepredetermined veto choices.

121

Figure 7-8: Scatter plot of Q Pipeline triggers in frequency vs. time. Green dots have
SNR > 10 and red dots have SNR > 20. The yellow bar marks a series of hardware
injections.
all four photodiodes at each instrument, useful for distinguishing transients causedby dust particles. There were no PdNMon triggers at the time of the event.

7.3.4

All-sky search and background estimates

KleineWelle+CorrPower online analysis
Figure 7-9 shows unshifted and time-shift background distributions estimated from100 time-shifts for the H1L1 high threshold kleineWelle+CorrPower online analysis
over the entire S5 run. The second plot includes a rudimentary H1H2 null-stream
veto based on Q/Ω Pipeline triggers, and is necessarily only on triple-coincident livetime. The equinox event is the loudest event at zero time-shift with Γ = 9.97, andthe expected background for the analysis above this Γ value is 3.7 events per S5122

Figure 7-9: Results from an S5 online search using coincident kleineWelle 5.3 triggers
from the gravitational-wave data with CorrPower [101] follow-up. The online search
was less sensitive than the oﬄine searches using Coherent Waveburst and the Q/Ω
Pipeline and was meant to quickly identify extraordinary burst events. On the left
is the distribution in CorrPower significance Γ of events in red and the expectedbackground in black. On the right is the result after the post-analysis application ofan H1−H2 null stream veto which was part of the Q Pipeline analysis for S5. In the
online search, the event does not stand out in the context of the entire S5 run.
search without the null stream, and 0.5 events with the null stream veto applied.
Originally during the online analysis, the event was compared against its 1-day background which showed a rate of 1 event per 100 calendar days, prompting the extensive
follow-up analysis. Otherwise the event does not stand out above background using
kleineWelle+CorrPower over the entire S5 run.
The other event in the post null-stream set is from December 4, 2006 (GPS849267817.78). It is low frequency (79 Hz in H1, 65 Hz in L1), and barely passes
the null-stream veto applied for this analysis. The SNR is 9.6 in H1 and 12.8 in L1.
There is no signal in H2. L1 is very noisy, and loses lock 20 minutes later. The event
is not recorded by Coherent Waveburst.
Coherent Waveburst
Coherent Waveburst is better able than kleineWelle+CorrPower to resolve GW-likesignals from noise transients, and is used along with Q/Ω Pipeline for the oﬄine
second-year all-sky burst search below 2048 Hz. Coherent Waveburst events are123

Figure 7-10: Sample background from the Coherent Waveburst pipeline from 1000
time-shifts of H1H2L1 data in the second calendar year of S5. On the left are eventsfrom 64-200 Hz, and on the right are events from 200-2048 Hz. Shaded events are
removed by various stages of data quality and vetoes. The candidate event at ∼110
Hz has ρ = 6.6.
divided into two frequency bands: 64–200 Hz and 200–2048 Hz. The events in eachband are ranked by their eﬀective significance, ρ, and are subject to the detection
thresholds of ρ > 6.0 for low frequency, and ρ > 4.2 for high frequency bands. This
gives a combined false-alarm probability of 6% for Coherent Waveburst on second-year
data. Category 2 Data Quality and vetoes (chapter 6) are applied before checking
for detection candidates, while category 3 data quality and vetoes are used to create
the clean data set for an upper limit. The number of low-frequency time-shift events
after category 2 cuts stronger than the equinox event with ρ ≥ 6.60 is 18 in 1000
time-shifts. After category 3 cuts the number is reduced to 10. Some amount oflive-time is lost for large time-shifts, and accounting for this loss the probability ofobserving a low-frequency background event in the detection set of the second yearof S5 with ρ ≥ 6.60 is 2% or once in every 28 years of S5 second-year live-time. After
category 3 cuts this is reduced to a probability 1% or once in every 43 years.
The two frequency bands for Coherent Waveburst result from the large populationof noise transients at low frequency. Figure 7-11 includes a histogram showing thefrequency distribution of time-shift events at various thresholds in ρ. The counts are
displayed on a logarithmic scale, and a low threshold they are largely dominated by124

Figure 7-11: Frequency and time-frequency distribution of sample background fromthe Coherent Waveburst pipeline from 1000 time-shifts of H1H2L1 data in the second
calendar year of S5. The frequency histogram on the left shows a broad excess of
low frequency background between 80−130 Hz. The right plot shows the individual
time-shift events between Nov 2006 and Oct 2007, with the candidate event in red
near the end of the run.
events between 85–95 Hz, which come primarily from the first half of the second-year.
However the strongest events don’t show the same narrow distribution. This can beseen in detail in the second scatter plot of frequency versus time for the time-shift
events above our follow-up threshold of ρ ≥ 6.0. Much of the background comes froma bad period in July 2007. During this time H2 suﬀered from poor sensitivity making
it a much poorer consistency check for moderate transients in H1.
The detection candidates themselves are subject to a myriad of tests by way ofthe detection checklist. These tests are not applied to the time-shift background
events because of the amount of work involved, and not reflected in the calculationof significance because of the ambiguity about what constitutes a definite veto andthe possibility of introducing bias in an un-blinded procedure. Without some consideration of the eﬀects of post-analysis, however, rates such as once every 26 years will
be misleading about the expected rate at which detection candidates above a certain
strength are presented as possible gravitational waves. In theory the best measurement of rates would come from application of the exact same rules to time-shift andunshifted events.
125

Another important factor when considering significance is the total number ofexperiments. With the division of Coherent Waveburst into two frequency bands,
the above-2048 Hz high-frequency burst search, and separate first and second year
analyses, the burst group has at least six independent all-sky searches for S5. In
addition, searches such as Q/Ω Pipeline and the search over LIGO/Virgo VSR1 data
cover the same live-time and are not independent from Coherent Waveburst LIGOonly searches in non-trivial ways. The burst group has brainstormed extensively
on ways to combine multiple searches, but currently the equinox event can only bedirectly compared with expected background distributions in its own restricted 64–
200 Hz search over S5 year 2. The 2% probability of observing a stronger event inthis set should thus be taken in appropriate context.
Q/Ω Pipeline
The equinox event did not pass the final cut on H1H2 correlated energy for the
Q/Ω Pipeline second year analysis. Figure 7-12 shows event scatter plots of H1H2
correlated vs. coherent energy, and H1H2 correlated vs. L1 energy for time-shift,unshifted events, and simulated signals, as well as the equinox event and semi-blind
inspiral hardware injection. The H1H2 plot on the left contains H1H2 events during
all time when H1 and H2 are in science mode with category 3 cuts applied, and thecut on H1H2 correlated energy is placed to maintain zero remaining H1H2 eventsin 10 time-shifts. The H1H2L1 plot on the right contains the small subset of H1H2
events which are also coincident with a trigger in L1. Here a lower threshold canbe set for H1H2 correlated energy to maintain a rate of 15 coincident events in 1000
time-shifts.
We obtain a measure of the significance of the equinox event in the H1H2L1 set byrelaxing the cut on H1H2 correlated energy to 10, and ranking all events by the joint
probability of observing a greater H1H2 correlated energy and a greater L1 energy.
Using this ranking, we expect 2.2 events of greater significance in the H1H2L1 set
below 200 Hz, or once per six months. This ranking does not take into accountthe H1H2 correlated vs. coherent energy consistency or the H1H2 correlated energy126

Figure 7-12: Sample background from 10 and 1000 time-shifts (black) and unshifted(red) events from the Q/Ω Pipeline H1H2 and H1H2L1 analyses for the second calendar year of S5. The three major energy measurements are shown: H1H2 correlated
energy (may be negative for anti-correlated signals), H1H2 coherent energy (energyof the optimal H1H2 sum), and L1 normalized signal energy. The candidate event(yellow star) is below the final threshold on correlated energy in each search. Also
shown is an earlier semi-blind inspiral injection.
vs. L1 energy consistency which are both shown to be quite good for the equinox
event itself. The Q/Ω pipeline itself was not tuned to perform well on events below
threshold, so it is not surprising that the equinox event, despite its strong waveformconsistency, is not well separated from background.

7.3.5

Event reconstruction

Coherent Event Display
The sky-maps in Figure 7-13 show the Coherent Waveburst sky statistic for H1H2L1and H1H2L1V1 detector configurations. The sky statistic is derived from the likelihood statistic and other output from Coherent Waveburst at diﬀerent sky positions.
Although the Virgo interferometer during VSR1 has much worse sensitivity at 100
Hz, it is able to reject regions of the sky where it has a favorable antenna patterncompared to LIGO.
Coherent Event Display [103] is also able to estimate the most likely coherent
waveform after choosing the best sky position. Figure 7-14 shows this single best-fit
127

Figure 7-13: Coherent Waveburst sky statistic for H1H2L1 and H1H2L1V1 networks.
The sky statistic is derived from the likelihood statistic and other output from Coherent Waveburst at diﬀerent sky positions.
detector
H1
H2
L1

hrss [Hz−1/2 ]SNR

874465554.715 3.16 × 10−22874465554.715 3.16 × 10−22874465554.710 3.04 × 10−22

13.6
5.3
11.9

GPS time

Table 7.4: Parameters of the Coherent Waveburst best-fit gravitational wave when
projected onto the interferometer geometry of each detector in local noise.

gravitational wave projected onto the arms of each detector, so that the H1 and H2
waveforms are the same by construction. From left to right are the calibrated strain
waveforms for H1, H2, and L1. Due to a plotting bug, the amplitude of each waveformis twice what it should be.
Also informative is the estimated waveform shown on top of the calibrated detector
output (Figure 7-15). The time series have been whitened with the same filter so thatthe noise is not dominated by lines. The H1 and L1 waveforms fit very well to theh(t) data. H2 shows some amount of excess noise.
Using the estimated waveform, the per-detector signal properties for Coherent
Waveburst are listed in Table 7.4. Other properties of the best-fit gravitational wave
are listed in Table 7.5. The addition of Virgo gives a slightly diﬀerent estimate forgalactic coordinates (α, δ) = (38.9◦ , 30.5◦ ).
128

Figure 7-14: Coherent Waveburst best-fit strain waveform projected onto the detector
geometry of each interferometer. Because this represents the projection of a coherent
gravitational wave, the reconstructed signal for the co-aligned H2 detector is the same
as for the H1 detector.
Omega Pipeline
Omega Pipeline’s Bayesian statistic [86] can calculate the probability of observing thedata given the presence of an elliptically-polarized sine-Gaussian gravitational wave
at a given sky location marginalized over random arrival time, amplitude, frequency,
Q, inclination and phase. Figure 7-16 shows arbitrarily normalized H1L1V1 probability distributions on the sky in linear and log scale. Omega Pipeline puts the mostprobable source location at (α, δ) = (87.6◦ , −54.9◦ ), which puts it in the southernhemisphere.
The Bayesian method has yet to produce a detection statistic for the equinox event,129

Figure 7-15: Coherent Waveburst best-fit (whitened) waveform plotted on top of thewhitened data streams for H1 (top-left), H2 (top-right), and L1 (bottom). The H1and L1 data show a very good fit.
or a statement about the physical consistency of the signal at the most probable sky
location.

7.3.6

Astrophysical analysis

Nearby electromagnetic events
There are a number of public listings of electromagnetic transients available. Table 7.6
lists transients from the Gamma-ray bursts Coordinates Network (GCN) which have
durationfrequencyαδρ0.039 s102.54 Hz

43.9◦

24.5◦

6.60

Table 7.5: Parameters of the Coherent Waveburst best-fit gravitational wave in Earthcentered coordinates.
130Figure 7-16: Ω Pipeline’s Bayesian sky probability distribution for the candidate
event. The right plot shows the log probability in order to better see the detailed
structure.
GCN

name

6807
6811
6821
6822
”
6823

GRB 070920
GRB 070920B
GRB 070923
SGR 1806-20”
GRB 070925

time

instrument

07/09/20 04:00:13
Swift
07/09/21 13:22:10
Swift
07/09/23 19:15:23
Swift
07/09/21 11:45:35 Konus-Wind
07/09/23 12:46:15”
07/09/25 17:26:59
Integral

type
long
long
short-hard
SGR burst
”
long

α◦

δ◦

100.968 72.250
0.127-34.844184.623 -38.294
272.164 -20.411””
253.218 -22.0355

Table 7.6: Electromagnetic events from the GCN catalog around 2007-09-22
been reported within a few days of the equinox event. None of the publicly reported transients are plausible counterparts to the equinox event, mostly being fromthe wrong sky location. International Astronomical Union (IAU) circulars were also
checked for the presence of nearby supernovae or other transients.
BH/BH merger interpretation
Fitting the waveform estimated by Coherent Waveburst to a ring-down gives ringdown parameters f = 97 Hz, Q = 2.7 for L1. Using the following relations for a
perturbed black hole,
f=

�
c3 �
1 − 0.63(1 − α)0.3
2πGM

Q = 2(1 − α)−0.45 ,
131

(7.1)
(7.2)

Figure 7-17: The Hilbert-Huang transform of the candidate event indicates a decreasein frequency from about 160 to 60 Hz in just a couple cycles.

gives total mass M = 140 M⊙ and spin α = 0.5.
One problem with the BH/BH merger interpretation is that the frequency seems
to decrease with time from about 160 to 60 Hz in just a couple cycles (Figure 717). This indication is from the Hilbert-Huang transform, an adaptive data analysis
strategy which provides a detailed description of the instantaneous frequency of the
waveform [105], In the following plots, time-frequency maps derived from the HHT
are shown for H1 and L1. These HHT-derived maps have a frequency accuracy related
to the instantaneous power of the waveform. The uncertainty in frequency over thecentral 5 ms of the waveform, where the power is high, is ±5 Hz, while at the edges,
where the power is lower, the uncertainty is ±30 Hz. A trend of decreasing frequency
in time is clearly seen. For a compact object coalescence, we would expect a chirp
waveform with increasing frequency.
We can compare the LIGO data with recent numerical simulations of non-spinning
equal-mass binary black-hole mergers from the GSFC group [106]. The simulated
waveforms are taken with total mass M = 140 M⊙ . At 70 Mpc and optimal orientation, the strain at Earth is shown in Figure 7-18. To match the peak strain observedin the 4km instruments, we reduce the strain by a factor of six and inject into LIGO
data one second before the equinox event. We make equivalent plots to Figure 7-1
by plotting the whitened LIGO data plus simulated BH merger signal at 60–2048 Hz
132

Figure 7-18: 70/70 M⊙ equal mass black hole merger waveform at 70 Mpc from the
GSFC group [106].

Figure 7-19: Time series of 70/70 M⊙ equal mass black hole merger simulation in
LIGO noise with arbitrary scaling. The low frequency inspiral and early merger isstrongly suppressed as it cannot be seen above the seismic noise wall.

133

Figure 7-20: Spectrogram of equal mass black hole merger simulation in LIGO noise.
and 60–140 Hz.
We see that the initial low frequency signal is quickly swamped by the noise curve,
and only the low-Q merger and ring-down show. A Q-scan (Figure 7-20) shows thebest match to be at slightly higher Q than for the actual equinox event. We also
see evidence of a small chirp. Curiously there a weak 100 Hz transient in H2 which
occurs less than 50 ms following the merger, similar to the additional noise preceding
the coherent component of the equinox event’s H2 waveform.
Source population study
The sky-maps in Figure 7-21 show blue-light luminosity L10 density per solid angle
in the sky over a certain distance range. The sources are taken from the CBC catalog[107], and smoothed over a resolution of 10 degrees. The first plot shows everything
within 1 Mpc where the major feature is the Andromeda galaxy M31. We also see two134other major nearby objects, M33 and LMC, intersecting the time-delay sky ring. The
fifth plot shows blue-light luminosity between 50 and 80 Mpc where the major featureis the large Perseus-Pisces supercluster (about 1016 M⊙ ). An optimally oriented 75/75
M⊙ non-spinning BH/BH merger at 70 Mpc would give an hpeak at the earth of about
2.5 × 10−20 according to numerical simulations [106]. This optimally oriented andzero inclination waveform is six times larger than the best fit waveform projected
onto the arms at either site as estimated by Coherent Waveburst.

7.3.7

Other events in S5

Outliers in time-shift analysis
Most of the loud time-shift events in the Coherent Waveburst analysis show similar
characteristics of the equinox event, namely low frequency of 70–120 Hz and low-Q.
This expectation of similar events in the background does not change the measured
significance of observing the event at zero time-shift, but the event would arguably bemore significant if it was clearly separate from the dominant background population.

Outliers at zero time-shift
There are no other outliers at zero time-shift in the Coherent Waveburst analysis.
The remaining events are all well below threshold.

Similar transients
Low-frequency transients are common in the LIGO instruments. On the day of theevent, there were three transients in H1 at the same frequency and Q which were
louder than the candidate event, and 30 stronger L1 transients at the same frequency
and Q.
More information about the presence of similar single detector transients in thehour of the event can be found in section 7.3.3.
135

Figure 7-21: L10 blue light luminosity density per solid angle over fixed distance
ranges out to 100 Mpc using the catalog of Kopparapu et al [107]. Also shown is asample of well known galaxies and galaxy clusters.
136

Figure 7-22: Sine-Gaussian f =100 Hz Q=9 hardware injection.

