Low-Latency Detection of Gravitational Waves for Electromagnetic Follow … · Existing ground-based gravitational wave detectors are currently being up-graded to their advanced con

PhD Thesis

Low-Latency Detection of Gravitational Wavesfor Electromagnetic Follow-up

Author:Shaun Hooper

Supervisors:Prof. David Blair

A/Prof. Linqing WenProf. Yanbei Chen

Dr Chad Hanna

This thesis is presented in partial fulfilment of the

requirements for the degree of Doctor of Philosophy of The

University of Western Australia.

School of Physics

2013

Preface

This section describes the author’s contribution to the work presented here,

and a summary of the layout in its presentation.

This thesis was undertaken between January 2009 and January 2013 at

the University of Western Australia, which includes three months between

March 2010 and June 2010 at the California Institute of Technology.

This thesis describes the design, implementation and testing of a new

search algorithm designed to detect the presence of gravitational waves from

low-mass binary coalescence in advanced detector data in real-time and with

near zero latency.

The author is not solely responsible for all work that contributed to this

thesis. Indeed, most gravitational wave scientists, including the author, are

members of a large >800 author collaboration known as the LIGO scientific

collaboration (LSC).

For the introductory chapters of this thesis, particularly the background

mathematical foundations of inspiral analysis in Chapter 2, the author has

taken inspiration from similar theses such as [1] and [2] to introduce the

reader to the details required to understand the development of the new low-

latency inspiral analysis pipeline presented in this thesis. Some descriptions

of linearised gravity in Chapter 2 overlap information found in standard texts

such as [3], where greater detail can be found.

Chapter 3 describes the design of the new search algorithm. The original

idea of using a summed IIR filter method for low-latency detection was in-

troduced to the author through the work of Yanbei Chen, Linqing Wen and

Jing Luan. Chapter 3 was published as a follow-up to a paper by Luan, et.

al. 2012 [4] that describes the use of a summed IIR method to search for

i

Newtonian waveforms. The author’s contribution to this chapter is, however,

original in its description and implementation of the design for higher order

waveforms. The author wrote this article, wrote the underlying experimental

programs, and analysed the results, with input from co-authors.

Chapter 5 is the result of an experiment that the author and Chad Hanna

performed on the LIGO computer cluster. The experimental results were

obtained from a computer application written by the author, but uses many

sub-routines from the LIGO algorithm library (LAL), which is a software

project contributed by many scientists. The author contributed significant

key sub-routines necessary to run the experiment in Chapter 5, such as the

IIR template bank construction, and the pipeline application itself (although

the design borrows heavily from similar pipelines written by Kipp Cannon,

Chad Hanna and Drew Keppel). The design and implementation of the

experiment and presentation of the results are the author’s work, with advice

and suggestions from the supervisors.

Chapter 6 contains the results of the pipeline that was part of a ma-

jor LIGO engineering run. The pipeline was similar to the one the author

wrote in Chapter 5, however again, much of the infrastructure to execute the

pipeline, such as the source of data, was supplied by the LSC. The design

and implementation of the experiment and presentation of the results are the

author’s work, with advice and suggestions from the supervisors.

ii

Abstract

Existing ground-based gravitational wave detectors are currently being up-

graded to their advanced configuration. When operational, the significant

increase in sensitivity will likely guarantee detection of gravitational waves.

With the imminent detection comes the question of what kind of electromag-

netic counterparts gravitational wave sources will have. One example has the

coalescence of neutron star binaries as a progenitor of short hard gamma-ray

bursts. Observing the rapidly fading electromagnetic counterpart of such

sources immediately after coalescence will provide information to verify as-

trophysical models and give greater insight to these highly energetic events.

Observation of the prompt optical and radio emission of gamma ray bursts

in real-time will require fast moving ground-based telescopes to respond to

triggers generated from gravitational wave detector searches.

This thesis describes the design, implementation and testing of a new

search algorithm designed to detect the presence of gravitational waves from

low-mass binary coalescence in advanced detector data in real-time and with

near zero latency. An introduction to the field of gravitational waves is given

in the first chapter, and specific gravitational wave data analysis techniques

are described in explicit detail in the second. The new algorithm, based on

the use of a bank of computationally efficient infinite impulse response filters

to search for an approximation of the inspiral phase of the gravitational wave-

form, is presented in the third and fourth chapters. With a good choice of

filter coefficients, the inspiral signals are shown to be approximated to greater

than 99%. The method was implemented in LIGO’s data analysis software

library, and made available to the greater community. The fifth chapter de-

scribes a search pipeline based on the new algorithm that was applied to real

iii

detector data from LIGO’s fifth science run, both with and without simu-

lated low-mass binary inspiral signals injected into the data. No significant

loss in detection efficiency or parameter estimation using the new algorithm

was found when compared to the theoretical limit. The sixth chapter demon-

strates the ability of the algorithm to recover signals in real-time and with

low-latency by searching for signals in LIGO’s second engineering run. The

pipeline was able to search for approximately 5000 templates in real-time and

report on multiple-detector coincident triggers for further follow-up with a

typical latency of ∼30 seconds. A final chapter describes how the aim of the

thesis was achieved, and outlines future work that can be developed from

this research.

iv

Acknowledgements

This thesis would not be possible without the help and guidance from all of

my supervisors; Prof Linqing Wen, Prof David Blair, Prof Yanbei Chen and

Chad Hanna. Throughout my candidature, there have been a number of fel-

low scientists that have contributed to the thesis. In particular, I owe a debt

of gratitude to Chad Hanna for his continuous encouragement. Similarly, I

would like to thank Kipp Cannon and Drew Keppel for their extensive help

on all matters related to gravitational wave research. In 2010 I was fortunate

enough to temporarily join the LIGO data analysis group at Caltech under

the guidance of Prof Alan Weinstein. Help from graduate students Stephen

Privitera, Leo Singer, Kari Hodge and Melissa Frei there was indispensable.