Figure 7-23: Gaussian 3 ms hardware injection.
Similar hardware injections
The most similar hardware injection in S5 is a sine-Gaussian waveform with f = 100
Hz, Q = 9, hrss = 6.4 × 10−21 Hz−1/2 (Figure 7-22). There is such an injection at
GPS 874465969.5 in the set six minutes following the equinox event. We also have a
3 ms Gaussian waveform injected at GPS 875253573.5 with hrss = 4.0 × 10−20 Hz−1/2(Figure 7-23). Because of the low frequency noise, the whitened signal resembles asine-Gaussian waveform. Unfortunately we do not have weaker hardware injections
for these parameters.

7.3.8

Conclusion of the follow-up procedure

Summary of findings prior to blind injection revelation
As soon as the equinox event was observed, it was clear that the low frequency,
low-Q morphology of both H1 and L1 signals made them very similar to the mostcommon transients observed. Also apparent is that the 1-2% chance of a stronger
137

event originating from background is marginal given the several independent burst
searches we run for S5.
The follow-up analysis of the event has brought in a large amount of additional
information to consider when evaluating our believability in this event as a true
gravitational wave. In favor of the event is,• We looked very hard and did not find any evidence of an instrumental cause ornoise transient• The event did not happen during noisy times in the run which represent a large
contribution to our background• The impressive consistency between H1L1 and H1H2 (minus the extra content
at the beginning of the H2 signal) is very unusual• The event has passed a very extensive checklist. It is likely that on deep inspection the event is a more convincing gravitational-wave candidate than events in
our expected background at the same ρ threshold.
Evidence found in post analysis which decrease our confidence in the event is,• The dominance of low frequency, low-Q events in our background suggests thatan optimal search might rank such events as less significant, allowing events
outside of this noisy region a higher ranking because they buy more sensitivity
at less false alarm rate. Thus an objectively better search may result in a lower
significance for this event. Generally we assume that a better search will provide
increased significance for a real event.
• The fact that the frequency and Q of the event is very consistent with our
background in morphology leaves open the possibility that observation of theevent is consistent with background and we have just made a mistake somewhereand underestimated the rate.
• The marked increase in weak H1 transients immediately around the event isvery suspicious. Though there is no clear connection other than time between
138

these transients and the event, the sudden and coincident excess transient rate
makes us wonder if H1 should be trusted during that short interval as it implies
an unexplained change in the instrument state as well as an elevated background
rate.
These considerations are beyond the scope of our blind analysis, and are highly
subjective. However that does not mean that subjective post-analysis will not yielda better separation of background from true gravitational waves. In the future, we ofcourse want to incorporate any useful information into the detection statistic itself.
Many could find their way into a likelihood statistic, which by construction provides
the best foreground-background separation given the parameters we measure.
Ultimately while very interesting, the event does not qualify as a gold-plated
detection candidate. A claim that the event is a true gravitational wave would be
subject to too high a chance of misidentification.
Closeout of blind injection challenge for S5With the completed analysis and follow-up in place, the blind injection challenge
was finally closed in March of 2010. It was then that the analysis teams learned
that the burst outlier was in fact a hardware injection, leaving no remaining true
outliers in the entire S5/VSR1 burst search. The blind injection was a coherent
low frequency broadband signal injected into the three LIGO detectors resulting inmatch-filter signal-to-noise ratios of 11.92 (H1), 6.15 (H2), and 10.31 (L1), close to
those measured by the burst pipelines (Table 7.1 and Table 7.4). Figure 7-24 andFigure 7-25 compares the reconstructed waveform and sky location provided by thecoherent methods.
The result of the blind injection challenge highlighted the problem of excess low
frequency transient background in the LIGO instruments. Even at a signal-to-noise
greater than ten in both 4 km instruments, we were not able to resolve the signal
suﬃciently from background in order to confidently make a detection. Such a signal
amplitude in Gaussian noise would have vanishingly small background even withouta coincidence requirement (Figure 5-1).
139

Figure 7-24: Comparison of injected and reconstructed strain waveform projections
for the S5 blind burst injection. The simulated burst gravitational wave was injected
coherently into the LIGO interferometers H1, H2, and L1 at the end of S5 as part ofthe blind injection challenge. Coherent Waveburst identified and reconstructed the
waveforms in blue using H1, H2, and L1 data. Because H1 and H2 share alignment,
they have identical waveforms thus H2 is not shown. Coherent Waveburst is able toreproduce the main part of the wave in the LIGO sensitive band (64-2048 Hz).

The data quality and auxiliary channel-derived vetoes applied to the bust search
as well as the signal-based vetoes which are part of the coherent multi-detector analysis go a long way to reducing the background to suitable levels for maintaining
sensitivity in an upper limit analysis. However, background levels need to be further
reduced by at least a couple orders of magnitude to obtain event significances (formoderate gravitational wave amplitudes) of 10−4 to 10−5 , approaching a more reasonable standard for detection. Doing so will require either another instrument, suchas tight coincidence with an electromagnetic counterpart, or several detectors operating at comparable sensitivity in order to produce a robust null-stream veto. For
S5/VSR1, the coherent null-stream vetoes, which rely on testing the residual after140

Figure 7-25: Comparison of injection and reconstructed sky location for the S5 blind
burst injection. Since the gravitational wave is seen only at two sites, it is largely
localized to a ring on the sky. The true location is represented as a white diamondat (α, δ) = (80.47, −73.06). Coherent Waveburst is run on H1H2L1 data and gives
a maximum likelihood sky location at (α, δ) = (43.9, 24.5). The Omega Bayesianstatistic was not part of the S5 blind analysis but ran on the event after it was
identified. On H1L1V1 data, the statistic gives a maximum likelihood sky position of(α, δ) = (87.6, −54.9) though this is not quite applicable because the injection was
not present in Virgo data.
removing the best-fit gravitational wave from the data, suﬀered from the poorer low
frequency performance of the Virgo instrument as well as the lower sensitivity of H2
and correlated H1-H2 environmental noise. Therefore a coincident set of moderately
loud low-frequency noise transients in H1 and L1 could be fit to the two degrees of
freedom in an arbitrary gravitational wave without incurring a null stream penalty
from the two less sensitive detectors. Since then, Virgo has made fast progress catching up to the LIGO low-frequency sensitivity, and Advanced LIGO should see H2upgraded to a 4 km baseline and running at full sensitivity.
The follow-up procedure also highlighted the diﬃculty for the burst analysis toassign the global significance of an outlier in terms of the probability of detecting abetter event from background over the entire search. The S5/VSR1 burst search wasparticularly aggressive over previous searches in maximizing coverage to reduce thechance of missing a rare event. The result was a large number of separate networks,
epochs, frequency bands, and analysis methods each of which had its own background
distribution unique to the particular configuration. The significance of an outlier was141

only defined in context of the limited configuration or at the global upper limit cuts
(which were set at too high a false-alarm-probability to be useful for detection). In
the following chapter, we introduce procedures for establishing a unified ranking of
events across multiple search configurations to address this problem.

142

Chapter 8
Unified ranking for
gravitational-wave events
A search method has the task of distinguishing signal events from background. In thesimplest of cases, this includes assigning each event to one of those categories. Events
classified as signal are those which pass the selection cuts, while events classified as
background are everything else. If N events from the data are classified as signal
and the expected contribution from true background to this measurement (falsealarm expectation µBG ) is known, the significance of this measurement can usually
be expressed by assuming Poisson statistics on the event counts (Appendix A),
Z(N,µBG ) = − ln [PPoiss (k ≥ N |µ = µBG )].

(8.1)

The significance of the measurement is the negative logarithm of the false-alarmprobability of measuring the observed number of events or more from backgroundalone.
The simple classification of events into signal and background can also be used to
set upper limits or confidence intervals on a true rate of signals. For the S5 all-sky
burst analysis, classical frequentist upper limits were established for the true rate ofevents classified as signal (above threshold) under the assumption of Poisson statistics.
The frequentist method attempts to set a confidence interval which contains the true143value at least 1 − α fraction of the time where 1 − α is the confidence level. Assuming
Poisson statistics and an observation of N events, this can be achieved by setting an
upper limit at the mean value µUL which solves,
α = PPoiss (x ≤ N |µ = µUL ).

(8.2)

For an observation N = 0, this gives an upper limit of µ ≤ 2.303 at 90% confidence,
meaning that for any true µ > 2.303, we would have measured zero (or fewer) events
less than 10% of the time. Thus the upper limit derived this way contains the true µ in
at least 90% of experiments. The frequentist upper limit is a limit on events classified
as signal, which includes in general contributions from foreground and backgroundsources. While background may be known and folded into the calculation, the limit
can be used as a conservative limit on the number of foreground events only (setting
the background contribution to zero).
A unified ranking of events implies a unique and global ordering of events frombackground-like to signal-like. A ranking can be based on signal-to-noise ratio, forexample, in the case of a matched-filter or excess power search (section 4.2). It is also
the output of a number of multivariate statistical classifiers. It is easily reduced tothe classification case by setting a single cut on the ranking variable r.
The ranking is more flexible than a simple categorization. Rate upper limits can beestablished from the loudest event statistic of Brady, Creighton, and Wiseman [108].
The loudest event statistic is based on the sensitivity of the search at a threshold
set at the rank of the loudest detected event. It therefore removes the need to set a
threshold by hand, which can be a daunting process in uncertain background. Suchis the case for a search on H1H2 data due to correlated environmental noise aﬀecting
the validity of the time-shift background estimate [74].
The loudest event statistic sets the threshold of the analysis at the value of theloudest observed event rmax . The upper limit is a statement about the number of
expected events above rmax and for the frequentist construction of the statistic gives
144the same limit as the classical frequentist upper limit for an observation N = 0,
α = PPoiss (k = 0|µ = µUL ).

(8.3)

This is because 1 − α is the probability that the loudest event occurs above thethreshold value that corresponds to a true µ = µUL . At 90% confidence, the upper
limit is 2.303 events with r ≥ rmax . In 90% of experiments, the loudest event will
have higher r than that which would have given a true expectation value of 2.303. In
those cases, 2.303 is greater than the actual expected number of events with r ≥ rmax .
In the event of a single outlier (which is the most likely case given the rarity ofstrong gravitational waves and shape of the background amplitude distribution), thelocal significance of the event can be derived from the probability of observing one ormore events from background with higher rank using 8.1 with N = 1,Z(1,µ) = − ln [1 − PPoiss (k = 0|µ)] = − ln [1 − eµ ].

(8.4)

Here µ is the integrated background distribution from the rank of the event to a rankof infinity, and we have used the fact that the Poisson distribution is a probability
distribution. The advantage of a single classification is that progressively better events
can be assigned correspondingly higher significance.

8.1

Trial factor issues in S5/VSR1

Evaluation of the global significance of an event was an issue in the S5/VSR1 burst
search as no universal ranking of events was defined as part of the blind analysis.
Instead, the analysis focused on classifying events as signal or background by setting
cuts dependent on a particular detector network, frequency range, analysis method,and data quality and veto configuration [8]. Events were only compared to one
another in the sense that they either passed or did not pass the predetermined analysis
cuts. Each search method generally did have a means to rank events within a single
configuration. The ranking was based on a single-to-noise like quantity and applied as
145the final fixed cut for that configuration. The significance of an event is thus defined
relative to background from the same configuration. To evaluate a global significance,
one can evaluate the probability of getting one or more events from any of the M
configurations with local background expectation less than µ,�

Z1,Global = − ln 1 −

M
�
i=1

�

e−µ .

(8.5)

This is unsurprisingly just the significance of a single event from a background expectation of µGlobal = M µ. For very small probabilities, the global false-alarm probability
is the individual false-alarm probability multiplied by the M independent trials.
Table 8.1 lists the configurations used for the S5/VSR1 burst search [8]. The
total live-time of 5.1 years is greater than the observation time because of overlap
between configurations. Exclusive networks are times when only the listed detectors
are operating. Inclusive networks include time when the unlisted detectors may be
operating, but only the detectors listed are part of the analysis. The configurations are
distinguished by their search method, frequency range, and detector network. Eachconfiguration has a diﬀerent set of hand-chosen cuts to classify events for the upper
limit analysis. In addition, a distinction is made between events flagged by category 3
data quality and vetoes and those not flagged. Flagged events are cut from the upper
limit set but are still considered for detection. In total this represents 40 diﬀerent“boxes” for the search, or 80 when counting the category 3 flag distinction.
The large number of configurations is a problem for the S5/VSR1 search as a naive
application of the trials factor would severely diminish the significance of an event. In
this straightforward approach, events which may come from a particularly insensitive
network configuration or a configuration with low live-time will rank alongside otherevents at the same local false-alarm probability. These configurations then contributeto the global background without adding much sensitivity. The cuts chosen by hand at
the global false-alarm probability used for the upper limit classification imply a certain
fixed rank of for events which live at the position of the cut. At those parameters,
they are often more reasonable than the false-alarm probability ranking. Insensitive146

pipeline

network

days

notes

cWB 64−200 Hz

H1H2L1V1

68.2

H1H2L1H1L1V1H1H2V1
H1H2
H1L1
H2L1
all (7)
H1H2L1H1H2V1
L1V1
H1H2

191.6
3.9
14.5
56.9
10.1
3.5
348.7
194.5
85.9
83.1
34.9

all (4)
H1H2L1V1
H1H2L1H1H2V1H1L1V1H2L1V1
H1H2
H1L1
H1V1
H2L1
H2V1
L1V1
H1H2L1V1
H1H2L1H1H2V1H1L1V1H2L1V1
H1H2
H1L1
H2L1
H1V1
H2V1
L1V1

all live-times for cWB 64−2048 Hz are
pre Cat2 flags, except for H1H2L1
including V1 time
excluding H2 time
excluding L1 time
excluding L1 time, including V1 time
excluding H2 time, including V1 time
excluding H1 time, including V1 timesame as below 200 Hz
including V1 time
including L1 time
including H1H2 time
excluding L1 and V1 time. Q/Ω was run
on inclusive networks, but it was decided
to remove the L1 and V1 time from theH1H2 network for this calculation.
same as below 200 Hz
all networks are exclusive

398.4
69.3
16.6
16.3
4.6
1.5
4.8
1.2
1.9
0.2
9.8
6.4
70 all networks are exclusive
128
15
4
1
35
6
3
1
0
6

cWB 200−2048 Hz
Q/Ω 64−200 Hz

Q/Ω 200−2048 Hz
EGC 300−5000 Hz

cWB 1280−6000 Hz

total

all (40)

1881.45.1 years

Table 8.1: List of configurations and live-times used for the S5/VSR1 burst search.
See text for details.
147networks are subject to higher thresholds, and a threshold on signal-to-noise ratio is
not sensitive to divisions in live-time.
We can list a few guiding principles to keep in mind when constructing an improved
global ranking of events to better handle the trials factor,• The ranking should be robust against divisions in live-time• The ranking should be robust against divisions in frequency or other signal
parameters• The ranking should show consistency with cuts chosen by hand, such as theupper limit cuts. That is, the cuts should be able to be described as a single
cut on rank.

8.2

Inverse false-alarm rate ranking

Inverse false-alarm rate (IFAR) provides an extremely simple way to rank events
that is by construction robust against divisions in live-time. In this ranking, events
are ranked by the rate at which louder events arise from background in the search
configuration where the event was found. It provides a live-time-scaled version of thefalse-alarm probability ranking. The IFAR ranking was used to rank events for the
S5 inspiral search [24, 25] and was also used to estimate the significance of the burst
outlier in the S5/VSR1 burst analysis followup [8].
Rankings such as IFAR that are based on predetermined monotonic functions oflocal false-alarm probability have a distinct practical advantage. For such a ranking,the global background at a given value of the rank variable is free of any measurement error by construction assuming only that local false-alarm probability could in
principle be measured to arbitrary precision. The error instead is contained in themeasurement of the rank variable itself for a particular event. The measurement,in turn, is based only on local data. To reduce error in the measurement of global
significance for a particular event, one must only better resolve the background distribution for the particular configuration where the event was observed, and not all148the background distributions of all configurations. This is very diﬀerent from a global
ranking which is instead based on some function of signal-to-noise ratio, for example.
In gravitational-wave transient searches, the background is estimated by running thefull analysis method over time-shifted data multiple times with diﬀerent time-shifts(subsection 4.3.3). The most sensitive search methods are computationally intensiveso there is great benefit from running additional time-shifts over just a subset of data.
The S5/VSR1 burst outlier provides an application of this procedure. The event
was detected in the H1H2L1 Coherent Waveburst 64−200 Hz analysis after application of category 2 and 3 data quality and vetoes. In 1000 time-shifts of L1 against
H1H2 data, there were 10 time-shift background events with a larger local ranking
statistic than the observed event, corresponding to a false-alarm probability of 1%
and a false-alarm rate of 1 per 43 years. Rather than trying to gather background
events from all 40 configurations using the same number of time-shifts in order tomake a global background distribution (most configurations were run with 100), we
can instead multiply the false-alarm rate by the total accumulated live-time of 5.1
years (Table 8.1). This gives us a total background expectation of 5.1/43 = 0.12, ora corresponding false-alarm probability of observing one or more events from background with a false-alarm rate of less than 1/43 yr−1 of 11% (8.4).
The calculation relies on the assumption that all configurations could in principle
resolve background to a rate of at least 1/43 yr−1 given a suﬃciently loud event
observed there. This might not be possible for configurations with a small live-time
as there are limits to the total number of independent and valid time-shifts that canbe performed. If a configuration cannot produce events resolvable to a suﬃciently low
false-alarm rate, they must assume the highest resolvable false-alarm rate. Insteadof estimating this threshold, we consider the global false-alarm probability to be aconservative estimate based on the assumption that all configurations can contribute.
Another assumption which factors into the assumption of Poisson statistics is thatthe background from various configurations are independent. For S5/VSR1, this wasbriefly verified by hand for certain overlapping data and background correlations weredeemed negligible. Correlations become much more important for the case of multiple
149detection.
The quality of a ranking based on false-alarm-probability can be improved byfolding in information about the relative expected sensitivities of the search configurations (one example is used in [100]). So long as the ranking is based on fixed
monotonic functions of false-alarm-probability (such as one derived from the application of configuration-dependent weight factors), the practical advantages of zeroerror in the expected background at any given rank remains. IFAR is an example ofthis where the weights are based on live-time. The general limitation to the approachis that the rank is only guaranteed to be good for the region of interest where thefunctions (or weights) have been calibrated. For a large search with many diverse
configurations, a simple ranking based on weighted false-alarm-probability provides
a balance between rank quality (separation of signal and background) and computational as well as human cost. If a population of target signals is defined, a choiceof weight factors could, for example, be set in order to provide equal detection eﬃciency divided by expected background contribution across all configurations whenconstrained to a single chosen global false-alarm-probability of interest.