Discussions with colleagues Shin Kee Chung, Yuan Liu, Qi Chu and Prof

Zhihui Du have been very beneficial. In reviewing this thesis, I would like to

thank Prof Ron Burman for his time and attention to detail.

Throughout my thesis, I have received help from the many professional

staff both at the UWA School of Physics, and the International Centre for

Radio Astronomy Research. I would like to thank Ian McArthur, Paul Ab-

bott, Jay Jay Jegathesan, Ruby Chan, Leanne Goodsell, Kathy Kok, Lee

Triplett, Micah Foster, Jeff Pollard, Michael Eilon, Mark Boulton and David

London for their professionalism.

The PhD experience would not have been the same (or as fun) if not

for my fellow students not already mentioned; Stefan Westerlund, Sunil Su-

smithan, Francis Torres, Zhu Xingjiang, Lucienne Dill, Timo Dill, Jacinta

Delhaize, Scott Meyer, Lee Kelvin, Morag Scrimgeour, Giovanna Zanardo,

Laura Hoppmann, Toby Potter, Mehmet Alpaslan, Florian Beutler, Rebecca

Lange, Jurek Malarecki, Gemma Anderson, Gar-Wing Truong and Chris

v

Perrella.

Other scientists that I have gained great insight from are; Jean-Charles

Dumas, Prof Ju Li, Eric Howell, Prof David Coward, Prof Gerhardt Meurer,

Prof Richard Dodson and Prof Chris Power.

Finally, I would like to thank the people of my personal life that have

helped me getting through the sometimes difficult experience of being a post-

graduate student. Although distant, I have counted on the support from my

family in Melbourne and New Zealand. So too have I from Wiebe & Shanti

Wilbers, whose friendship I consider close to family. Last but not least I

thank my wife, Shannon, for her patience and continuing support and love.

vi

Contents

Preface i

Abstract iv

Acknowledgements v

Table of contents x

List of figures xii

List of tables xiii

List of abbreviations xv

Useful formula xvii

1 Introduction 1

1.1 Background to gravitational waves . . . . . . . . . . . . . . . 2

1.2 Sources of gravitational waves . . . . . . . . . . . . . . . . . . 2

1.3 Indirect observation of gravitational waves . . . . . . . . . . . 6

1.4 Direct detection of gravitational waves . . . . . . . . . . . . . 7

1.5 Multi-messenger astronomy . . . . . . . . . . . . . . . . . . . 11

1.5.1 Gamma ray bursts . . . . . . . . . . . . . . . . . . . . 11

1.5.2 GRB triggered GW search . . . . . . . . . . . . . . . . 14

1.5.3 GW triggered EM search . . . . . . . . . . . . . . . . . 15

1.6 Motivation for low-latency GW detection method . . . . . . . 16

1.7 Goals of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

vii

1.8 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Gravitational Waves 21

2.1 Linearised gravity . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Plane wave solution . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Transverse traceless gauge . . . . . . . . . . . . . . . . 23

2.2 Detection of gravitational waves . . . . . . . . . . . . . . . . . 24

2.2.1 Noises in interferometer . . . . . . . . . . . . . . . . . 28

2.3 Inspiral gravitational waves . . . . . . . . . . . . . . . . . . . 30

2.3.1 Geometry of binary system . . . . . . . . . . . . . . . . 31

2.3.2 Orientation of the binary relative to an observer . . . . 34

2.3.3 Orbital frequency as a function of time . . . . . . . . . 37

2.3.4 Higher order multipole corrections . . . . . . . . . . . . 38

2.4 Inspiral waveform . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Intrinsic and extrinsic parameters . . . . . . . . . . . . 43

2.5 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5.1 Matched Filter . . . . . . . . . . . . . . . . . . . . . . 44

2.5.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . 46

2.5.3 Template bank . . . . . . . . . . . . . . . . . . . . . . 47

2.5.4 Matched filter as a function of unknown time of coa-

lescence . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5.5 Matched filter of unknown phase . . . . . . . . . . . . 49

2.5.6 Signal to noise ratio . . . . . . . . . . . . . . . . . . . 50

2.5.7 Discrete time domain filtering . . . . . . . . . . . . . . 51

2.5.8 Infinite Impulse Response Filter . . . . . . . . . . . . . 53

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Low-Latency GW Detection Method 57

3.0 Paper abstract . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.1 The Inspiral Waveform . . . . . . . . . . . . . . . . . . 62

3.2.2 Two-Phase Matched Filter . . . . . . . . . . . . . . . . 65

viii

3.2.3 Discrete Time Domain Filtering . . . . . . . . . . . . . 67

3.2.4 Infinite Impulse Response Filter . . . . . . . . . . . . . 68

3.2.5 Approximation to an inspiral waveform . . . . . . . . . 69

3.2.6 Summed Parallel IIR filtering . . . . . . . . . . . . . . 72

3.3 Implementation for Performance Testing . . . . . . . . . . . . 72

3.3.1 IIR bank construction . . . . . . . . . . . . . . . . . . 72

3.3.2 Detector Data Simulation . . . . . . . . . . . . . . . . 73

3.3.3 Detection Efficiency . . . . . . . . . . . . . . . . . . . . 75

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.1 Inspiral Waveform Overlap . . . . . . . . . . . . . . . . 76