8.3

Likelihood-ratio ranking

The inverse false-alarm rate ranking is not explicitly independent of divisions in the
search parameters other than time. For example, dividing part of a search into two
frequency bands will split the background, reducing the false-alarm rate assigned toevents and thus increasing their rank with respect to a diﬀerent portion of the searchwhich did not undergo any divisions. The frequency bands and other divisions in the
S5/VSR1 search which define the search configurations in Table 8.1 are not basedon maximizing global eﬃciency with the use of an IFAR ranking, but rather they
are based on practical considerations, such as making easier the implementation of
frequency dependent cuts or the tuning and application of specific analysis methods.
The likelihood ratio, introduced in Equation 4.34, provides the optimal ranking
of independent events parametrized by a set of measurements x. It is defined as the150

ratio of the probability of measuring x under one hypothesis H1 to the probability ofmeasuring x under an alternate hypothesis H0 ,Λ(x) =

P (x|H1 ).
P (x|H0 )(8.6)

The Neyman-Pearson lemma states that a threshold on likelihood ratio is the mostpowerful test to distinguish between the two hypotheses. In our case, H1 is the
hypothesis that the measured parameters arise in the presence of a gravitationalwave signal while for H0 they are a product of background.
The measurements x can be any product of the data, but the power of thelikelihood-ratio test will depend on the ability of the chosen parameters x to represent essential features in the two populations as well as our ability to measure P (x)accurately. In the extreme case, x can be the raw data itself which would provide avery powerful test were the probability densities practical to measure. In our case, wehave just two parameters: one discreet parameter i indexing the search configuration,and one continuous parameter x which is the local detection statistic for the search.
Optimizing the detection statistic is left to the search method, and our task is incombining events from diﬀerent search configurations in a meaningful way.
With just one continuous parameter, it is practical to calculate the likelihood ratio
directly. We use analytic fits to a sample signal population from Monte Carlo simulations and background from time-shift experiments. The ratio of these functional fits
generates a mapping from the local detection statistic to the likelihood ratioΛ i (x)which is then used to globally rank the events.

8.3.1

Signal population

The parameters x measured for hypothesis H1 should reflect those expected from a
realistic signal population. The search for gravitational-wave bursts generally targets unknown or poorly modeled signals covering a large parameter space, and tests
detection eﬃciency over a set of ad-hoc waveforms meant to sample the space suﬃciently. Simulated waveforms used for the S5/VSR1 burst analysis between 64−2048151

Hz include [8],• Sine-Gaussian waveforms:
�(2πf0 t)2
h+ (t) = h0 sin (2πf0 t) exp −
,
2Q2

(8.7)

h× (t) = 0,

(8.8)

�

with discretely sampled f = 70 . . . 2000 Hz and Q = 3 . . . 100.
• Gaussian waveforms:�t2
h+ (t) = h0 exp − 2 ,
τ

(8.9)

h× (t) = 0,(8.10)

�with duration τ = 0.1, 1.0, 2.5, or 4.0 ms.
• Band-limited white noise signals consisting of Gaussian-windowed white noise
limited to frequency bands centered about 100, 250, 1000, and 2000 Hz withbandwidths of 10, 100, and 1000 Hz and durations τ = 10 and 100 ms. The
signals have uncorrelated h+ (t) and h× (t) with equal rms amplitudes.
There are roughly equal numbers of waveforms injected of each type. Discrete
amplitudes are spaced uniformly in log(hrss ) (4.3) from about 2 × 10−22 to 3 × 10−19
to span the sensitive range of the instrument. The sudden cutoﬀ at high amplitude is somewhat arbitrary. For a more realistic population, we use the amplitude
distribution of a homogeneous population of standard candles given h ∝ 1/r:
dN
∝ r2
dr

(8.11)

dN ∝ −r4 dh(8.12)

dN ∝ −h−4 dh.

(8.13)

152

The original discrete amplitude distribution is uniform in log(h):dN ∝ −h−1 dh,(8.14)

so to compensate, we assign a weight of (2×10−21 /hrss )3 to each injection when counting. Other weights could, for example, make the population uniform in gravitationalwave energy, but we choose here to keep the remaining parameter density intact.
Each configuration runs over the same Monte Carlo sample of simulated signals
which are defined over the entire run and spaced randomly in time, sky position,and polarization angle. Since we are only concerned about relative and not absolutefactors in the likelihood ratio for the purposes of ranking events, we use the total
weighted number density of detected signals dNi (x)/dx for each configuration as thenumerator in the likelihood ratio. This amounts to the expected number density ofdetected events at a certain value of the detection statistic x from a homogeneous
population of standard candles with some arbitrary total rate.

8.3.2

Background population

The population of sample background is provided by time-shift experiments performed by each analysis method. Because the number of time-shifts analyzed undereach configuration may be diﬀerent, we normalize the total time-shift backgroundcounts so that they represent the expected background distribution from a single
live-time. This is approximately equal to dividing by the number of time-shifts,
though edge eﬀects create small corrections.

8.3.3

Analytic fits

Analytic fits to the signal and background distributions are used in order to get asmooth mapping from configuration-dependent detection statistic x to the likelihood
ratioΛ i (x). This is complicated by the fact that the analytic form of each distribution
is not known and that the background distribution transitions at some unspecified
point from a bulk region of high statistics to the tail of the distribution with low
153statistics. With a functional form which will necessarily be overly simplistic, classicalfitting techniques applied to the background distribution will tend to use up all thefree parameters fitting the well-sampled bulk of the distribution while it is the tails
which are important for most analyses.

We are not so interested in getting the correct analytic form of the distribution,
fitting the exact parameters, or getting the best statistical match, however. Instead
the goal is to have the smallest error onΛ i (x) over a large range of x with particularfocus on the tail of the distribution. Moreover, the statistical errors of the signal distribution are no longer Poisson because of the amplitude-dependent weighting applied
to each injection. For this reason, we fit to the discrete binned signal and background
distribution based on minimizing the least-squared fractional diﬀerence between fit
and measurements. At the tails of the background distribution, defined by the point
at which the bin counts fall below some tune-able threshold value (∼100), the costfunction transitions smoothly to negative log-likelihood assuming Poisson errors.

For the signal population which typically maintains good statistics across therange of x, the cost function for fit µ[n] to measurements y[n] across the N bins is,

gsignal =

N �
�
n=1

y[n] − µ[n]max (y[n], µ[n])�2(8.15)

Using the maximum of y[n] and µ[n] is necessary to make the function symmetricand avoid bias in the fit. For the background population which generally consists ofa bulk distribution at low x with high statistics, and a tail at high x that transitions
to zero, the cost function is,
 �
�2

M
y[n]
−
µ[n]
1

N 
+ ln 2πM�
2 max (y[n], µ[n])
2
gBG =


n=1 − ln [PPoiss (y[n], µ[n])]154y[n] ≥ M
y[n] < M.

(8.16)

At M , the cost function transitions from Poisson/Gaussian statistics to fractional χ2 ,���(∆M )2
1− ln [PPoiss (M + ∆M, M )] � − ln √
exp −
2M
2πM
�
�2
M ∆M
1
�
+ ln 2πM .
2M
2
�(8.17)(8.18)The parameters for the analytic fits are chosen to minimize the cost functions using
the Nelder-Mead Simplex algorithm from SciPy [109].
Next we must choose functional forms for the analytic fits. The detection statistics
can usually be cast in a form related to the signal amplitude. For Coherent Waveburst,
we use the average network signal-to-noise ρ, and for the Q/Ω Pipeline we use the�square root of the correlated H1H2 normalized energy ZHcorr+ for the case of networks
including H1H2. For the L1V1 network, we use the the square root of the normalized√energy at L1, ZL1 . Since the detection statistic x scales with signal amplitude,

we expect the simulated population to trace the h−4 power-law distribution of ahomogeneous population (8.13). To account for method-dependent eﬀects, we allow
the power-law index b to vary and for the detection variable x to be shifted awayfrom the origin. The functional form of the binned signal population is therefore fit
to a shifted power-law with three free parameters,µsignal [n] = a(x[n] + c)−b .

(8.19)The background distribution is fit to a bulk distribution plus a power-law tail.
The distribution probably contains the superposition of several diﬀerent tails from
diﬀerent epochs and sources, but due to limited statistics we try to fit as best as
possible to one. The single-interferometer noise transient distribution (Figure 5-1) cangive an indication of what to expect from a coincident search. However the coherent
methods should be able to systematically eliminate very loud events through nullstream vetoes. Therefore the power-law most likely cuts oﬀ at very high amplitude.
In practice, more time-shifts should be run if there is the need to rank an event
outside of the range of the available time-shift background samples. The bulk of the155background is either best fit to an exponential distribution or a second power-law
depending on how far the events go into the noise. We fit to both forms: exponential
bulk distribution + power-law tail and power-law bulk distribution + power-law tail.
The fit with the lower cost function is used. Thus, the background distribution takes
one of the following forms,µBG [n] = ea−bx[n] + cx[n]−dµBG [n] = ax[n]−b + cx[n]−d .

or(8.20)(8.21)

Guesses are used to guide the signal and background distribution fits. The freevariables are also all limited to positive values.

8.3.4

Results

Likelihood ratio maps are generated (Figure 8-1 to 8-5) for the subset of searches
in Table 8.1 which cover the 64−2048 Hz frequency range and have reliable background estimates (no H1H2 only search due to correlated environmental eﬀects) andappreciable live-time. As in the IFAR calculation, events are separated by the configurations listed in the table as well as by whether or not they fall within times flaggedby category 3 data quality and vetoes which cover about 15% of the total live-time(category 2 and below data quality and vetoes are applied unconditionally). Datapassing category 3 data quality and vetoes are referred to as “clean” data, while theexclusive set of flagged data is referred to as “flagged.”
In general the likelihood-ratio mappings match our expectations from relative configuration sensitivities and background rates. The 64−200 Hz band has lower spectral
noise density than the 200−2048 Hz frequency range (Figure 3-4), so that at fixedsignal-to-noise ratio, the higher frequency simulations have larger strain amplitude
and thus smaller signal weight factors (Equation 8.13). However, the relative absenceof loud expected background in the upper frequency range more than compensates
for this eﬀect from reduced sensitivity so that the high frequency events are assigned
higher likelihood ratios. We also see that events from flagged times are generally156

pipeline(range [Hz])network
configUL

cWB(64−200)
Q/Ω(64−200)

H1H2L1V1
H1H2L1H1H2L1H1H2V1
L1V1
H1H2L1V1
H1H2L1H1H2L1H1H2V1
L1V1

4.50
6.00
6.08
4.69
5.66
4.10
4.20
3.61
3.74
5.48

cWB
(200−2048)
Q/Ω(200−2048)

pipeline threshold
ULf
LR
LRf
∞
∞
∞
∞
∞
∞
∞
∞
∞
∞

4.41 7.64
5.92 8.32
8.83 9.90
7.32 28.33
45.79∞
4.12 4.21
4.15 4.44
3.55 4.40
4.01 5.84
∞
∞

total

false-alarm [%]ULLReﬃciency [#]
UL
LR

1.07
1.68
1.50
0.90
0.40
0.00
0.84
1.10
0.00
0.50

1.60
2.10
0.00
0.00
0.00
0.53
2.00
1.80
0.00
0.00

28.3
56.4
8.3
1.0
0.0
32.8
132.7
119.6
8.1
0.0

30.8
62.3
1.3
0.3
0.0
37.2
155.5
135.7
6.1
0.0

7.99%

8.03%

239.4

270.5

Table 8.2: Comparison of the hand-chosen thresholds used for the S5/VSR1 upper
limit (UL) and the ones derived from setting a likelihood-ratio threshold (LR) which
gives the equivalent total false-alarm probability of ∼8%. The likelihood ratio allocates background away from insensitive configurations toward more sensitive ones. It
is also able to automatically choose higher thresholds (LRf ) to use for data flaggedby category 3 data quality and vetoes which were originally excluded from the upper
limit analysis (ULf = ∞). The false-alarm percent is represented as 100 times the
background expectation for each configuration rather than a probability of non-zero
events (Equation 8.4). The eﬃciency represents the number of signals above thresholdfrom an ah-hoc homogeneous population of sources with an arbitrarily chosen total
rate. The total eﬃciency is lower than the combined eﬃciency from all configurations
because of overlap which, for counting purposes, is checked for simulations but notfor sample background events. At this choice of total false-alarm probability, thelikelihood-ratio thresholds give a 13% increase in expected signals from the ad-hoc
homogeneous population of sources. It also gives an automatic tuning at any choiceof total false-alarm probability by defining a universal ranking of events.

157

assigned lower rank than those from the corresponding clean data configurations.
There is an occasional exception to this rule at high amplitude where there may
not be enough live-time in the flagged data to resolve the tail of the background
distribution. In practice, more time-shifts would need to be run should an event occur
outside the sampled region. A peculiar case arises when the expected background
distribution falls more slowly with signal-to-noise ratio than the homogeneous signal
population (e.g. the first three plots in Figure 8-2). This causes the likelihoodratio mapping to turn over for loud events because we expect true loud signals to
be very rare. The best way to correct this behavior is to find ways to further clean
the background so that its distribution falls oﬀ more quickly with signal strength.
Alternatively, adding in diﬀerent signal populations (such as a disk population) may
lead to broader distributions, or some ad-hoc preferential weighting applied to wellresolved signals may be used.
The blind injection from the burst S5/VSR1 analysis has a Coherent Waveburst
ρ = 6.60 in the H1H2L1 “clean” data set after application of category 3 flags anddata quality. The likelihood ratio for ρ = 6.60 in this configuration is 26.7. At thisthreshold, the background expectation is 0.0217 giving a false-alarm probability of
2.2%. The IFAR ranking over the same configurations gives a background expectation of 0.08, which is the same as the false-alarm probability set by the original
upper limit cuts (Table 8.2). The likelihood-ratio test shows the event to be moresignificant because it comes from a network and configuration of high sensitivity andlow background.
The expected number of total background events across all configurations at thehand-tuned upper limit cuts defined in the original S5/VSR1 analysis below 2048 Hz is
0.080. A likelihood-ratio threshold of 7.97 (arbitrary units) gives the same backgroundexpectation and is set to apply the equivalent thresholds on pipeline statistic presented
in Table 8.2. At the original upper limit thresholds, 239.4 weighted injections pass
all cuts. 270.5 weighted injections pass the likelihood-ratio cut reflecting a 13% gainin the expected signal rate from the mock population at this particular false-alarm
threshold.
158

After removing the blind injection from the search, the surviving highest-rank
un-shifted event is from the Coherent Waveburst H1H2L1 200−2048 Hz search overthe clean data set (all data quality and veto cuts applied). The event has ρ = 3.87
which corresponds to a likelihood ratio of 0.572 for this configuration (Figure 8-2).
The global background expectation at this likelihood-ratio threshold is 1.29 (0.18
contribution from this search configuration alone), so an event of this strength is
not unexpected. Using the loudest event prescription of Brady, Creighton, and Wiseman [108] we can set a frequentist upper limit of 2.303 expected events withΛ > 0.572
at 90% confidence. At this threshold, 385.7 events from the mock population survive,corresponding to a 61.1% increase in population detection eﬃciency over the fixed
thresholds used for the original S5/VSR1 all-sky upper limit (Table 8.2).
As the upper limit for the population itself scales inversely with detection eﬃciency, a 61.1% increase in eﬃciency implies a 37.9% reduced upper limit on the
intrinsic number of events in the population (such as rate per volume). In the original analysis [8], individual upper limits were established for various populations ofsignals that shared a single morphology and fixed amplitude at the Earth. Such populations are not explored here. The original upper limit thresholds were also chosen
in the context of additional contributions to false-alarm-probability from networks
and frequency ranges not used in this study (Table 8.1), and a reduced set of configurations would imply slightly lower thresholds assuming a fixed total false-alarmprobability.