3.4.2 Ability to Recover SNR . . . . . . . . . . . . . . . . . 77

3.4.3 Detection Efficiency . . . . . . . . . . . . . . . . . . . . 78

3.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 80

3.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 81

3.7 Noise Spectral Density . . . . . . . . . . . . . . . . . . . . . . 82

4 Multi-rate SPIIR method 83

4.1 Multi-rate SPIIR filtering . . . . . . . . . . . . . . . . . . . . 83

4.2 Multiple templates . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Offline SPIIR pipeline 91

5.1 The SPIIR application . . . . . . . . . . . . . . . . . . . . . . 93

5.1.1 Internal structure of gstlal iir inspiral . . . . . . . 94

5.2 Data for offline run . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 IIR Bank generation . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Simulated inspiral signals . . . . . . . . . . . . . . . . . . . . . 103

5.5 Behaviour in non-Gaussian data . . . . . . . . . . . . . . . . . 105

5.6 Ranking triggers . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7 The offline SPIIR pipeline . . . . . . . . . . . . . . . . . . . . 110

5.8 Confirmation of false alarm rate estimation . . . . . . . . . . . 112

5.9 Sensitivity of search . . . . . . . . . . . . . . . . . . . . . . . . 113

5.10 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 117

ix

5.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Online SPIIR pipeline 123

6.1 SPIIR online pipeline . . . . . . . . . . . . . . . . . . . . . . . 125

6.1.1 GraCEDb . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 LIGO’s second engineering run . . . . . . . . . . . . . . . . . 129

6.3 Analysis setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 ER2 search parameter space . . . . . . . . . . . . . . . . . . . 133

6.5 Results of search . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.6 Blind software injections . . . . . . . . . . . . . . . . . . . . . 142

6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Conclusion 149

7.1 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 Thesis aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Bibliography 157

x

List of Figures

1.1 Gravitational wave spectrum. . . . . . . . . . . . . . . . . . . 4

1.2 Operating schedule for GW detectors . . . . . . . . . . . . . . 9

1.3 Volume of space seen by LIGO . . . . . . . . . . . . . . . . . . 10

2.1 Schematic of GW detector interferometer . . . . . . . . . . . . 25

2.2 Sky coordinates of incoming GW relative to detector frame . . 26

2.3 Best strain sensitivities (ASD) for initial LIGO . . . . . . . . . 30

2.4 Binary coordinate system . . . . . . . . . . . . . . . . . . . . 32

2.5 Binary coordinate system with respect to an observer . . . . . 35

2.6 Trajectories of compact binary coalescence . . . . . . . . . . . 40

3.1 A schematic overview of the SPIIR method . . . . . . . . . . . 61

3.2 Flow chart of digital single-pole IIR filter . . . . . . . . . . . . 69

3.3 Illustrative diagram of summed sinusoids . . . . . . . . . . . . 71

3.4 Overlap as a function of number of sinusoids . . . . . . . . . . 76

3.5 Example SPIIR output . . . . . . . . . . . . . . . . . . . . . . 77

3.6 ROC curve of IIR method . . . . . . . . . . . . . . . . . . . . 79

4.1 Multirate SPIIR . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Multirate multi-template SPIIR . . . . . . . . . . . . . . . . . 89

5.1 Flow of data through gstlal iir inspiral . . . . . . . . . . 95

5.2 Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 IIR template bank generation . . . . . . . . . . . . . . . . . . 100

5.4 Parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Offline IIR overlap . . . . . . . . . . . . . . . . . . . . . . . . 103

xi

5.6 Chi-square-SNR distribution . . . . . . . . . . . . . . . . . . . 107

5.7 Post-gstlal iir inspiral procedure . . . . . . . . . . . . . . 111

5.8 Inverse FAR distribution . . . . . . . . . . . . . . . . . . . . . 113

5.9 Detection efficiency . . . . . . . . . . . . . . . . . . . . . . . . 115

5.10 Search Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.11 Search Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.12 Chirp mass accuracy . . . . . . . . . . . . . . . . . . . . . . . 119

5.13 Time accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.1 Flow of online pipeline . . . . . . . . . . . . . . . . . . . . . . 128

6.2 Low-latency data transfer . . . . . . . . . . . . . . . . . . . . 131

6.3 Online analysis DAG . . . . . . . . . . . . . . . . . . . . . . . 133

6.4 ER2 Parameter space . . . . . . . . . . . . . . . . . . . . . . . 134

6.5 Online IIR overlap . . . . . . . . . . . . . . . . . . . . . . . . 135

6.6 Number of IIR filters . . . . . . . . . . . . . . . . . . . . . . . 137

6.7 False alarm rate . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.8 Event rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.9 Latency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.10 Latency histogram . . . . . . . . . . . . . . . . . . . . . . . . 142

xii

List of Tables

1.1 List of known neutron star-neutron star systems . . . . . . . . 7

2.1 Intrinsic and extrinsic inspiral parameters . . . . . . . . . . . 43

2.2 Approximate template duration . . . . . . . . . . . . . . . . . 53

4.1 Computational cost of the multi-rate SPIIR method . . . . . . 87

6.1 Simulated injections . . . . . . . . . . . . . . . . . . . . . . . 144

xiii

xiv

List of abbreviations

ASD amplitude spectral density

EM electromagnetic

ER2 Engineering Run 2

GraCEb Gravitational-wave Candidate Event Database

GRB gamma-ray burst

GW gravitational waves

gwf Gravitational wave frame (files)

ISCO innermost stable circular orbit

LAL LIGO Algorithm Library

LIGO Laser Interferometer Gravitational-wave Observatory

LLOID Low-Latency On-line Inspiral Data

MBTA Multi-Band Template Analysis

NS neutron star

PSD power spectral density

SNR signal to noise

xv

xvi

Useful formulae

Quantity Formula UnitStrain h = ∆L/L unitlessGeometrized solar mass T = GM/c3 timeTotal mass M = m1 +m2 massReduced mass µ = m1m2/M massChirp mass M = η3/5M massSymmetric mass ratio η = m1m2/M

2 unitless

ISCO frequency fISCO = c3/(6√

6πGM)

hertz

xvii

Chapter 1

Introduction

In this introduction chapter, I will describe why there is a scientific need for a

new method to detect perturbations of space-time in real-time and with little

delay — i.e low-latency. Firstly I will introduce the predictions of Einstein’s

general theory of relativity, including the propagation of the perturbations,

commonly known as gravitational waves (GWs). A discussion of potential

sources will follow, noting that perhaps the most promising candidate for de-

tection will be GWs from the inspiral phase of solar mass coalescing compact

binaries. Detecting the inspiral phase of GWs produced by low-mass coalesc-

ing compact binaries with low-latency will be the focus of this thesis. The

confidence of the existence of GWs will be described by detailing their indi-

rect observation. The latest efforts to detect GWs directly will be outlined,

with an emphasis on the next generation ground based laser interferometer

detectors, which are expected to have an unprecedented level of sensitivity.

The scientific benefit of directly detecting the presence of GWs in real-time

and with a low-latency processing time will be demonstrated by describing

the GW sources that may have transient electromagnetic counterparts. The

goals of this thesis will be stated, and an overview of the rest of the thesis

will be given, describing the lay-out of the research done.

1

2 CHAPTER 1. INTRODUCTION

1.1 Background to gravitational waves

Einstein’s general theory of relativity is a remarkable improvement to the

Newtonian theory of gravitation. So far, general relativity has accounted

for all the observations of both the special theory of relativity and Newton’s

law of gravitation, and additionally explains observations that the Newto-

nian theory can’t. For example general relativity accurately accounts for the

previously unexplained precession of the perihelion (closest approach to the

Sun) of the planet Mercury. One of the early experimental confirmations of

general relativity occurred during a solar eclipse in 1919 when Eddington ob-

served the light from distant stars displaced by the Sun during a total eclipse

[5]. It was known as early as 1801 that Newtonian gravity would predict a

deflection, however this value was only half of that predicted by general rela-

tivity. Eddington’s observation was the definitive turning point in confirming

general relativity. Since then there have been many experiments that have all

verified general relativity to ever increasing accuracy [6]. Instead of describ-

ing gravity as an interaction between massive bodies at a distance, general

relativity describes space-time itself as curved. The curvature is caused by

the presence of matter, and can be specified by the Einstein field equations.

Most of the observations verifying general relativity have been made where

the curvature is slight, the so called weak field limit. One prediction of gen-

eral relativity is that the motion of non-spherically symmetric bodies with

a time-varying quadrupole moment will emit gravitational waves (GWs) —

perturbations or ripples of space-time. Only in the strong field limit, where

the curvature is the greatest are GWs likely to be detected.

1.2 Sources of gravitational waves

Gravitational radiation can be described (in an order of magnitude esti-

mate) as an analog of electromagnetic (EM) radiation. For EM radiation,

the power outputted is proportional to the second time derivative of the elec-

tric and magnetic multipole moments. The strongest EM moment is the elec-

tric dipole moment, followed by the magnetic dipole moment and the electric

1.2. SOURCES OF GRAVITATIONAL WAVES 3

quadrupole moment. In the gravitational analog, mass moments of inertia are

analogous to the electric and magnetic moments. Hence the strongest source

of gravitational radiation would be the second time derivative of the mass

dipole, which is the change in total momentum. But this must vanish due to

the conservation of momentum. The next strongest type of radiation is the

gravitational analog of the magnetic moment, the total angular momentum.

The second time derivative of this moment also vanishes due to the conserva-

tion of angular momentum. To produce any kind of GW, a source must have

a time-varying mass quadrupole moment I, i.e., have a non-axisymmetric

time-varying mass distribution. It can be shown that the power outputted

by a GW is proportional to (G/c5)...I 2 [3]. A very large quadrupole moment

will be required to overcome the very small G/c5 ∼ 10−53 W−1 factor. Hence

any terrestrial or laboratory generation of GWs is very unlikely (see Section

36.3 of [3] for a good example of why). However a large quadrupole moment

comparable to c5/G can be expected when studying astrophysical systems

where the quadrupole moment is relativistic, i.e. v approaches c.

As the GWs propagate outwards from their source they distort local

space-time by alternately stretching and squeezing it. The frequency of the

stretching and squeezing is known as the gravitational wave frequency, and

the fractional distortion change the strain, commonly denoted by the sym-

bol h. In general GW scientists classify four main types of astrophysical

gravitational wave sources characterised by their expected GW signature, or

gravitational waveform. Figure 1.1 shows the spectrum of expected gravita-

tional wave sources and the sensitive bandwidths of proposed and existing

GW detectors. Details of how ground-based GW detectors operate will be

given in Section 2.2.