159

Figure 8-1: Coherent Waveburst H1H2L1V1 likelihood-ratio maps. Dashed blue
curves represent the population of sample background events from 200 time-shifts.
Red curves represent the number of signals from the single homogeneous mock population, scaled appropriately to the number of time-shifts and relative bin size (for thecase of the discrete histogram). Analytic fits to the discrete distribution are used to
calculate the likelihood-ratio mapping (signal/background) for the configuration.
160

Figure 8-2: Coherent Waveburst H1H2L1 likelihood-ratio maps Dashed blue curves
represent the population of sample background events from 1000 time-shifts. Red
curves represent the number of signals from the single homogeneous mock population,
scaled appropriately to the number of time-shifts and relative bin size (for the case of
the discrete histogram). Analytic fits to the discrete distribution are used to calculate
the likelihood-ratio mapping (signal/background) for the configuration.
161

Figure 8-3: Q/Ω Pipeline H1H2L1 likelihood-ratio maps Dashed blue curves represent the population of sample background events from 1000 time-shifts. Red curves
represent the number of signals from the single homogeneous mock population, scaled
appropriately to the number of time-shifts and relative bin size (for the case of thediscrete histogram). Analytic fits to the discrete distribution are used to calculate
the likelihood-ratio mapping (signal/background) for the configuration.
162

Figure 8-4: Q/Ω Pipeline H1H2V1 likelihood-ratio maps Dashed blue curves represent the population of sample background events from 1000 time-shifts. Red curves
represent the number of signals from the single homogeneous mock population, scaled
appropriately to the number of time-shifts and relative bin size (for the case of thediscrete histogram). Analytic fits to the discrete distribution are used to calculate
the likelihood-ratio mapping (signal/background) for the configuration.
163

Figure 8-5: Q/Ω Pipeline L1V1 likelihood-ratio maps Dashed blue curves represent
the population of sample background events from 1000 time-shifts. Red curves represent the number of signals from the single homogeneous mock population, scaled
appropriately to the number of time-shifts and relative bin size (for the case of thediscrete histogram). Analytic fits to the discrete distribution are used to calculate
the likelihood-ratio mapping (signal/background) for the configuration.
164

Chapter 9
Conclusions
The prospects for direct detection of gravitational radiation famously predicted by
Einstein’s theory of general relativity have improved enormously in the last few years
with the successful commissioning of first generation kilometer-scale laser interferometric detectors. Having met its initial target of one full year of three-detector
coincident observation at design sensitivity in October 2007, the US led Laser Interferometric Gravitational-wave Observatory (LIGO) is currently running in an upgraded configuration in anticipation of the move to Advanced LIGO. It is joinedby the French-Italian Virgo detector which provides comparable sensitivity to LIGO
across much of the observation band. Together, the three sites provide not only anultra wide-field view of the gravitational-wave sky but also the important ability toresolve a source so that we can learn the most about the origins and dynamics of apotential gravitational-wave event.
Being prepared for a detection, then, is of prime importance for current analyses.
Searches for gravitational-wave transients, which for ground-based detectors comeprimarily from the rare, violent motions and interactions of compact objects, arecurrently limited by the presence of non-Gaussian noise transients which populate
the data to a suﬃcient degree that accidental coincidences dominate the background.
Identifying and removing these noise transients are then key to maintaining sensitivity
to weak gravitational wave signals. Furthermore, being able to characterize anddeveloping a familiarity for the expected background is an important step to establish165

confidence in any outlier event.
In this work, we have developed kleineWelle, an eﬃcient method for transient identification based on the dyadic wavelet transform. Due to its simplicity and streamlined
data handling, kleineWelle is able to scan all the fast and semi-fast channels at each
LIGO site in real-time, producing lists of triggers which characterize the transient
population at hundreds of test points in the interferometer and local environment.
This information is fed back graphically to the instrument operators to aid in monitoring stationarity and correlations between channels, and is also used for real-time
identification of instrumental disturbances for rapid rejection of online search background.
An automated procedure for scanning over the large collection of noise transientsand selecting and tuning an eﬃcient set of auxiliary channels for use as vetoes forburst search background was developed and applied to the all-sky un-triggered searchfor gravitational-wave bursts during S5/VSR1. The selection produces a list of vetoconditions ordered from most eﬀective to least eﬀective at flagging noise transientsobserved in the gravitational-wave data using observed transient behavior in auxiliary channels. The automated classification greatly aids in tuning these veto choices,which allowed the S5 search to use a much richer set of auxiliary information to
reject instrumental disturbances than what was permitted by time constraints in previous LIGO searches. The “event-by-event” vetoes were applied as a final step forbackground rejection in the S5/VSR1 burst analysis, and remove around 30−50%
of remaining single-detector noise transients with a 1% per-instrument reduction in
live-time. The performance on random coincident, coherent background varies considerably from about 10% for weak signals to around 50% for background outliers.
In the second half of this work, we introduced the follow-up procedure used tothoroughly test potential gravitational-wave candidate burst events surviving our otherwise blind analysis. The follow-up built upon an earlier example from the LIGO
S2 burst search, but developed into a much more involved investigation with thepresence of a single statistically significant outlier near the end of the S5/VSR1 run.
The outlier event was subject to an extensive detection checklist where it was deter166

mined that there was no compelling evidence to suggest the signal was caused by an
instrumental artifact. Ultimately the level of background in our blind analysis was
such that the moderately loud event could not be considered suﬃciently significant
for a detection claim. The event remained as a marginally significant outlier in the
final analysis until it was ultimately revealed to have been a blind hardware injection
applied during the run to test the end-to-end search, specifically including follow-up
procedures and the readiness to report a detection. The unique opportunity to vet a
possibly real signal was a valuable exercise and experience for the analysis teams.
The follow-up procedure for S5 highlighted the importance of having a well definedprocedure for evaluating the significance of an outlier in the context of a larger burst
search encompassing several detector combinations, methods, and other divisions of
the search. These divisions prevent an event from one sub-search from being directlycomparable with an event from another. We have introduced two procedures which
can be used to handle this trials factor with special consideration to the practicalities of transient gravitational-wave searches. The first is a unified comparison basedon local false-alarm-rate, which has already been applied successfully in searches forcompact binary coalescence [24, 25]. It is noted that the global background expectation for rankings based on fixed monotonic functions of local false-alarm probability(such as false-alarm-rate) can be readily calculated from local background only, which
is of great practical merit. When applied to the burst outlier, there is an 11% chanceof an event with lower false-alarm-rate to arise from one of the many burst search
configurations, or an 8% chance for the subset of burst searches below 2048 Hz.
The second unified ranking is based on the likelihood ratio for each event when
events are assumed to be independent and characterized only by the detection statistic
assigned to them by the corresponding search method. The likelihood ratio requires
the calculation of a signal and background population. We have implemented a procedure to cast existing burst simulations into a more realistic homogeneous population
model, and developed procedures to calculate robust analytic fits for both signal
and time-shift background distributions from which the likelihood-ratio mapping isreadily derived. When applied to the burst outlier, there is a 2.2% chance of an167

event with higher likelihood ratio to arise from background in one of the burst search
configurations for events below 2048 Hz.
The likelihood-ratio ranking provides a 13% increase in detection eﬃciency for themock signal population at the same total false-alarm probability as the original handtuned upper limit thresholds. The unified ranking also allows for the use of a loudestevent based upper limit determined by the highest rank event in the analysis. Applied
to the S5/VSR1 burst events, the threshold set at the highest rank event provides a
61% increase in detection eﬃciency for the mock signal population over the original
fixed upper limit thresholds. The frequentist upper limit of 2.303 detectable events
at 90% confidence is the same for all three cases (hand-chosen fixed thresholds, fixed
likelihood-ratio threshold, threshold at loudest event), so these increases in detection
eﬃciency translate into 12% and 38% reductions in the population rate upper limit
respectively.
While the likelihood ratio calculated this way is optimal under the assumption of
independence among events, this is a poor assumption for the S5/VSR1 burst configurations. Events from gravitational-wave signals for search configurations which
overlap in parameter space are generally dependent, indicating that better rankings
could be calculated if the correlations between overlapping searches could be measured. For the time being, this is made diﬃcult due to technical diﬀerences in timeshift background generation. In the case of search configurations which are largely
independent, the likelihood ratio can give a fast and eﬃcient means to combine events
once the appropriate maps are calculated.
As the network of ground-based gravitational-wave detectors grows and improvesin sensitivity, the confidence of a coincident observation as well as the ability to resolve an incoming gravitational wave is greatly enhanced. Joint analysis becomesincreasingly more complicated, however, due to the additional non-stationarity introduced by each instrument. Automated procedures for characterizing each detector
and interpreting and unifying results from a variety of diﬀerent search configurations
will play an important role in the first detection of a gravitational-wave event.

168

Appendix A

Statistical significance of Poissonand χ2 processes

A.1

Poisson distribution

The Poisson distribution represents the probability of measuring k discrete occurancesfrom a Poisson process with expectation value µ. Often times we are dealing witha random process with a mean rate λ. Then, for a given observation time, L, theexpectation value is µ = λ L. The Poisson distribution has a discrete probability
density function,
PPoiss (k|µ) = f (k; µ) =µk e−µ.
k!

(A.1)

The cumulative distribution function is the probability of measuring k or fewer occurances,g(k; µ) =

n=k
�
n=0

f (n; µ) =Γ(k + 1, µ)Γ(k + 1, µ)== Q(k + 1, µ)k!
Γ(k + 1)
169

(A.2)

where we have defined the gamma function, the upper incomplete gamma function,and the regularized upper incomplete gamma function, Q,
Γ(z) =

� ∞tz−1 e−t dt(A.3)

tz−1 e−t dt

(A.4)

0

Γ(z, x) =� ∞x

Q(z, x) =Γ(z, x)
.
Γ(z)

(A.5)

For the purposes of measuring statistical significance of some measurement excess,we are generally interested in the probability of measuring k or more occurances,
n=∞
�
n=k

f (n; µ) = 1 − g(k − 1; µ) = 1 −Γ(k, µ)= 1 − Q(k, µ) = P (k, µ)
Γ(k)

(A.6)

where we have defined the regularized lower incomplete gamma function, P , which
can be expressed in terms of the lower incomplete gamma function γ,γ(z, x) =� x
0P (z, x) =tz−1 e−t dt =Γ( z) − Γ(z, x)γ(z, x)= 1 − Q(z, x).
Γ(z)

(A.7)
(A.8)

We can then define the significance of this excess as the negative natural logarithmof this probability of observing k or more occurances,
Z = − ln P (k, µ).

A.2

(A.9)

Gaussian approximation

For large number statistics (µ � 1), we can make use of the Gaussian approximation(σ 2 = µ) to the Poisson distribution,1f (k; µ) ≈ √
exp
2πµ
170

�

−(k − µ)22µ

�(A.10)

and use the number of sigmas away from mean as a proxy for f (k, µ) itself,k−µnσ ≡ √ .
µ(A.11)

This does not help us integrate the Poisson distribution for calculating a significance.
In fact we need to be careful not to compare the discrete Poisson cumulative distribution to a continuous integration of the Gaussian distribution. In addition theapproximation breaks down for large sigma regardless of sample size, making it inappropriate for estimating significance in the regime where the sum in equation A.6 isdominated by the first n = k term. In general the Gaussian approximation can give
the Poisson probability for large expectation value µ when the observed number ofoccurances k is not too far from the mean. Using the error function to estimate theprobability of observing k or more occurances will only be a good approximation in
the same regime.

A.3

Calculating Poisson significanceCalculating Poisson significance can be tricky because at small excesses, various approximations break down, and at large excesses the probabilities can become so smallthat numerical precision or floating-point boundaries get in the way. Although it
might not actually matter if we can distinguish between significance values of several
hundred, and the ridiculously small probabilities they correspond to, it is still nice tohave a method of calculation which behaves well numerically. Because of the extreme
probabilities involved, calculating directly the log-probability avoids many problems.

A.3.1

Gamma function approximations

The Lanczos approximation is used to calculate log-gamma,Γ(a + 1) =√

�

1
2π a + g +2171�a+1/2ea+g+1/2 Ag (a),(A.12)

where g is a fixed constant, and Ag is a simple series with precalculated coeﬃcients.
Code for calculating lnΓ( a + 1) is available in Numerical Recipies.
The lower incomplete gamma function is approximated using the series−x aγ(a, x) = e x

∞
�

Γ(a)
xn
Γ(a
+
1
+
n)
n=0

(A.13)

which can be made eﬃcient with the recursion relation,
Γ(a + 1) = aΓ(a)γ(a, x) = e−x xa(A.14)∞
�xn
a(a + 1) . . . (a + n)n=0ln γ(a, x) = −x + a ln x + ln
ln P (a, x) = ln∞�xna(a + 1) . . . (a + n)n=0γ(a, x)= ln γ(a, x) − ln Γ(a).
Γ(a)(A.15)(A.16)(A.17)The upper incomplete gamma function is approximated using the continued fraction�
1 1−a 1 2−a 2...
Γ(a, x) = e xx+ 1+ x+ 1+ x+��
1 1−a 1 2−a 2ln Γ(a, x) = −x + a ln x + ln...
x+ 1+ x+ 1+ x+Γ(a, x)= lnΓ( a, x) − ln Γ(a).
ln Q(a, x) = ln
Γ(a)
−x a

�

(A.18)(A.19)(A.20)The series expansion converges more rapidly on the domain x < a + 1 while thecontinuous fraction representation converges more rapidly for x > a + 1. Therefore if
we want to calculate the regularized lower incomplete gamma function for x > a + 1,
we first calculate Q(a, x) and use P (a, x) = 1−Q(a, x). For calculation of ln P or ln Q
at the extreme tails of the distribution (where we are often interested), it is necessary
to move the logarithm inside the incomplete gamma function when making use of thecode presented in Numerical Recipies. This generally requires only making a smallchange to take the log of the return statement.
172

A.3.2Extreme tail of the Poisson CDF

We can revisit the question of what happens at the extreme tail of the upper Poisson
CDF; equation A.6 for k � µ. Here the upper Poisson CDF is dominated by the
first term in the sum. Using the Poisson distribution function we can estimate thesignificance,−Z ≈ lnµk e−µk!

(A.21)≈ k ln µ − µ − ln k!

(A.22)≈ k ln µ − µ − ln k − ln (k − 1)!

(A.23)≈ k ln µ − µ − ln k − ln Γ(k).

(A.24)

Stirling’s approximation can be used in place of ln k! in equation A.22 to give,�1− Z ≈ k ln µ − µ − k +
2

�

1
1
ln k + k − ln 2π −
+O
2
12k

�

1
k3

�

.

(A.25)

We can compare equation A.24 with the series expansion for ln P (k, µ),∞�µn− ln Γ(k)(A.26)k(k + 1) . . . (k + n)
n=0
�
�
µ
µ2
1
= −µ + k ln µ + ln++
+ . . . − ln Γ(k)(A.27)k k(k + 1) k(k + 1)(k + 2)
�
�
1
µ
µ2
= −µ + k ln µ + ln
1++
+ . . . − ln Γ(k)(A.28)k(k + 1) (k + 1)(k + 2)��µµ2
= −µ + k ln µ − ln k − ln Γ(k) + ln 1 +++ ...
(k + 1) (k + 1)(k + 2)

−Z = ln P (k, µ) = −µ + k ln µ + ln

(A.29)
� 3 ��
µµ2µ2µ= −µ + k ln µ − ln k − ln Γ(k) +
+
−
.
+
O
(k + 1) (k + 1)(k + 2) 2(k + 1)2
k3
�

(A.30)
So our approximation for the significance in equation A.24 is too large by the amountshown by the series. This makes sense because it only includes the dominant term inthe sum for the upper Poisson CDF, and thus overestimates the significance.
173

A.4

χ2 distributions

The χ2 distribution often arises in excess power statistics as the values summed areoften the squares of Gaussian random variables. A bonus of having robust meansfor calculating regularized incomplete gamma distributions is that they are readily
applicable for calculating significances based on χ2 statistics. The random value, x,
representing the sum of squares of k random normal processes (zero-mean and unity
variance) will be χ2 distributed with k degrees of freedom,f (x; k)dx = χ2k (x)dx(A.31)The probability of observing x greater than some value E is represented by the upper
cumulative χ2 distribution,
Prob(x ≥ E; k) =� Ex

χ2k (x)dx =Γ(k/2, E/2)
= Q(k/2, E/2).
Γ(k/2)(A.32)

Thus the significance of a measurement E given k degrees of freedom is
Z = − ln Q(k/2, E/2)

174

(A.33)

Appendix B
KleineWelle channel configuration
B.1

S5 channels and frequency range

channel
H0:PEM-BSC10 ACC1Y
H0:PEM-BSC10 MAGX
H0:PEM-BSC10 MAGX
H0:PEM-BSC10 MAGY
H0:PEM-BSC10 MAGY
H0:PEM-BSC10 MAGZ
H0:PEM-BSC10 MAGZ
H0:PEM-BSC10 MIC
H0:PEM-BSC1 ACCY
H0:PEM-BSC1 MAG1X
H0:PEM-BSC1 MAG1X
H0:PEM-BSC1 MAG1Y
H0:PEM-BSC1 MAG1Y
H0:PEM-BSC1 MAG1Z
H0:PEM-BSC1 MAG1Z
H0:PEM-BSC2 ACCX
H0:PEM-BSC2 ACCY
H0:PEM-BSC3 ACCX
H0:PEM-BSC4 ACCX
H0:PEM-BSC4 ACCY
H0:PEM-BSC5 ACCX
H0:PEM-BSC5 MAGX
H0:PEM-BSC5 MAGX
H0:PEM-BSC5 MAGY
H0:PEM-BSC5 MAGY
H0:PEM-BSC5 MAGZ
H0:PEM-BSC5 MAGZ
H0:PEM-BSC5 MIC
H0:PEM-BSC6 ACCY
H0:PEM-BSC6 MAGX
H0:PEM-BSC6 MAGX
H0:PEM-BSC6 MAGY
H0:PEM-BSC6 MAGY
H0:PEM-BSC6 MAGZ
H0:PEM-BSC6 MAGZ
H0:PEM-BSC6 MIC

flow

fhigh

10
10
0.5
10
0.5
10
0.5
10
10
10
0.5
10
0.5
10
0.5
10
10
10
10
10
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0.5
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0.5
10
0.5
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10
0.5
10
0.5
10