As we will show, of the four different sources of GWs, those from solar

mass coalescing compact binaries are perhaps the most promising source for

detection since their gravitational waveform is known to high precision, and

they have a GW frequency that enters the sensitive bandwidth of ground

based GW detectors. Detection of GWs from the inspiral phase of com-

pact binary coalescence will be the primary focus of this thesis. A thorough

derivation of the inspiral gravitational waveform will be given in Section 2.3.


Figure 1.1: Gravitational wave spectrum. The amplitude spectral density(strain) of different sources of GWs are shown as a function of their fre-quency. The sensitivity limits of three different kinds of GW detectors:the radio telescope based Parkes Pulsar Timing Array (PPTA), the pro-posed satellite mission Laser Interferometer Space Antenna (LISA) and theground based Laser Interferometer Gravitational-wave Observatory (LIGO)are shown. Credit: PPTA Collaboration [7].

For now, a qualitative description of the four main sources follows:

Compact Binary Coalescence/Inspiral One example of transient source

of GWs is that of two closely orbiting compact (dense) bodies. As the

bodies orbit each other, there is a large time varying mass quadrupole

moment, from which general relativity predicts that GWs will be emit-

ted by the system. The GWs carry energy away from the binary system

and, to obey the conservation of energy, the orbital separation of the

two bodies decreases, as does the orbital period. The frequency of the

1.2. SOURCES OF GRAVITATIONAL WAVES 5

GW emitted is directly related to the orbital frequency, which is mono-

tonically increasing in time as the orbital separation shrinks. This is

known as the inspiral phase and the waveform predicted is sometimes

called a chirping waveform. As the bodies coalesce, the amplitude of

the gravitational strain h increases approximately as a power law with

time. Given the intrinsic parameters of the system (e.g. component

masses) the inspiral gravitational waveform can be modelled analyt-

ically with a high degree of accuracy using post-Newtonian methods

[8, 9, 10]. The inspiral gravitational waveform will be derived in Sec-

tion 2.3. It is predicted that beyond a boundary known as the inner-

most stable circular orbit, the bodies will plunge in toward each other

and cataclysmically merge (the merger phase). Finally, the resulting

mass will oscillate in the ringdown phase. The entire process is called

compact binary coalescence.

Examples of compact bodies are neutron stars (NS), black holes (BH)

and white dwarfs. A typical NS has a mass roughly equivalent to our

Sun, but a radius on the order of 10 kilometres. The masses of the bod-

ies will dictate the particular signature of the inspiral phase (described

in detail in Section 2.3). For NS-NS binaries (with component masses

around 1− 3M), the GW frequency near coalescence will be around

102 − 103 Hz. For BH-NS or BH-BH binaries, the GW frequency will

be much lower.

Burst Any other transient GW signal of short duration is called a GW

burst event. Generally the morphology of the signal is highly uncertain.

There are a variety of potential sources; for example the non-symmetric

core collapse of a supernova or NS glitch (such as a starquake), or

perhaps the merger phase of a compact binary coalescence event. See

[11, and references therein] for a review on burst sources and how GW

analysts search for them.

Continuous wave Any source of GWs that produces a quasi-monochromatic

gravitational waveform signal and is distinguishable from the back-

ground is called a continuous wave source. An example of a continuous


source could be a rapidly rotating NS that has a slight non-spherical

distortion. As NSs are compact objects that spin very fast, any slight

non-axisymmetric symmetry would produce a very strong quadrupole

radiation. For a full review of continuous wave sources see [12, and

references therein].

Stochastic background As the EM spectrum has a background of unre-

solved sources, so one would expect something similar for the GW

spectrum. This background could have originated from cosmologi-

cal sources such as inflation, cosmic strings and pre-Big-Bang models.

GWs from Galactic white-dwarf binaries or slow spinning Galactic pul-

sars could also account for a stochastic background. Searches to define

this background are generally done by cross-correlating the strain data

recorded from different GW detectors. See [13, 14, 15, 16] for important

stochastic background searches.

1.3 Indirect observation of gravitational waves

The first observational evidence of the existence of GWs came from a double

neutron star (NS-NS) system. The pulsar binary system PSR B1913+16

was discovered and observed by Russell Hulse and Joseph Taylor [17]. Radio

observations of the pulsar indicated that it is in a binary system, where the

companion body is another NS, and has an orbital period of 7.75 hr. Decades

of observing the timing of the radio pulses showed that orbital period is

slowing with a rate of decrease within 0.2% of the rate predicted by general

relativity [18, 19]. This discovery earned Hulse and Taylor the 1993 Nobel

Prize in Physics.

Although this indirect detection of GWs is significant in verifying general

relativity, indirect detection in this way relies solely on serendipity — to

discover this system, at least one NS had to be a pulsar with its beaming angle

passing the Earth. To date, there have been only six confirmed discoveries of

NS-NS systems [20]. Table 1.1 gives details of them. Although the effect of

period decrease can be attributed with high accuracy to that predicted by the

1.4. DIRECT DETECTION OF GRAVITATIONAL WAVES 7

Table 1.1: List of known neutron star-neutron star systems

PSR Yeardiscov-ered

OrbitalPeriod( hr)

m1

( M)m2

( M)Distance( kpc)

Reference

B1913+16 1974 7.75 1.4398 1.3886 9.9 [18, 21]B1534+12 1991 10.1 1.3332 1.3452 1.02 [22, 23]B2127+11C 1991 8.05 1.358 1.354 9.58 [24, 25]J0737-3079 2003 2.45 1.24 1.35 0.5-0.6 [26]J1756-2251 2005 7.67 1.40 1.18 2.5 [27]J1906+0746 2006 3.98 1.248 1.365 5 [28]

generation of GWs, the frequency of the signal is outside the bandwidth of

ground based GW detectors. As will be shown in Section 2.3, the amplitude

of the strain increases when the binary is close to coalescence. However none

of the known NS-NS systems will coalesce for at least a few millions of years.