512
512
32
512
32
512
32
512
512
512
32
512
32
512
32
512
512
512
512
512
512
512
32
512
32
512
32
512
512
512
32
512
32
512
32
512

H0:PEM-BSC7 ACCX
H0:PEM-BSC7 MIC
H0:PEM-BSC8 ACCY
H0:PEM-BSC8 MIC
H0:PEM-BSC9 ACC1X
H0:PEM-BSC9 MAGX
H0:PEM-BSC9 MAGX
H0:PEM-BSC9 MAGY
H0:PEM-BSC9 MAGY
H0:PEM-BSC9 MAGZ
H0:PEM-BSC9 MAGZ
H0:PEM-BSC9 MIC
H0:PEM-COIL MAGX
H0:PEM-COIL MAGX
H0:PEM-COIL MAGZ
H0:PEM-COIL MAGZ
H0:PEM-EX SEISX
H0:PEM-EX SEISY
H0:PEM-EX SEISZ
H0:PEM-EX V1
H0:PEM-EX V2
H0:PEM-EY SEISX
H0:PEM-EY SEISY
H0:PEM-EY SEISZ
H0:PEM-EY V1
H0:PEM-EY V2
H0:PEM-HAM1 ACCX
H0:PEM-HAM1 ACCZ
H0:PEM-HAM3 ACCX
H0:PEM-HAM7 ACCX
H0:PEM-HAM7 ACCZ
H0:PEM-HAM9 ACCX
H0:PEM-IOT1 MIC
H0:PEM-IOT7 MIC
H0:PEM-ISCT10 ACCX
H0:PEM-ISCT10 ACCY
H0:PEM-ISCT10 ACCZ
H0:PEM-ISCT10 MIC
H0:PEM-ISCT1 ACCX

175

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512
512
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512
512
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512
512
512
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512

H0:PEM-ISCT1 ACCY
H0:PEM-ISCT1 ACCZ
H0:PEM-ISCT1 MIC
H0:PEM-ISCT4 ACCX
H0:PEM-ISCT4 ACCY
H0:PEM-ISCT4 ACCZ
H0:PEM-ISCT4 MIC
H0:PEM-ISCT7 ACCX
H0:PEM-ISCT7 ACCY
H0:PEM-ISCT7 ACCZ
H0:PEM-ISCT7 MIC
H0:PEM-LVEA2 V1
H0:PEM-LVEA2 V2
H0:PEM-LVEA2 V3
H0:PEM-LVEA MAGX
H0:PEM-LVEA MAGX
H0:PEM-LVEA MAGY
H0:PEM-LVEA MAGY
H0:PEM-LVEA MAGZ
H0:PEM-LVEA MAGZ
H0:PEM-LVEA MIC
H0:PEM-LVEA SEISX
H0:PEM-LVEA SEISY
H0:PEM-LVEA SEISZ
H0:PEM-MX SEISX
H0:PEM-MX SEISY
H0:PEM-MX SEISZ
H0:PEM-MX V1
H0:PEM-MX V2
H0:PEM-MY SEISX
H0:PEM-MY SEISY
H0:PEM-MY SEISZ
H0:PEM-MY V1
H0:PEM-MY V2
H0:PEM-PSL1 ACCX
H0:PEM-PSL1 ACCZ
H0:PEM-PSL1 MIC
H0:PEM-PSL2 ACCX
H0:PEM-PSL2 ACCZ
H0:PEM-PSL2 MIC
H0:PEM-RADIO CS 1
H0:PEM-RADIO CS 2
H0:PEM-RADIO LVEA
H1:ASC-BS P
H1:ASC-BS Y
H1:ASC-ETMX P
H1:ASC-ETMX Y
H1:ASC-ETMY P
H1:ASC-ETMY Y
H1:ASC-ITMX P
H1:ASC-ITMX Y
H1:ASC-ITMY P
H1:ASC-ITMY Y
H1:ASC-QPDX DC
H1:ASC-QPDX P
H1:ASC-QPDX Y
H1:ASC-QPDY DC
H1:ASC-QPDY P
H1:ASC-QPDY Y
H1:ASC-RM P
H1:ASC-RM Y
H1:ASC-WFS1 QP
H1:ASC-WFS1 QY
H1:ASC-WFS2 IP
H1:ASC-WFS2 IY
H1:ASC-WFS2 QP
H1:ASC-WFS2 QY
H1:ASC-WFS3 IP

10
10
10
10
10
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10
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10
10
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10
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256

H1:ASC-WFS3 IY
H1:ASC-WFS4 IP
H1:ASC-WFS4 IY
H1:IOO-MC F
H1:IOO-MC F
H1:LSC-AS AC
H1:LSC-AS I
H1:LSC-AS I
H1:LSC-AS Q
H1:LSC-DARM CTRL
H1:LSC-DARM CTRL EXC DAQ
H1:LSC-DARM CTRL EXC DAQ
H1:LSC-DARM ERR
H1:LSC-DARM ERR
H1:LSC-ETMX EXC DAQ
H1:LSC-ETMX EXC DAQ
H1:LSC-MC L
H1:LSC-MICH CTRL
H1:LSC-MICH CTRL
H1:LSC-POB I
H1:LSC-POB I
H1:LSC-POB Q
H1:LSC-POB Q
H1:LSC-POBS DC
H1:LSC-PRC CTRL
H1:LSC-PRC CTRL
H1:LSC-REFL DC
H1:LSC-REFL I
H1:LSC-REFL I
H1:LSC-REFL Q
H1:LSC-REFL Q
H1:SUS-BS OPLEV PERROR
H1:SUS-BS OPLEV YERROR
H1:SUS-ETMX OPLEV PERROR
H1:SUS-ETMX OPLEV YERROR
H1:SUS-ETMY OPLEV PERROR
H1:SUS-ETMY OPLEV YERROR
H1:SUS-ITMX OPLEV PERROR
H1:SUS-ITMX OPLEV YERROR
H1:SUS-ITMY OPLEV PERROR
H1:SUS-ITMY OPLEV YERROR
H1:SUS-MMT3 OPLEV PERROR
H1:SUS-MMT3 OPLEV YERROR
H1:SUS-RM OPLEV PERROR
H1:SUS-RM OPLEV YERROR
H1:TCS-ITMX PD1AC
H1:TCS-ITMX PD2AC
H1:TCS-ITMY PD1AC
H1:TCS-ITMY PD2AC
H2:ASC-BS P
H2:ASC-BS Y
H2:ASC-ETMX P
H2:ASC-ETMX Y
H2:ASC-ETMY P
H2:ASC-ETMY Y
H2:ASC-ITMX P
H2:ASC-ITMX Y
H2:ASC-ITMY P
H2:ASC-ITMY Y
H2:ASC-QPDX DC
H2:ASC-QPDX P
H2:ASC-QPDX Y
H2:ASC-QPDY DC
H2:ASC-QPDY P
H2:ASC-QPDY Y
H2:ASC-RM P
H2:ASC-RM Y

176

4
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1024
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1024
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1024
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1024
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1024
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256
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1024
2048
1024
1024
4096
1024
1024
1024
2048
1024
4096
1024
2048
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2048
1024
4096
1024
4096
1024
1024
2048
1024
1024
4096
1024
2048
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256
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256
256
256

H2:ASC-WFS1 QP
H2:ASC-WFS1 QY
H2:ASC-WFS2 IP
H2:ASC-WFS2 IY
H2:ASC-WFS2 QP
H2:ASC-WFS2 QY
H2:ASC-WFS3 IP
H2:ASC-WFS3 IY
H2:ASC-WFS4 IP
H2:ASC-WFS4 IY
H2:IOO-MC F
H2:IOO-MC F
H2:LSC-AS AC
H2:LSC-AS I
H2:LSC-AS I
H2:LSC-AS Q
H2:LSC-DARM CTRL
H2:LSC-DARM CTRL EXC DAQ
H2:LSC-DARM CTRL EXC DAQ
H2:LSC-DARM ERR
H2:LSC-DARM ERR
H2:LSC-ETMX CAL
H2:LSC-ETMX CAL
H2:LSC-ETMX CAL EXC DAQ
H2:LSC-ETMX CAL EXC DAQ
H2:LSC-ETMX EXC DAQ
H2:LSC-ETMX EXC DAQ
H2:LSC-ETMY CAL
H2:LSC-ETMY CAL
H2:LSC-ETMY CAL EXC DAQ
H2:LSC-ETMY CAL EXC DAQ
H2:LSC-MC L
H2:LSC-MC L
H2:LSC-MICH CTRL
H2:LSC-MICH CTRL
H2:LSC-POB I
H2:LSC-POB I
H2:LSC-POB Q
H2:LSC-POB Q
H2:LSC-POY DC
H2:LSC-PRC CTRL
H2:LSC-PRC CTRL
H2:LSC-REFL AC
H2:LSC-REFL I
H2:LSC-REFL I
H2:LSC-REFL Q
H2:LSC-REFL Q
H2:SUS-BS OPLEV PERROR
H2:SUS-BS OPLEV YERROR
H2:SUS-ETMX OPLEV PERROR
H2:SUS-ETMX OPLEV YERROR
H2:SUS-ETMY OPLEV PERROR
H2:SUS-ETMY OPLEV YERROR
H2:SUS-FMX OPLEV PERROR
H2:SUS-FMX OPLEV YERROR
H2:SUS-FMY OPLEV PERROR
H2:SUS-FMY OPLEV YERROR
H2:SUS-ITMX OPLEV PERROR
H2:SUS-ITMX OPLEV YERROR
H2:SUS-ITMY OPLEV PERROR
H2:SUS-ITMY OPLEV YERROR
H2:SUS-MMT3 OPLEV PERROR
H2:SUS-MMT3 OPLEV YERROR
H2:SUS-RM OPLEV PERROR
H2:SUS-RM OPLEV YERROR
H2:TCS-ITMX PD1AC
H2:TCS-ITMX PD2AC
H2:TCS-ITMY PD1AC

4
4
4
4
4
4
4
4
4
4
64
1024
64
64
1024
64
64
64
1024
64
1024
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1024
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1024
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2048
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2048
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2048
1024
2048
1024
2048
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2048
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2048
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4096
1024
4096
1024
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2048
1024
1024
4096
1024
4096
256
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256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256

H2:TCS-ITMY PD2AC
L0:PEM-BSC1 ACCX
L0:PEM-BSC1 ACCY
L0:PEM-BSC1 ACCZ
L0:PEM-BSC2 ACCX
L0:PEM-BSC2 ACCY
L0:PEM-BSC2 ACCZ
L0:PEM-BSC3 ACCX
L0:PEM-BSC3 ACCY
L0:PEM-BSC3 ACCZ
L0:PEM-BSC4 ACCX
L0:PEM-BSC4 ACCY
L0:PEM-BSC4 ACCZ
L0:PEM-BSC4 MIC
L0:PEM-BSC5 ACCX
L0:PEM-BSC5 ACCY
L0:PEM-BSC5 ACCZ
L0:PEM-BSC5 MIC
L0:PEM-COIL MAGX
L0:PEM-COIL MAGX
L0:PEM-COIL MAGZ
L0:PEM-COIL MAGZ
L0:PEM-EX BAYMIC
L0:PEM-EX MAGX
L0:PEM-EX MAGX
L0:PEM-EX MAGY
L0:PEM-EX MAGY
L0:PEM-EX MAGZ
L0:PEM-EX MAGZ
L0:PEM-EX SEISX
L0:PEM-EX SEISY
L0:PEM-EX SEISZ
L0:PEM-EX V1
L0:PEM-EY BAYMIC
L0:PEM-EY MAGX
L0:PEM-EY MAGX
L0:PEM-EY MAGY
L0:PEM-EY MAGY
L0:PEM-EY MAGZ
L0:PEM-EY MAGZ
L0:PEM-EY SEISX
L0:PEM-EY SEISY
L0:PEM-EY SEISZ
L0:PEM-EY V1
L0:PEM-HAM1 ACCX
L0:PEM-HAM1 ACCZ
L0:PEM-HAM2 ACCX
L0:PEM-HAM2 ACCZ
L0:PEM-ISCT1 ACCX
L0:PEM-ISCT1 ACCY
L0:PEM-ISCT1 ACCZ
L0:PEM-ISCT1 MIC
L0:PEM-ISCT4 ACCX
L0:PEM-ISCT4 ACCY
L0:PEM-ISCT4 ACCZ
L0:PEM-ISCT4 MIC
L0:PEM-LVEA BAYMIC
L0:PEM-LVEA MAGX
L0:PEM-LVEA MAGX
L0:PEM-LVEA MAGY
L0:PEM-LVEA MAGY
L0:PEM-LVEA MAGZ
L0:PEM-LVEA MAGZ
L0:PEM-LVEA MIC
L0:PEM-LVEA SEISX
L0:PEM-LVEA SEISY
L0:PEM-LVEA SEISZ

177

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32
512
32
512
32
512
512
512
512

L0:PEM-LVEA V1
L0:PEM-RADIO LVEA
L1:ASC-BS P
L1:ASC-BS Y
L1:ASC-ETMX P
L1:ASC-ETMX Y
L1:ASC-ETMY P
L1:ASC-ETMY Y
L1:ASC-ITMX P
L1:ASC-ITMX Y
L1:ASC-ITMY P
L1:ASC-ITMY Y
L1:ASC-QPDX DC
L1:ASC-QPDX P
L1:ASC-QPDX Y
L1:ASC-QPDY DC
L1:ASC-QPDY P
L1:ASC-QPDY Y
L1:ASC-RM P
L1:ASC-RM Y
L1:ASC-WFS1 QP
L1:ASC-WFS1 QY
L1:ASC-WFS2 IP
L1:ASC-WFS2 IY
L1:ASC-WFS2 QP
L1:ASC-WFS2 QY
L1:ASC-WFS3 IP
L1:ASC-WFS3 IY
L1:ASC-WFS4 IP
L1:ASC-WFS4 IY
L1:IOO-MC F
L1:IOO-MC F
L1:LSC-AS AC
L1:LSC-AS I
L1:LSC-AS I
L1:LSC-AS Q
L1:LSC-DARM CTRL
L1:LSC-DARM CTRL EXC DAQ
L1:LSC-DARM CTRL EXC DAQ
L1:LSC-DARM ERR

B.2

10
10
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
64
1024
64
64
1024
64
64
64
1024
64

512
512
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
256
1024
2048
1024
1024
4096
1024
1024
1024
2048
1024

L1:LSC-DARM ERR
L1:LSC-ETMX CAL
L1:LSC-ETMX CAL
L1:LSC-ETMX EXC DAQ
L1:LSC-ETMX EXC DAQ
L1:LSC-ETMY CAL
L1:LSC-ETMY CAL
L1:LSC-MICH CTRL
L1:LSC-MICH CTRL
L1:LSC-POB I
L1:LSC-POB I
L1:LSC-POB Q
L1:LSC-POB Q
L1:LSC-PRC CTRL
L1:LSC-PRC CTRL
L1:LSC-REFL AC
L1:LSC-REFL I
L1:LSC-REFL I
L1:LSC-REFL Q
L1:LSC-REFL Q
L1:SEI-ETMX STS2 X
L1:SEI-ETMY STS2 Y
L1:SEI-LVEA STS2 X
L1:SEI-LVEA STS2 Y
L1:SEI-LVEA STS2 Z
L1:SUS-BS OPLEV PERROR
L1:SUS-BS OPLEV YERROR
L1:SUS-ETMX OPLEV PERROR
L1:SUS-ETMX OPLEV YERROR
L1:SUS-ETMY OPLEV PERROR
L1:SUS-ETMY OPLEV YERROR
L1:SUS-ITMX OPLEV PERROR
L1:SUS-ITMX OPLEV YERROR
L1:SUS-ITMY OPLEV PERROR
L1:SUS-ITMY OPLEV YERROR
L1:SUS-MMT3 OPLEV PERROR
L1:SUS-MMT3 OPLEV YERROR
L1:SUS-RM OPLEV PERROR
L1:SUS-RM OPLEV YERROR

1024
64
1024
64
1024
64
1024
64
1024
64
1024
64
1024
64
1024
64
64
1024
64
1024
1
1
1
1
1
4
4
4
4
4
4
4
4
4
4
4
4
4
4

4096
1024
4096
1024
2048
1024
4096
1024
2048
1024
4096
1024
4096
1024
2048
1024
1024
4096
1024
2048
128
128
128
128
128
256
256
256
256
256
256
256
256
256
256
256
256
256
256