There have been many studies into the actual coalescence rate of compact

binaries (see [29, references therein]). Coalescence rates are usually quoted

in either per Milky Way Equivalent Galaxy per Myr or per Mpc3 per Myr.

There are significant uncertainties in the astrophysical rates of compact bi-

nary coalescence estimates owing to the small sample size of known galactic

NS-NS binaries and poor constraints for population-synthesis models. At

present, the latest estimate is 1 per Myr per Mpc3 [29, 30]. The actual de-

tection rate will depend on the properties of the GW detection instrument(s)

used. It must be noted that the uncertainties can amount to 1 or 2 orders of

magnitude, hence making statements about the expected number of events

observed highly variable.

1.4 Direct detection of gravitational waves

The trouble with measuring the strain is that for a category of sources in

the nearby universe h ∼ 10−22! So far, there has been no direct detection of

GWs by measuring the strain h.

Despite the incredibly small strain h predicted by general relativity, there

has been concerted worldwide effort to directly detect GWs over the last 50


years. The first generation of GW detectors built in the 1960s and 1970s were

solid metal cylindrical bars — so called bar detectors. First built by Joesph

Weber [31], these bar detectors were isolated from the effects of the sur-

rounding environment by using seismic isolation suspensions inside vacuum

chambers to prevent acoustic interference. Piezoelectric transducers were

glued to the surface of the bar designed to measure any vibrations induced

in this driven harmonic oscillator by GWs (results in [31, 32]). Weber’s re-

search spurred on much activity in this field, and soon several groups around

the world built their own bar detectors.

Later a different kind of technology was built to analyse the minute differ-

ence in length a GW causes. This was based on Michelson laser interferome-

ters. This kind of GW detector is designed to measure the actual differential

fractional change in arm length as opposed to the amplitude of oscillations

induced in the bar detectors. More details on how interferometric GW de-

tectors operate, and their sources of noise, will be given in Section 2.2. GW

detectors built as interferometers are known as second generation. The first

interferometer built for detection of GWs had arm lengths of one metre [33].

However this detector was too small to have a sensitivity that could measure

typical GWs [34].

Several ground based kilometre scale GW interferometric detectors have

been built in the US and Europe. The US effort, known as the Laser In-

terferometer Gravitational-wave Observatory (LIGO [35]) has built two 4

kilometre long Michelson interferometers in Hanford, Washington, and Liv-

ingston, Louisiana [36]. The French/Italian consortium Virgo [37] has built a

3 kilometre interferometer near Cascina, Italy [38]. There is also the smaller

300 metre TAMA300 detector based at the Tokyo Astronomical Observa-

tory, Japan [39]. The sensitivity of this detector not as high as the larger

LIGO/Virgo detectors, but aims to act as a test bed for developing advanced

detector hardware. Similarly, there is also the 600 metre GEO600, built by

the German/British and located outside of Hannover, Germany [40].

With arms on the kilometre scale, these type of detectors have a sensi-

tive bandwidth in the 40 Hz–2000 Hz range. This is because they have been

optimised for detecting GWs from compact binary coalescence events where

1.4. DIRECT DETECTION OF GRAVITATIONAL WAVES 9

the binary sources have masses in the range of 1M–20M(i.e. NS-NS and

NS-BH binaries). The first configuration of the LIGO detectors, known as

initial LIGO was built in the late 1990s/early 2000s. There have since been

six science runs, known as S1, S2, etc. The inaugural S1 ran for 17 days in

2002, and data was collected from both LIGO detector sites [41]. Since then

subsequent science runs with ever increasing sensitivities have taken place

[42], some in coincidence with GEO600, TAMA300 and Virgo detectors. For

an overview of the operating schedule, see Figure 1.2. For a complete list

of publications by the LIGO Scientific Collaboration including observational

results and conference proceedings, see [43] and the the LIGO document

control center [44].

Figure 1.2: The operating schedule of the various ground-based GW detectorsin the initial detector era. Credit: [45]

Currently both the LIGO and Virgo detectors are offline, as they are

undergoing a major hardware change to the advanced detector configuration.

Both the Advanced LIGO[46] and Advanced Virgo [47] are expected to be

operational from 2015. Once built, Advanced LIGO is expected to have a

10 fold sensitivity improvement compared to initial LIGO [48]. Hence in the

era of advanced detectors GWs produced from compact binary coalescence

events will be detectable within a volume of space one thousand times larger

than that of initial LIGO, out to ∼200 Mpc–300 Mpc [49] (see Figure 1.3).

With this increase in sensitivity the estimated detection rate for GWs from

NS-NS binaries could be between 1 and 400 per year (see table 5 of [29] for


Figure 1.3: The volume of space that Advanced LIGO is sensitive to isexpected to be ten times that of initial LIGO. Credit: [50, 51]

a full discussion on the difficulties of predicting detection rates).

A number of other gravitational wave detection experiments are either

underway or planned. A pulsar timing array measures the arrival time dif-

ferences of pulses emitted from millisecond pulsars due to GWs (e.g. [52]).