S6 channels and frequency range

channel
H0:PEM-BSC10 ACC1Y
H0:PEM-BSC10 MAGX
H0:PEM-BSC10 MAGY
H0:PEM-BSC10 MAGZ
H0:PEM-BSC10 MIC
H0:PEM-BSC1 ACCY
H0:PEM-BSC1 MAG1X
H0:PEM-BSC1 MAG1Y
H0:PEM-BSC1 MAG1Z
H0:PEM-BSC2 ACCX
H0:PEM-BSC2 ACCY
H0:PEM-BSC3 ACCX
H0:PEM-BSC4 ACCX
H0:PEM-BSC4 ACCY
H0:PEM-BSC5 MIC
H0:PEM-BSC6 MIC
H0:PEM-BSC7 ACCX
H0:PEM-BSC7 MIC
H0:PEM-BSC8 ACCY

flow

fhigh

8
1
1
1
8
8
1
1
1
8
8
8
8
8
8
8
8
8
8

1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024

178

H0:PEM-BSC8 MIC
H0:PEM-BSC9 ACC1X
H0:PEM-BSC9 MAGX
H0:PEM-BSC9 MAGY
H0:PEM-BSC9 MAGZ
H0:PEM-BSC9 MIC
H0:PEM-COIL MAGX
H0:PEM-COIL MAGZ
H0:PEM-EX PWR1
H0:PEM-EX SEISX
H0:PEM-EX SEISX
H0:PEM-EX SEISY
H0:PEM-EX SEISY
H0:PEM-EX SEISZ
H0:PEM-EX SEISZ
H0:PEM-EX V1
H0:PEM-EY SEISX
H0:PEM-EY SEISX
H0:PEM-EY SEISY
H0:PEM-EY SEISY
H0:PEM-EY SEISZ
H0:PEM-EY SEISZ

8
8
1
1
1
8
1
1
8
1
8
1
8
1
8
8
1
8
1
8
1
8

1024
1024
1024
1024
1024
1024
1024
1024
1024
8
128
8
128
8
128
1024
8
128
8
128
8
128

H0:PEM-EY V1
H0:PEM-HAM1 ACCX
H0:PEM-HAM1 ACCZ
H0:PEM-HAM3 ACCX
H0:PEM-HAM6 ACCX
H0:PEM-HAM6 ACCY
H0:PEM-HAM6 ACCZ
H0:PEM-HAM6 MIC
H0:PEM-HAM7 ACCX
H0:PEM-HAM7 ACCZ
H0:PEM-HAM9 ACCX
H0:PEM-IOT1 MIC
H0:PEM-IOT7 MIC
H0:PEM-ISCT10 ACCX
H0:PEM-ISCT10 ACCY
H0:PEM-ISCT10 ACCZ
H0:PEM-ISCT10 MIC
H0:PEM-ISCT1 ACCX
H0:PEM-ISCT1 ACCY
H0:PEM-ISCT1 ACCZ
H0:PEM-ISCT1 MIC
H0:PEM-ISCT4 ACCPER
H0:PEM-ISCT4 ACCZ
H0:PEM-ISCT4 MIC
H0:PEM-ISCT7 ACCX
H0:PEM-ISCT7 ACCY
H0:PEM-ISCT7 ACCZ
H0:PEM-ISCT7 MIC
H0:PEM-LSC1 MAGX
H0:PEM-LSC1 MAGY
H0:PEM-LSC1 MAGZ
H0:PEM-LVEA2 V1
H0:PEM-LVEA2 V2
H0:PEM-LVEA2 V3
H0:PEM-LVEA MAGX
H0:PEM-LVEA MAGY
H0:PEM-LVEA MAGZ
H0:PEM-LVEA MIC
H0:PEM-LVEA MIC
H0:PEM-LVEA PWR1
H0:PEM-LVEA SEISX
H0:PEM-LVEA SEISX
H0:PEM-LVEA SEISY
H0:PEM-LVEA SEISY
H0:PEM-LVEA SEISZ
H0:PEM-LVEA SEISZ
H0:PEM-MX SEISX
H0:PEM-MX SEISX
H0:PEM-MX SEISY
H0:PEM-MX SEISY
H0:PEM-MX SEISZ
H0:PEM-MX SEISZ
H0:PEM-MY SEISX
H0:PEM-MY SEISX
H0:PEM-MY SEISY
H0:PEM-MY SEISY
H0:PEM-MY SEISZ
H0:PEM-MY SEISZ
H0:PEM-OMC1 MAGX
H0:PEM-OUT PWR1
H0:PEM-PSL1 ACCX
H0:PEM-PSL1 ACCY
H0:PEM-PSL1 ACCZ
H0:PEM-PSL1 MIC
H0:PEM-PSL2 ACCX
H0:PEM-PSL2 ACCZ
H0:PEM-PSL2 MIC
H0:PEM-RACK 1Y22 MAGZ

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
1
1
1
8
8
8
1
1
1
32
1024
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
8
8
8
8
8
8
8
1

1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
2048
4096
1024
8
128
8
128
8
128
8
128
8
128
8
128
8
128
8
128
8
128
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024

179

H0:PEM-RADIO CS 1
H0:PEM-RADIO CS 2
H0:PEM-RADIO LVEA
H0:PEM-RADIO LVEA H1
H1:ASC-BS P
H1:ASC-BS Y
H1:ASC-ETMX P
H1:ASC-ETMX Y
H1:ASC-ETMY P
H1:ASC-ETMY Y
H1:ASC-ITMX P
H1:ASC-ITMX Y
H1:ASC-ITMY P
H1:ASC-ITMY Y
H1:ASC-QPDX DC
H1:ASC-QPDX P
H1:ASC-QPDX Y
H1:ASC-QPDY DC
H1:ASC-QPDY P
H1:ASC-QPDY Y
H1:ASC-RM P
H1:ASC-RM Y
H1:ASC-WFS1 QP
H1:ASC-WFS1 QY
H1:ASC-WFS2 IP
H1:ASC-WFS2 IY
H1:ASC-WFS2 QP
H1:ASC-WFS2 QY
H1:ASC-WFS3 IP
H1:ASC-WFS3 IY
H1:ASC-WFS4 IP
H1:ASC-WFS4 IY
H1:IOO-MC F
H1:ISI-OMC CONT RX IN1 DAQ
H1:ISI-OMC CONT RY IN1 DAQ
H1:ISI-OMC CONT RZ IN1 DAQ
H1:ISI-OMC CONT X IN1 DAQ
H1:ISI-OMC CONT Y IN1 DAQ
H1:ISI-OMC CONT Z IN1 DAQ
H1:ISI-OMC DISPPF H1 IN1 DAQ
H1:ISI-OMC DISPPF V1 IN1 DAQ
H1:ISI-OMC GEOPF H1 IN1 DAQ
H1:ISI-OMC GEOPF H2 IN1 DAQ
H1:ISI-OMC GEOPF H3 IN1 DAQ
H1:ISI-OMC GEOPF V1 IN1 DAQ
H1:ISI-OMC GEOPF V2 IN1 DAQ
H1:ISI-OMC GEOPF V3 IN1 DAQ
H1:LSC-AS AC
H1:LSC-AS DC
H1:LSC-DARM CTRL
H1:LSC-DARM CTRL
H1:LSC-DARM CTRL EXC DAQ
H1:LSC-DARM CTRL EXC DAQ
H1:LSC-DARM ERR
H1:LSC-DARM ERR
H1:LSC-ETMX CAL
H1:LSC-ETMX CAL
H1:LSC-ETMX CAL EXC DAQ
H1:LSC-ETMX CAL EXC DAQ
H1:LSC-ETMX EXC DAQ
H1:LSC-ETMX EXC DAQ
H1:LSC-ETMY CAL
H1:LSC-ETMY CAL
H1:LSC-ETMY CAL EXC DAQ
H1:LSC-ETMY CAL EXC DAQ
H1:LSC-MC L
H1:LSC-MICH CTRL

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
32
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
32

1024
1024
1024
1024
256
256
256
256
256
256
256
256
256
256
128
128
512
128
128
128
256
256
256
256
256
256
256
256
256
256
256
256
2048
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
128
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
2048

H1:LSC-POB I
H1:LSC-POB I
H1:LSC-POB Q
H1:LSC-POB Q
H1:LSC-POBS DC
H1:LSC-PRC CTRL
H1:LSC-REFL DC
H1:LSC-REFL I
H1:LSC-REFL I
H1:LSC-REFL Q
H1:LSC-SPOB I
H1:OMC-DUOTONE OUT DAQ
H1:OMC-DUOTONE OUT DAQ
H1:OMC-HTR DRV OUT DAQ
H1:OMC-LSC Q OUT DAQ
H1:OMC-NULLSTREAM OUT DAQ
H1:OMC-PD SUM OUT DAQ
H1:OMC-PD SUM OUT DAQ
H1:OMC-PZT LSC OUT DAQ
H1:OMC-PZT VMON AC OUT DAQ
H1:OMC-PZT VMON DC OUT DAQ
H1:OMC-QPD1 P OUT DAQ
H1:OMC-QPD1 SUM IN1 DAQ
H1:OMC-QPD1 Y OUT DAQ
H1:OMC-QPD2 P OUT DAQ
H1:OMC-QPD2 SUM IN1 DAQ
H1:OMC-QPD2 Y OUT DAQ
H1:OMC-QPD3 P OUT DAQ
H1:OMC-QPD3 SUM IN1 DAQ
H1:OMC-QPD3 SUM IN1 DAQ
H1:OMC-QPD3 Y OUT DAQ
H1:OMC-QPD4 P OUT DAQ
H1:OMC-QPD4 SUM IN1 DAQ
H1:OMC-QPD4 Y OUT DAQ
H1:OMC-TT1 SUSPIT IN1 DAQ
H1:OMC-TT1 SUSPOS IN1 DAQ
H1:OMC-TT1 SUSYAW IN1 DAQ
H1:OMC-TT2 SUSPIT IN1 DAQ
H1:OMC-TT2 SUSPOS IN1 DAQ
H1:OMC-TT2 SUSYAW IN1 DAQ
H1:PSL-FSS MIXERM F
H1:SUS-BS OPLEV PERROR
H1:SUS-BS OPLEV YERROR
H1:SUS-BS SENSOR SIDE
H1:SUS-BS SUSPIT IN
H1:SUS-BS SUSPOS IN
H1:SUS-BS SUSYAW IN
H1:SUS-ETMX COIL LL
H1:SUS-ETMX COIL LR
H1:SUS-ETMX COIL UL
H1:SUS-ETMX COIL UR
H1:SUS-ETMX OPLEV PERROR
H1:SUS-ETMX OPLEV YERROR
H1:SUS-ETMX SENSOR SIDE
H1:SUS-ETMX SUSPIT IN
H1:SUS-ETMX SUSPOS IN
H1:SUS-ETMX SUSYAW IN
H1:SUS-ETMY COIL LL
H1:SUS-ETMY COIL LR
H1:SUS-ETMY COIL UL
H1:SUS-ETMY COIL UR
H1:SUS-ETMY OPLEV PERROR
H1:SUS-ETMY OPLEV YERROR
H1:SUS-ETMY SENSOR SIDE
H1:SUS-ETMY SUSPIT IN
H1:SUS-ETMY SUSPOS IN
H1:SUS-ETMY SUSYAW IN
H1:SUS-ITMX COIL LL

32
1024
32
1024
8
32
8
32
1024
32
8
32
1024
8
256
32
32
1024
8
32
8
8
8
8
8
8
8
8
32
1024
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8

2048
4096
2048
4096
512
2048
512
2048
4096
2048
128
2048
4096
256
2048
2048
2048
4096
256
2048
256
1024
1024
1024
1024
1024
1024
1024
2048
4096
1024
1024
1024
1024
512
512
512
512
512
512
1024
256
256
256
32
32
32
1024
1024
1024
1024
256
256
256
32
32
32
1024
1024
1024
1024
256
256
256
32
32
32
1024

180

H1:SUS-ITMX COIL LR
H1:SUS-ITMX COIL UL
H1:SUS-ITMX COIL UR
H1:SUS-ITMX OPLEV PERROR
H1:SUS-ITMX OPLEV YERROR
H1:SUS-ITMX SENSOR SIDE
H1:SUS-ITMX SUSPIT IN
H1:SUS-ITMX SUSPOS IN
H1:SUS-ITMX SUSYAW IN
H1:SUS-ITMY COIL LL
H1:SUS-ITMY COIL LR
H1:SUS-ITMY COIL UL
H1:SUS-ITMY COIL UR
H1:SUS-ITMY OPLEV PERROR
H1:SUS-ITMY OPLEV YERROR
H1:SUS-ITMY SENSOR SIDE
H1:SUS-ITMY SUSPIT IN
H1:SUS-ITMY SUSPOS IN
H1:SUS-ITMY SUSYAW IN
H1:SUS-MMT3 OPLEV PERROR
H1:SUS-MMT3 OPLEV YERROR
H1:SUS-RM OPLEV PERROR
H1:SUS-RM OPLEV YERROR
H1:SUS-RM SENSOR SIDE
H1:SUS-RM SUSPIT IN
H1:SUS-RM SUSPOS IN
H1:SUS-RM SUSYAW IN
H1:TCS-ITMX PD ISS OUT AC
H1:TCS-ITMY PD ISS OUT AC
L0:PEM-BSC1 ACCX
L0:PEM-BSC1 ACCY
L0:PEM-BSC1 ACCZ
L0:PEM-BSC2 ACCX
L0:PEM-BSC2 ACCY
L0:PEM-BSC2 ACCZ
L0:PEM-BSC3 ACCX
L0:PEM-BSC3 ACCY
L0:PEM-BSC3 ACCZ
L0:PEM-BSC4 ACCX
L0:PEM-BSC4 ACCY
L0:PEM-BSC4 ACCZ
L0:PEM-BSC4 MIC
L0:PEM-BSC5 ACCX
L0:PEM-BSC5 ACCY
L0:PEM-BSC5 ACCZ
L0:PEM-BSC5 MIC
L0:PEM-COIL MAGX
L0:PEM-COIL MAGZ
L0:PEM-EX BAYMIC
L0:PEM-EX MAGX
L0:PEM-EX MAGY
L0:PEM-EX MAGZ
L0:PEM-EX SEISX
L0:PEM-EX SEISX
L0:PEM-EX SEISY
L0:PEM-EX SEISY
L0:PEM-EX SEISZ
L0:PEM-EX SEISZ
L0:PEM-EX V1
L0:PEM-EY BAYMIC
L0:PEM-EY MAGX
L0:PEM-EY MAGY
L0:PEM-EY MAGZ
L0:PEM-EY SEISX
L0:PEM-EY SEISX
L0:PEM-EY SEISY
L0:PEM-EY SEISY

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
1
1
8
1
1
1
1
8
1
8
1
8
8
8
1
1
1
1
8
1
8

1024
1024
1024
256
256
256
32
32
32
1024
1024
1024
1024
256
256
256
32
32
32
256
256
256
256
256
32
32
32
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
8
128
8
128
8
128
1024
1024
1024
1024
1024
8
128
8
128

L0:PEM-EY SEISZ
L0:PEM-EY SEISZ
L0:PEM-EY V1
L0:PEM-HAM1 ACCX
L0:PEM-HAM1 ACCZ
L0:PEM-HAM2 ACCX
L0:PEM-HAM2 ACCZ
L0:PEM-HAM6 ACCX
L0:PEM-HAM6 ACCY
L0:PEM-HAM6 ACCZ
L0:PEM-HAM6 MIC
L0:PEM-ISCT1 ACCX
L0:PEM-ISCT1 ACCY
L0:PEM-ISCT1 ACCZ
L0:PEM-ISCT1 MIC
L0:PEM-LVEA BAYMIC
L0:PEM-LVEA MAGX
L0:PEM-LVEA MAGY
L0:PEM-LVEA MAGZ
L0:PEM-LVEA MIC
L0:PEM-LVEA MIC
L0:PEM-LVEA SEISX
L0:PEM-LVEA SEISX
L0:PEM-LVEA SEISY
L0:PEM-LVEA SEISY
L0:PEM-LVEA SEISZ
L0:PEM-LVEA SEISZ
L0:PEM-PSL1 ACCX
L0:PEM-PSL1 ACCY
L0:PEM-PSL1 ACCZ
L0:PEM-PSL1 MIC
L0:PEM-RADIO LVEA
L0:PEM-RADIO ROOF
L1:ASC-BS P
L1:ASC-BS Y
L1:ASC-ETMX P
L1:ASC-ETMX Y
L1:ASC-ETMY P
L1:ASC-ETMY Y
L1:ASC-ITMX P
L1:ASC-ITMX Y
L1:ASC-ITMY P
L1:ASC-ITMY Y
L1:ASC-QPDX DC
L1:ASC-QPDX P
L1:ASC-QPDX Y
L1:ASC-QPDY DC
L1:ASC-QPDY P
L1:ASC-QPDY Y
L1:ASC-RM P
L1:ASC-RM Y
L1:ASC-WFS1 QP
L1:ASC-WFS1 QY
L1:ASC-WFS2 IP
L1:ASC-WFS2 IY
L1:ASC-WFS2 QP
L1:ASC-WFS2 QY
L1:ASC-WFS3 IP
L1:ASC-WFS3 IY
L1:ASC-WFS4 IP
L1:ASC-WFS4 IY
L1:IOO-MC F
L1:ISI-OMC CONT RX IN1 DAQ
L1:ISI-OMC CONT RY IN1 DAQ
L1:ISI-OMC CONT RZ IN1 DAQ
L1:ISI-OMC CONT X IN1 DAQ
L1:ISI-OMC CONT Y IN1 DAQ
L1:ISI-OMC CONT Z IN1 DAQ