The sensitive bandwidth of this experiment is in the low frequency regime of

10−9 Hz–10−6 Hz. There are also proposals for a space based laser interfer-

ometer, for example the eLISA mission [53]. This experiment would have a

sensitive bandwidth in the 10−4 Hz–10−1 Hz range.

1.5. MULTI-MESSENGER ASTRONOMY 11

1.5 Multi-messenger astronomy

With the imminent detection of GWs, scientists have begun to ask what sort

of EM counterparts are coincident with a GW event [54]. Connecting the

detection of a GW event with an EM counterpart will break the degeneracy

of inferred binary parameters (for GWs from compact binary coalescence

events). Observing GWs that originate from extra-Galactic host galaxies

will give a measure of absolute distance, thereby allowing an independent

measure of the Hubble constant [54, 55, 56, 57, 58, 59, 60, 61]. Perhaps one

of the most promising EM counterparts of a compact binary coalescence event

is that of gamma ray bursts (GRBs). So called “multi-messenger” astronomy,

where both GW and EM information are collected, will give maximum insight

to the physics of such highly energetic events.

1.5.1 Gamma ray bursts

In this section I briefly summarise the observations and basic underlying

models that cause GRBs. For excellent reviews on the topic of GRBs, see

[62] and [63].

Gamma ray bursts (GRBs) are intense flashes of γ-rays in the MeV band,

that for a short time radiate in an otherwise empty γ-ray sky. The flash over-

whelms any other γ-ray source, including the Sun. GRBs were first observed

between 1969 and 1972 by the Vela military satellites designed to monitor vi-

olations of the nuclear test ban treaty [64]. However it was quickly discovered

that the bursts were coming from not the Earth, but the opposite direction,

the sky. Over the next decades, a series of satellites was launched to observe

this new astrophysical phenomenon, and many theoretical models of GRBs

were founded to explain the observations (see [62] for a complete history).

However it was not until the 1991 launch of the Compton Gamma Ray Ob-

servatory [65] (see [66] for results) that greater insight was obtained. On

board, the Burst and Transient Source Experiment detected more than 3000

isotropically distributed bursts, suggesting a cosmological rather than Galac-

tic distribution. Later, in 1996 the Beppo-SAX [67] satellite was launched,

and was able to localise the X-ray emission from some GRBs. It also dis-


covered the previously predicted “afterglow”, which appears as fading softer

X-ray, optical and radio emissions [68]. This in turn allowed host galaxies to

be identified, and redshifts to be observed. The High-Energy Transient Ex-

plorer (HETE-2) satellite [69] was launched in 2000, and continued providing

afterglow positions.

Further advances were made after the 2004 launch of the Swift satel-

lite [70] due to its onboard array of multi-wavelength instrumentation, and

greater sensitivity. Upon detecting a GRB, it is able to rapidly (within about

100 s) slew to the direction of the source, and record multi-wavelength spec-

tra and light curves. The most recent GRB satellite to be launched was the

Fermi [71] mission in 2008, which has provided the most powerful window

into these high energy events. Other operating GRB satellites are the Eu-

ropean space agency’s International Gamma-Ray Astrophysics Laboratory

[72] launched 2002, and Astrorivelatore Gamma a Immagini Leggero [73]

launched in 2007.

Despite some recent classification issues [74], there is enough evidence to

show that GRBs can be divided into two distributions based on their dura-

tion [75]. Those with burst durations longer than 2 seconds are characterised

as long soft GRBs, and those with durations less than 2 seconds are char-

acterised as short hard GRBs. The two populations are thought to have

different progenitor models:

Long GRB It has been thought that the core collapse of a rapidly rotating

massive star could be the progenitor of long GRBs [62, 76, 77].

Short GRB Merging compact bodies (NS-NS or BH-NS) have been pro-

posed as the progenitors of short GRBs (e.g. [63, 78, 79, 80, etc]

amongst others). Following the coalescence of the binary system, where

the characteristic inspiral GW signal is produced, the two bodies merge,

and form an accretion disk around a central body (perhaps a black hole)

(see [81] and references). The rapid accretion (<1 s [54]) powers a col-

limated relativistic jet, which produces a GRB. The creation of this

central engine launches a relativistic outflow of energy 1048 erg–1050 erg

on timescales of 0.1 s–1 s. This scenario has been computationally con-


firmed by [81]. The delay between the final GW emission and the onset

of the GRB is estimated to be as short as 0.1 seconds or as long as tens

to hundreds of seconds [82, 83].

With the discovery beginning in 1997 of optical afterglows from GRBs

[84], an internet distribution service was created to automatically send out

alerts to interested parties for follow-up observations. The GRB Coordinates

Network was based on the coordinates and distribution network from the

earlier Burst and Transient Source Experiment [85], and has enabled many

afterglow emissions to be observed by ground based telescopes.

The afterglow is thought to arise from the collision of the relativistic

expansion of ejecta into the surrounding medium. As the shock wave decel-

erates though the medium, the EM afterglow becomes progressively weaker,

decaying on the order of hours, though the radio afterglow may last for weeks

[63].

The first detection of a prompt optical emission was made by the Robotic

Optical Transient Search Experiment (ROTSE)[86]. Since then, a number of

prompt optical emissions or flashes have been observed in long GRBs [87].

Here prompt emission refers to observations made when the GRB is still

active. One theory is that EM emissions associated with GRBs occur when

outgoing matter collides with the matter ejected by earlier shocks (see [62,

and references therein]). This can potentially create a reverse shock, and

produce a bright prompt optical flash in a single burst of very short duration

[88].