1
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
1
1
1
32
1024
1
8
1
8
1
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
32
8
8
8
8
8
8

8
128
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
1024
2048
4096
8
128
8
128
8
128
1024
1024
1024
1024
1024
1024
256
256
256
256
256
256
256
256
256
256
128
128
128
128
128
128
256
256
256
256
256
256
256
256
256
256
256
256
2048
1024
1024
1024
1024
1024
1024

181

L1:ISI-OMC DISPPF H1 IN1 DAQ
L1:ISI-OMC DISPPF V1 IN1 DAQ
L1:ISI-OMC GEOPF H1 IN1 DAQ
L1:ISI-OMC GEOPF H2 IN1 DAQ
L1:ISI-OMC GEOPF H3 IN1 DAQ
L1:ISI-OMC GEOPF V1 IN1 DAQ
L1:ISI-OMC GEOPF V2 IN1 DAQ
L1:ISI-OMC GEOPF V3 IN1 DAQ
L1:LSC-AS AC
L1:LSC-AS DC
L1:LSC-DARM CTRL
L1:LSC-DARM CTRL
L1:LSC-DARM CTRL EXC DAQ
L1:LSC-DARM CTRL EXC DAQ
L1:LSC-DARM ERR
L1:LSC-DARM ERR
L1:LSC-ETMX CAL
L1:LSC-ETMX CAL
L1:LSC-ETMX CAL EXC DAQ
L1:LSC-ETMX CAL EXC DAQ
L1:LSC-ETMX EXC DAQ
L1:LSC-ETMX EXC DAQ
L1:LSC-ETMY CAL
L1:LSC-ETMY CAL
L1:LSC-ETMY CAL EXC DAQ
L1:LSC-ETMY CAL EXC DAQ
L1:LSC-MC L
L1:LSC-MICH CTRL
L1:LSC-POB I
L1:LSC-POB I
L1:LSC-POB Q
L1:LSC-POB Q
L1:LSC-PRC CTRL
L1:LSC-REFL AC
L1:LSC-REFL DC
L1:LSC-REFL I
L1:LSC-REFL I
L1:LSC-REFL Q
L1:LSC-SPOB I
L1:OMC-ASC ANG Y IN1 DAQ
L1:OMC-ASC POS X IN1 DAQ
L1:OMC-ASC POS Y IN1 DAQ
L1:OMC-DUOTONE OUT DAQ
L1:OMC-DUOTONE OUT DAQ
L1:OMC-NULLSTREAM OUT DAQ
L1:OMC-NULLSTREAM OUT DAQ
L1:OMC-PD SUM OUT DAQ
L1:OMC-PD SUM OUT DAQ
L1:OMC-PZT LSC OUT DAQ
L1:OMC-PZT VMON AC OUT DAQ
L1:OMC-PZT VMON DC OUT DAQ
L1:OMC-QPD1 P OUT DAQ
L1:OMC-QPD1 SUM OUT DAQ
L1:OMC-QPD1 Y OUT DAQ
L1:OMC-QPD2 P OUT DAQ
L1:OMC-QPD2 SUM OUT DAQ
L1:OMC-QPD2 Y OUT DAQ
L1:OMC-QPD3 P OUT DAQ
L1:OMC-QPD3 SUM IN1 DAQ
L1:OMC-QPD3 Y OUT DAQ
L1:OMC-QPD4 P OUT DAQ
L1:OMC-QPD4 SUM IN1 DAQ
L1:OMC-QPD4 Y OUT DAQ
L1:OMC-TT1 SUSPIT IN1 DAQ
L1:OMC-TT1 SUSPOS IN1 DAQ
L1:OMC-TT1 SUSYAW IN1 DAQ
L1:OMC-TT2 SUSPIT IN1 DAQ

8
8
8
8
8
8
8
8
8
8
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
1024
32
32
32
1024
32
1024
32
8
8
32
1024
32
8
8
8
8
32
1024
32
1024
32
1024
8
32
8
32
32
32
32
32
32
8
8
8
8
8
8
8
8
8
8

1024
1024
1024
1024
1024
1024
1024
1024
1024
512
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
4096
2048
2048
2048
4096
2048
4096
2048
1024
128
2048
4096
2048
128
1024
1024
1024
2048
4096
2048
4096
2048
4096
1024
2048
256
2048
2048
2048
2048
2048
2048
1024
1024
1024
1024
1024
1024
64
64
64
64

L1:OMC-TT2 SUSPOS IN1 DAQ
L1:OMC-TT2 SUSYAW IN1 DAQ
L1:PSL-FSS MIXERM F
L1:SEI-BS RX
L1:SEI-BS RY
L1:SEI-BS RZ
L1:SEI-BS X
L1:SEI-BS Y
L1:SEI-BS Z
L1:SEI-ETMX RX
L1:SEI-ETMX RY
L1:SEI-ETMX RZ
L1:SEI-ETMX STS2 X
L1:SEI-ETMX X
L1:SEI-ETMX Y
L1:SEI-ETMX Z
L1:SEI-ETMY RX
L1:SEI-ETMY RY
L1:SEI-ETMY RZ
L1:SEI-ETMY STS2 Y
L1:SEI-ETMY X
L1:SEI-ETMY Y
L1:SEI-ETMY Z
L1:SEI-ITMX RX
L1:SEI-ITMX RY
L1:SEI-ITMX RZ
L1:SEI-ITMX X
L1:SEI-ITMX Y
L1:SEI-ITMX Z
L1:SEI-ITMY RX
L1:SEI-ITMY RY
L1:SEI-ITMY RZ
L1:SEI-ITMY X
L1:SEI-ITMY Y
L1:SEI-ITMY Z
L1:SEI-LVEA STS2 X
L1:SEI-LVEA STS2 Y
L1:SEI-LVEA STS2 Z
L1:SEI-MC1 RX
L1:SEI-MC1 RY
L1:SEI-MC1 RZ
L1:SEI-MC1 X
L1:SEI-MC1 Y
L1:SEI-MC1 Z
L1:SEI-MC2 RX
L1:SEI-MC2 RY
L1:SEI-MC2 RZ
L1:SEI-MC2 X
L1:SEI-MC2 Y
L1:SEI-MC2 Z
L1:SEI-OUT RX

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8

64
64
1024
128
128
128
128
128
128
128
128
128
256
128
128
128
128
128
128
256
128
128
128
128
128
128
128
128
128
128
128
128
128
128
128
256
256
256
128
128
128
128
128
128
128
128
128
128
128
128
128

182

L1:SEI-OUT RY
L1:SEI-OUT RZ
L1:SEI-OUT X
L1:SEI-OUT Y
L1:SEI-RM RX
L1:SEI-RM RY
L1:SEI-RM RZ
L1:SEI-RM X
L1:SEI-RM Y
L1:SEI-RM Z
L1:SUS-BS OPLEV PERROR
L1:SUS-BS OPLEV YERROR
L1:SUS-BS SENSOR SIDE
L1:SUS-BS SUSPIT IN
L1:SUS-BS SUSPOS IN
L1:SUS-BS SUSYAW IN
L1:SUS-ETMX OPLEV PERROR
L1:SUS-ETMX OPLEV YERROR
L1:SUS-ETMX SENSOR SIDE
L1:SUS-ETMX SUSPIT IN
L1:SUS-ETMX SUSPOS IN
L1:SUS-ETMX SUSYAW IN
L1:SUS-ETMY OPLEV PERROR
L1:SUS-ETMY OPLEV YERROR
L1:SUS-ETMY SENSOR SIDE
L1:SUS-ETMY SUSPIT IN
L1:SUS-ETMY SUSPOS IN
L1:SUS-ETMY SUSYAW IN
L1:SUS-ITMX OPLEV PERROR
L1:SUS-ITMX OPLEV YERROR
L1:SUS-ITMX SENSOR SIDE
L1:SUS-ITMX SUSPIT IN
L1:SUS-ITMX SUSPOS IN
L1:SUS-ITMX SUSYAW IN
L1:SUS-ITMY OPLEV PERROR
L1:SUS-ITMY OPLEV YERROR
L1:SUS-ITMY SENSOR SIDE
L1:SUS-ITMY SUSPIT IN
L1:SUS-ITMY SUSPOS IN
L1:SUS-ITMY SUSYAW IN
L1:SUS-MMT3 OPLEV PERROR
L1:SUS-MMT3 OPLEV YERROR
L1:SUS-RM OPLEV PERROR
L1:SUS-RM OPLEV YERROR
L1:SUS-RM SENSOR SIDE
L1:SUS-RM SUSPIT IN
L1:SUS-RM SUSPOS IN
L1:SUS-RM SUSYAW IN
L1:TCS-ITMX PD ISS OUT AC
L1:TCS-ITMY PD ISS OUT AC

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8

128
128
128
128
128
128
128
128
128
128
256
256
32
32
32
32
256
256
256
32
32
32
256
256
256
32
32
32
256
256
32
32
32
32
256
256
32
32
32
32
256
256
256
256
32
32
32
32
1024
1024

Appendix C
Auxiliary channel veto statistics
KleineWelle auxiliary (AUX) channel individual veto performance on noise transients
also identified by kleineWelle running on the gravitational-wave (GW) channel during
the second calendar year of S5. For each auxiliary channel, the configuration withe
highest statistical significance is reported if it is greater than 10. Columns are,channel
flow
fhigh
thr
±ms
dt [%]
�35 [%]
�100 [%]
signifauxiliary channel transient source
lower frequency for AUX trigger generation
upper frequency for AUX trigger generation
kleineWelle significance threshold for AUX triggers used for veto
veto window in milliseconds between AUX and GW-channel transient
fractional dead-time incurred by veto
veto eﬃciency on GW noise transients with trigger significance > 35
veto eﬃciency on GW noise transients with trigger significance > 35
statistical significance of veto correlation (−ln P)

C.0.1

Veto performance on H1 transientschannel

flow

fhigh

thr

64
64
64
64
64
64
64
64
64
4
4

1024
1024
1024
1024
1024
1024
1024
1024
1024
256
256

200
35
50
50
200
200
35
50
50
400
400

H1:LSC-AS AC
H1:LSC-REFL DC
H1:LSC-PRC CTRL
H1:LSC-POB I
H1:LSC-MC L
H1:LSC-REFL I
H1:LSC-REFL Q
H1:LSC-POB Q
H1:LSC-MICH CTRL
H1:ASC-ETMX P
H1:ASC-ETMY P

183

±ms
25
25
25
25
25
25
25
25
25
50
50

dt[%]

�35

�100

signif

0.01
0.02
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.01
0.01

39.89
39.15
38.98
38.91
37.23
37.16
36.91
37.04
36.95
3.76
3.79

31.81
32.02
34.55
34.51
30.93
30.74
32.13
31.93
31.81
0.69
0.62

1414511327681302091301011216511212461163271155911153309348
9241

channel

flow

fhigh

thr

H1:ASC-ETMX Y
H1:LSC-POB I
H1:LSC-PRC CTRL
H1:ASC-ITMX Y
H1:ASC-WFS1 QP
H1:ASC-ETMY Y
H1:ASC-ITMX P
H1:ASC-ITMY Y
H1:ASC-RM Y
H1:ASC-ITMY P
H1:ASC-RM P
H1:ASC-QPDY DC
H1:ASC-WFS1 QY
H1:LSC-REFL I
H1:ASC-WFS2 QP
H1:ASC-WFS2 IY
H1:ASC-WFS3 IY
H1:ASC-WFS2 IP
H1:LSC-MICH CTRL
H1:ASC-WFS2 QY
H1:ASC-WFS3 IP
H1:LSC-POB Q
H1:ASC-QPDY P
H1:ASC-QPDY Y
H1:LSC-REFL Q
H1:ASC-WFS4 IY
H1:ASC-QPDX DC
H1:ASC-WFS4 IP
H1:ASC-BS Y
H1:ASC-QPDX Y
H1:ASC-QPDX P
H1:ASC-BS P
H1:SUS-ITMX OPLEV YERROR
H1:SUS-RM OPLEV YERROR
H0:PEM-ISCT4 ACCZ
H0:PEM-ISCT4 ACCX
H0:PEM-ISCT4 ACCY
H1:SUS-ETMX OPLEV YERROR
H0:PEM-RADIO LVEA
H1:SUS-ITMX OPLEV PERROR
H0:PEM-LVEA MIC
H1:TCS-ITMY PD1AC
H1:SUS-ETMX OPLEV PERROR
H0:PEM-MX V2
H0:PEM-BSC9 ACC1X
H0:PEM-ISCT4 MIC
H0:PEM-ISCT7 ACCZ
H0:PEM-BSC7 MIC
H0:PEM-PSL1 ACCX
H0:PEM-ISCT7 ACCX
H1:IOO-MC F
H0:PEM-ISCT1 ACCX

4
1024
1024
4
4
4
4
4
4
4
4
4
4
1024
4
4
4
4
1024
4
4
1024
4
4
1024
4
4
4
4
4
4
4
4
4
10
10
10
4
10
4
10
4
4
10
10
10
10
10
10
10
64
10

256
4096
2048
256
256
256
256
256
256
256
256
256
256
4096
256
256
256
256
2048
256
256
4096
256
256
2048
256
256
256
256
256
256
256
256
256
512
512
512
256
512
256
512
256
256
512
512
512
512
512
512
512
1024
512

50
35
35
50
100
50
50
50
50
50
100
35
100
35
50
100
35
100
35
50
100
35
35
35
35
50
50
50
35
400
100
35
35
35
1600
800
400
35
50
35
35
35
35
35
400
200
35
35
35
35
800
35

184

±ms
200
50
200
200
200
200
200
200
200
200
200
200
200
150
200
200
200
200
200
200
200
50
200
200
100
200
200
200
200
200
200
200
200
200
25
25
25
200
200
200
200
200
200
200
200
25
200
200
200
200
25
200

dt[%]

�35

�100

signif

0.02
0.00
0.00
0.02
0.01
0.02
0.01
0.02
0.02
0.01
0.00
0.07
0.02
0.00
0.01
0.00
0.11
0.00
0.00
0.07
0.00
0.00
0.02
0.01
0.00
0.04
0.03
0.07
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.06
0.00
0.00
0.53
0.01
4.13
0.00
0.00
0.04
0.00
0.01
0.02
0.00
0.03

1.25
0.53
0.63
1.21
0.88
1.07
0.98
1.10
1.09
0.94
0.68
1.33
0.85
0.57
0.69
0.54
1.38
0.44
0.40
1.01
0.46
0.27
0.62
0.39
0.15
0.46
0.39
0.52
0.18
0.09
0.13
0.13
0.08
0.08
0.01
0.01
0.01
0.03
0.16
0.03
0.03
0.74
0.04
4.68
0.01
0.01
0.11
0.03
0.05
0.07
0.01
0.08

0.65
1.54
1.47
0.71
0.86
0.54
0.66
0.56
0.65
0.65
0.46
2.72
0.66
0.71
0.50
0.20
1.47
0.35
0.93
0.65
0.52
0.83
1.26
0.76
0.46
0.47
0.51
0.50
0.12
0.07
0.08
0.12
0.06
0.02
0.01
0.01
0.01
0.01
0.10
0.01
0.01
0.52
0.02
4.66
0.01
0.01
0.10
0.01
0.02
0.07
0.01
0.03

2016
1900
1872
1861
1855
1704
1651
1645
1615
1547
1471
1339
1299
1208
1206
1191
1144
1044
1004
924
918
828
760
601
527
360
320
299
297
278
192
154
149
50
47
46
42
32
32
28
24
23
23
21
21
19
18
18
18
17
17
16

channel

flow

fhigh

thr

H0:PEM-BSC9 MIC
H0:PEM-LVEA2 V1
H0:PEM-HAM9 ACCX
H0:PEM-ISCT7 ACCY
H0:PEM-EX SEISZ
H0:PEM-BSC9 MAGX
H1:SUS-MMT3 OPLEV PERROR
H0:PEM-LVEA SEISY
H1:TCS-ITMY PD2AC
H1:SUS-BS OPLEV PERROR
H0:PEM-ISCT1 ACCY
H0:PEM-HAM7 ACCZ
H0:PEM-BSC5 ACCX
H1:IOO-MC F
H0:PEM-COIL MAGZ
H0:PEM-PSL2 ACCZ
H0:PEM-HAM7 ACCX
H0:PEM-ISCT10 ACCX
H0:PEM-LVEA2 V3
H0:PEM-EX SEISY

10
10
10
10
10
10
4
10
4
4
10
10
10
1024
10
10
10
10
10
10

512
512
512
512
512
512
256
512
256
256
512
512
512
2048
512
512
512
512
512
512

50
100
50
35
50
35
50
35
400
35
100
35
50
400
35
35
35
50
200
100

±ms
200
200
150
200
200
50
200
200
100
150
200
200
200
25
200
200
200
200
200
200

C.0.2

Veto performance on H2 transientschannel

flow

fhigh

thr

H2:LSC-PRC CTRL
H2:LSC-POB I
H2:LSC-REFL Q
H2:LSC-MICH CTRL
H2:LSC-POB Q
H2:LSC-AS AC
H2:LSC-POB I
H2:ASC-QPDY Y
H2:ASC-ETMX P
H2:ASC-ITMX P
H2:ASC-ETMY P
H2:ASC-ETMY Y
H2:ASC-WFS2 QP
H2:ASC-ITMX Y
H2:ASC-ETMX Y
H2:ASC-QPDY P
H2:ASC-ITMY P
H2:ASC-ITMY Y
H2:ASC-QPDY DC
H2:LSC-PRC CTRL
H2:ASC-WFS2 QY
H2:ASC-WFS1 QY
H2:ASC-WFS1 QP
H2:ASC-RM Y
H2:ASC-WFS4 IY

64
64
64
64
64
64
1024
4
4
4
4
4
4
4
4
4
4
4
4
1024
4
4
4
4
4

1024
1024
1024
1024
1024
1024
4096
256
256
256
256
256
256
256
256
256
256
256
256
2048
256
256
256
256
256

50
50
35
35
35
35
35
35
100
200
200
200
200
200
100
35
200
200
50
35
200
100
100
100
35

185

±ms
25
25
150
200
200
200
200
150
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

dt[%]

�35

�100

signif

0.00
0.02
0.00
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.00
4.74
0.03
0.01
0.12
0.02
0.00

0.02
0.05
0.03
0.07
0.02
0.01
0.02
0.02
0.00
0.01
0.01
0.03
0.04
0.01
5.14
0.07
0.02
0.18
0.04
0.01

0.02
0.10
0.01
0.09
0.04
0.02
0.00
0.01
0.01
0.01
0.00
0.01
0.04
0.03
5.13
0.05
0.01
0.23
0.05
0.00

15
15
15
14
14
14
13
13
13
12
12
12
12
11
11
11
10
10
10
10

dt[%]

�35

�100

signif

0.00
0.00
0.01
0.03
0.03
0.02
0.00
0.01
0.08
0.02
0.02
0.03
0.01
0.01
0.05
0.03
0.04
0.04
0.03
0.00
0.01
0.04
0.11
0.06
0.01

28.90
28.66
16.51
17.17
17.03
4.92
3.10
3.73
5.78
4.35
3.81
3.91
3.57
3.55
4.42
4.14
4.18
4.15
3.74
2.12
3.02
3.61
4.57
3.86
2.56

35.91
35.69
18.81
17.98
17.98
5.21
4.46
2.03
3.64
2.57
2.23
2.52
2.18
2.02
2.46
2.26
2.62
2.77
2.44
3.00
1.79
2.29
3.05
2.73
1.84

62487
61867
28677
26049
24915
6291
5875
5302
5249
5023
4386
4382
4378
4355
4329
4320
4245
4109
4095
4014
3599
3477
3401
3373
3289

channel

flow

fhigh

thr

H2:SUS-ITMX OPLEV YERROR
H2:ASC-QPDX P
H2:SUS-ITMX OPLEV PERROR
H2:ASC-WFS3 IY
H2:ASC-WFS4 IP
H2:SUS-ITMY OPLEV PERROR
H2:ASC-BS P
H2:ASC-WFS2 IP
H2:SUS-ITMY OPLEV YERROR
H2:LSC-REFL AC
H2:ASC-WFS2 IY
H2:SUS-ETMY OPLEV YERROR
H2:ASC-RM P
H2:SUS-ETMX OPLEV PERROR
H2:ASC-WFS3 IP
H2:SUS-ETMX OPLEV YERROR
H2:LSC-MC L
H2:LSC-REFL I
H2:SUS-ETMY OPLEV PERROR
H2:ASC-BS Y
H2:IOO-MC F
H2:LSC-REFL I
H2:TCS-ITMY PD2AC
H2:IOO-MC F
H2:LSC-MC L
H2:ASC-QPDX Y
H2:LSC-REFL Q
H2:TCS-ITMY PD1AC
H2:ASC-QPDX DC
H2:LSC-POB Q
H2:LSC-MICH CTRL
H2:SUS-RM OPLEV YERROR
H0:PEM-ISCT4 ACCY
H0:PEM-ISCT4 ACCX
H0:PEM-ISCT4 ACCZ
H0:PEM-BSC5 ACCX
H0:PEM-PSL2 ACCZ
H2:SUS-RM OPLEV PERROR
H0:PEM-MX SEISY
H0:PEM-ISCT10 ACCY
H0:PEM-MX SEISZ
H0:PEM-MX SEISX
H0:PEM-ISCT4 MIC
H2:SUS-FMX OPLEV PERROR
H0:PEM-BSC6 MAGZ
H0:PEM-BSC5 MIC
H0:PEM-PSL2 ACCX
H0:PEM-MY SEISX
H0:PEM-BSC10 MAGZ
H2:SUS-FMY OPLEV PERROR
H0:PEM-MY SEISY
H0:PEM-MY SEISZ

4
4
4
4
4
4
4
4
4
64
4
4
4
4
4
4
64
64
4
4
64
1024
4
1024
1024
4
1024
4
4
1024
1024
4
10
10
10
10
10
4
10
10
10
10
10
4
10
10
10
10
10
4
10
10

256
256
256
256
256
256
256
256
256
1024
256
256
256
256
256
256
1024
1024
256
256
1024
4096
256
2048
2048
256
4096
256
256
4096
2048
256
512
512
512
512
512
256
512
512
512
512
512
256
512
512
512
512
512
256
512
512

35
35
35
35
35
35
100
50
35
35
100
35
100
35
35
35
200
200
35
50
400
35
400
100
35
200
35
400
400
35
35
35
1600
1600
1600
800
200
35
400
35
800
800
35
50
50
35
50
35
50
50
200
200

186

±ms
150
200
150
200
200
200
200
200
150
25
200
200
200
150
200
200
25
25
200
200
25
25
100
25
25
200
25
100
200
25
25
150
200
200
200
200
50
100
200
200
150
200
200
200
200
200
200
200
100
200
200
200

dt[%]

�35

�100

signif

0.01
0.03
0.01
0.04
0.01
0.01
0.06
0.72
0.01
0.00
0.10
0.00
0.03
0.02
0.01
0.01
0.00
0.00
0.02
0.11
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
1.17
0.01
0.01
0.00
0.10
0.00
0.00
0.00

2.30
3.13
2.29
3.27
2.43
2.38
3.07
6.85
2.13
1.48
3.45
1.64
2.59
2.25
1.79
1.71
1.08
1.07
1.96
2.26
0.57
0.44
0.60
0.60
0.36
0.46
0.24
0.64
0.21
0.15
0.07
0.12
0.08
0.08
0.08
0.02
0.02
0.05
0.03
0.04
0.01
0.01
0.06
0.03
1.58
0.06
0.06
0.04
0.19
0.01
0.01
0.01

1.47
1.82
1.39
2.44
1.77
1.48
1.45
6.33
1.42
1.51
2.49
1.24
1.77
1.52
1.38
1.21
0.93
0.94
1.31
1.22
0.40
0.36
0.13
0.43
0.32
0.27
0.35
0.16
0.18
0.24
0.11
0.18
0.03
0.03
0.03
0.01
0.01
0.07
0.01
0.01
0.00
0.00
0.02
0.03
1.91
0.04
0.05
0.02
0.26
0.01
0.01
0.01

3238
3073
3017
2981
2825
2769
2570
2552
2526
2489
2477
2460
2432
2352
2027
2000
1986
1962
1835
1300
1083
832
755
755
688
683
548
538
347
345
164
148
61
59
57
34
34
32
31
30
27
24
22
21
20
18
17
16
13
12
12
10

channel
H2:SUS-MMT3 OPLEV PERROR

flow

fhigh

thr

4

256

35

±ms
200

C.0.3

Veto performance on L1 transientschannel

flow

fhigh

thr

L1:LSC-PRC CTRL
L1:LSC-MICH CTRL
L1:LSC-POB I
L1:LSC-POB Q
L1:ASC-WFS2 QY
L1:ASC-WFS2 IY
L1:ASC-RM Y
L1:ASC-ITMY Y
L1:ASC-ITMX Y
L1:ASC-ITMY P
L1:ASC-ITMX P
L1:ASC-WFS2 IP
L1:ASC-WFS1 QY
L1:ASC-RM P
L1:ASC-WFS2 QP
L1:LSC-PRC CTRL
L1:LSC-REFL Q
L1:ASC-WFS3 IY
L1:LSC-REFL I
L1:ASC-QPDX Y
L1:ASC-BS Y
L1:ASC-WFS1 QP
L1:ASC-ETMY Y
L1:ASC-ETMX Y
L1:LSC-POB I
L0:PEM-LVEA MAGY
L1:ASC-BS P
L1:LSC-MICH CTRL
L1:LSC-REFL I
L1:ASC-QPDX P
L1:ASC-QPDY Y
L1:SUS-ETMY OPLEV YERROR
L1:LSC-AS AC
L1:LSC-POB Q
L1:SUS-ITMX OPLEV YERROR
L1:ASC-QPDY DC
L1:ASC-ETMX P
L1:ASC-WFS4 IY
L1:ASC-WFS3 IP
L1:SUS-ITMY OPLEV YERROR
L1:ASC-QPDY P
L1:ASC-ETMY P
L1:LSC-REFL AC
L0:PEM-LVEA MAGZ

64
64
64
64
4
4
4
4
4
4
4
4
4
4
4
1024
64
4
64
4
4
4
4
4
1024
10
4
1024
1024
4
4
4
64
1024
4
4
4
4
4
4
4
4
64
10

1024
1024
1024
1024
256
256
256
256
256
256
256
256
256
256
256
2048
1024
256
1024
256
256
256
256
256
4096
512
256
2048
4096
256
256
256
1024
4096
256
256
256
256
256
256
256
256
1024
512

35
35
35
35
50
100
50
50
50
50
50
50
100
35
35
35
35
35
800
100
35
400
200
200
35
200
35
35
100
100
50
35
35
35
35
100
400
35
35
35
50
400
50
200

187

±ms
200
100
25
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
25
200
200
200
200
200
25
25
200
200
25
200
200
200
200
25
200
200
200
200
200
200
200
200
200
25

dt[%]

�35

�100

signif

0.00

0.02

0.01

10

dt[%]

�35

�100

signif

0.08
0.03
0.01
0.05
0.03
0.01
0.04
0.05
0.05
0.05
0.05
0.04
0.03
0.13
0.04
0.00
0.07
0.02
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.04
0.00
0.00
0.03
0.00
0.02
0.14
0.01
0.01
0.00
0.08
0.03

7.48
4.92
3.63
4.79
3.40
2.88
3.32
3.41
3.42
3.11
3.10
2.85
2.46
3.60
2.55
1.05
2.29
1.39
0.62
0.69
0.67
0.46
0.45
0.36
0.29
0.31
0.48
0.35
0.20
0.22
0.35
0.24
0.47
0.12
0.16
0.33
0.12
0.27
0.60
0.16
0.20
0.10
0.43
0.28

8.69
6.14
4.77
6.09
3.07
3.01
3.15
3.19
3.23
2.92
2.91
3.16
2.58
3.36
2.79
1.64
3.21
1.69
1.17
0.97
0.83
0.75
0.67
0.61
0.54
0.58
0.62
0.62
0.50
0.36
0.61
0.34
0.99
0.30
0.25
0.53
0.16
0.16
0.53
0.27
0.32
0.10
0.31
0.52

21864
16964
14773
14066
10514
10223
9656
9279
8982
8030
7945
7586
7250
7034
6777
4906
4815
3963
3209
2901
2231
2208
2130
1721
1385
1372
1305
1241
926
794
714
693
624
539
507
468
417
417
350
339
327
326
314
303

channel

flow

fhigh

thr

L1:ASC-WFS4 IP
L0:PEM-LVEA MAGY
L0:PEM-LVEA MAGX
L1:LSC-REFL Q
L1:SUS-MMT3 OPLEV YERROR
L1:SUS-MMT3 OPLEV PERROR
L0:PEM-BSC4 ACCZ
L1:IOO-MC F
L1:IOO-MC F
L1:SUS-ETMY OPLEV PERROR
L1:SUS-ETMX OPLEV YERROR
L1:SUS-ITMX OPLEV PERROR
L1:ASC-QPDX DC
L1:SUS-RM OPLEV YERROR
L0:PEM-HAM1 ACCX
L0:PEM-ISCT1 ACCY
L0:PEM-HAM1 ACCZ
L0:PEM-BSC4 ACCY
L0:PEM-BSC1 ACCX
L1:SUS-BS OPLEV YERROR
L0:PEM-EY MAGY
L0:PEM-BSC5 ACCX
L0:PEM-BSC2 ACCY
L0:PEM-HAM2 ACCX
L0:PEM-EY MAGX
L0:PEM-LVEA MAGZ
L0:PEM-EY MAGZ
L0:PEM-RADIO LVEA
L0:PEM-ISCT1 ACCX
L1:SUS-BS OPLEV PERROR
L0:PEM-EX SEISZ

4
0.5
10
1024
4
4
10
64
1024
4
4
4
4
4
10
10
10
10
10
4
10
10
10
10
10
0.5
10
10
10
4
10

256
32
512
2048
256
256
512
1024
2048
256
256
256
256
256
512
512
512
512
512
256
512
512
512
512
512
32
512
512
512
256
512

400
100
50
35
35
50
35
35
35
800
35
35
400
35
35
35
35
35
35
35
400
50
800
50
100
200
1600
35
50
35
35

188

±ms
200
200
25
25
200
200
200
50
50
200
200
150
200
200
200
200
200
200
200
200
25
100
100
200
50
200
150
25
200
200
100

dt[%]

�35

�100

signif

0.03
0.02
0.01
0.00
1.15
0.99
0.07
0.73
0.54
0.00
0.01
0.00
0.00
0.01
0.01
0.02
0.02
0.01
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.09
0.00
0.08
0.02
0.01
0.00

0.25
0.18
0.14
0.04
1.90
1.66
0.29
1.19
0.91
0.02
0.09
0.04
0.01
0.04
0.03
0.06
0.06
0.04
0.01
0.04
0.00
0.03
0.01
0.01
0.01
0.14
0.00
0.12
0.04
0.02
0.02

0.08
0.46
0.49
0.17
1.99
1.66
0.12
1.22
0.86
0.01
0.12
0.03
0.02
0.03
0.01
0.03
0.02
0.03
0.01
0.01
0.00
0.01
0.00
0.00
0.01
0.34
0.01
0.20
0.01
0.01
0.02

271
217
210
209
175
157
144
105
89
73
72
57
50
35
34
24
22
20
19
19
14
13
13
13
13
12
12
11
11
10
10

Appendix D
Burst detection checklist
Zero-level sanity
Record GPS/UTC time of event
Read detector logbook for surrounding days
Verify instrument state is correctly flagged
Check baseline sensitivity and range of detectors
Check times of nearest hardware signal injections
How close is event to the end of a science segment?
Data acquisition and integrity
Check timing across instruments
Record nearest reboots or software/configuration changes
Check hardware injection excitation channels
Verify data checksum and consistency between RAW and reduced data sets
Examine any other test points for stored data
Calibration
Record calibration constants and errors about event
Check robustness of event properties to diﬀerences in calibration
Is the event seen in uncalibrated data?
Check for glitches in calibration excitation channels
189

Event properties and methods
Construct detailed spectrograms of events in diﬀerent detectors
Check for time-frequency consistency
Run Coherent Event display
Extract best-fit waveform and sky location
Compare coherent and incoherent extracted parameters
Check all null streams
Record which methods and detector networks saw the event
Check event amplitude and background for diﬀerent methods
Are observations consistent with method and network sensitivity?
Compare parameter estimation from diﬀerent methods
Verify method robustness to stride boundary, prefiltering, specific thresholds
Could the event arise from nonstationary lines?
What is the search method sensitivity at the time of the event?
Check signal content at all frequencies at time of event
Is anything seen in the inspiral or ringdown searches?
State of the instrument and instrumental vetoes
Check stationary of the instrument: power spectrum and transient rates
Check levels of upconversion of low frequency noise sources
Run Q-scan (nearby transients and spectrograms) on all channels in full frames
Check for associated instrumental and environmental disturbances
Which data quality flags were in eﬀect at and near the time of the event?
Check records for earthquakes, trucks, trains, wind, storms, airplanes
Look at seismic spectrograms and rms levels at time of event
Check correlations with auxiliary channel transients (kleineWelle)
Establish the significance and safety of any correlation
Are auxiliary channel coincidences consistent with known transfer functions?
Check for fluctuations in thermal compensation laser power
Check for cosmic ray events
190

Verify signals in four photodiodes are the same (dust veto)
Check for abnormalities in channel trend data
Listen to gravitational-wave stream in audio about event time
Investigate any other possible non-astrophysical causes of the event
Background estimation
Run additional time-shifts on local data to measure local backgroundEstablish how often single-detector waveform arises from background
Check event frequency, bandwidth against known background distribution
Check background isotropy for coherent methods at reconstructed sky location
Check for non-stationarities in background rate and properties near event
Estimate any trials factor, and establish global significance of the event
Other detectors
Any other interferometers or bars online, would they see the event?
Check GCN, IAU, and other public transient catalogs
Any known sub-prime EM or particle events?
Astrophysical analysis
Which sources overlap with the sky location?
How do the extracted waveforms compare with astrophysical waveforms?
Does the amplitude/energy scale make sense?
Characterize waveform morphology, is there a chirp?

191

192

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