Consider that to observe the rapidly decaying and not very bright optical

flash of any GRB, one must already have a telescope pointing a direction that

has the flash within its field of view, or the telescope must slew to the loca-

tion provided by a pre-existing trigger. Although the former is not likely to

occur, this is exactly what happened in the particular case of GRB 080319B.

Two wide field instruments “Pi of the Sky” and Tortora were observing the

afterglow of GRB 080319A when by chance, another GRB, GRB 080319B

occurred sufficiently close in time (∼30 minutes) and sky location (∼10 )

[89]. The optical emission was quickly variable, rising within 10 seconds after

the beginning of the GRB, and lasted for about 50 seconds [87].


However, in general, to catch the prompt emission of a GRB requires

fast moving telescopes responding to triggers. Conventional astronomical

telescopes are not designed to make rapid, follow up searches. However since

the invention of the GRB Coordinates Network in 1997, several specialised

rapidly moving robotic ground-based telescopes (e.g. ROTSE, Telescopes a

Action Rapide pour les Objets Transitoires (TAROT) [90], Zadko [91]) and

wide field cameras have been developed for this purpose.

Roughly one quarter to a half of all observed short GRBs are followed

by an X-ray emission beginning ∼10 s after the initial GRB and lasting for

hundreds of seconds [92]. Moreover, it was recently discovered that some

short GRBs are followed by an X-ray “plateau”, which is not expected from

the standard fireball model describing GRBs [93, 94, 95]. Hence the scientific

benefits of EM-GW coincident detection can be based on the following ideas:

• GW observations of a short GRB could confirm the central engine of

the GRB to be an inspiral event

• multi-wavelength EM emission associated with the prompt emission

of GRB will lead to a better understanding of the central engine and

provide physical conditions to improve the models describing the un-

derlying processes

In order to make the connection between the two, one must either use the

GRB to a trigger a GW search, or use a GW detection to trigger an EM

follow-up.

1.5.2 GRB triggered GW search

Considering that both progenitor models of short and long GRBs have GW

emission, there have been several studies performed using EM observations

as triggers to search for coincident GW events in both LIGO and Virgo data

[36, 96, 97, 98, 99, 100, 101, 102, 103]. GRB-triggered searches have targeted

GWs produced from both inspiral and burst sources. These studies have

adopted a time window of a few minutes for long GRBs, and a few seconds

for short GRBs around the GRB trigger time. It has been estimated that


triggered searches will increase the sensitivity of GW detectors by about 50%

and the detection rate will increase by a factor of 3 [104].

Two specific cases, those of the discovery of short GRB 070201 [98, 105]

and GRB 051103 [103, 106] are quite interesting in that the GRB location

overlapped nearby galaxies (M31 and M81 respectively). A NS-NS merger

would be detectable if it occurred within this time period at this distance. In

both cases, a NS-NS binary merger was ruled out with very high confidence.

1.5.3 GW triggered EM search

Since the γ-ray emissions from GRBs is likely beamed [54], not all compact

binary coalescence events may have an observable GRB counterpart. How-

ever their optical and perhaps radio afterglows may be observable off axis,

although it may be somewhat dimmer. Indeed, such an observation would

yield a confirmation of the jet model, show the beaming distribution and

greatly help theoretical models of relativistic outflows [107, and references

therein].

A GW triggered GRB search would entail first localising the source di-

rection. Even with optimistic configurations of a worldwide GW detector

network, the error box could be as large as tens of square degrees [54, 108,

109, 110, 111]. However this large sky error-area can be partially mitigated

by restricting counterpart searches to transients that occur within nearby

galaxies that are within the LIGO-Virgo horizon distance [112].

This idea was first explored by the Locating and Observing Optical Coun-

terparts to Unmodelled Pulses in GW study [113]. During the latest LIGO-

Virgo run (S6/VSR3) in 2010, there were several GW search detection pipelines

operating with the goal of using GW detection as triggers to send out as-

tronomical alerts. These were Coherent WaveBurst, Omega and Multi-Band

Template Analysis (MBTA) [114, 115, 116]. Both Coherent WaveBurst and

Omega were searches for un-modelled burst sources based on time-frequency

analysis.

An important concept here is the notation of low-latency GW detection.

Here the term latency is defined as the time between finding a GW signal in


the detector’s data and the (wall clock) time that the GW passed through

the detector. This time needs to be as short as possible so that the fading

optical counterpart can be observed. In this thesis, we aim to produce an

inspiral search pipeline that can achieve sub-minute latencies.

1.6 Motivation for low-latency GW detection

method

In the history of LIGO, inspiral GW search strategies have primarily fo-

cused on accurate detection of GW events. As will be shown in Chapter 2,

traditional inspiral searches are based on cross-correlating expected inspiral

waveforms and the detector data. In general, this process is computationally

expensive. The computational cost can be reduced by using Fourier domain

correlations. However this necessarily introduces a latency and the analysis

is usually done offline.

As we move toward the advanced detector era where the sensitivity of

the detectors increases, searches for longer inspiral waveforms will demand

significantly more computational resources.

Recently two independent GW search methods have been developed to

search for inspiral signals with low-latency. The Virgo group has produced

the low-latency pipeline MBTA [116], and LIGO is also working on a new

method, Low-Latency On-line Inspiral Data (LLOID) analysis method [117].

A latency of less than 3 minutes until the availability of a trigger using

the MBTA pipeline has been achieved in initial LIGO data [116]. The

LLOID pipeline achieves low-latency by reducing the computational cost of

the pipeline using a number of strategies discussed in [117, 118].

Observing the EM counterparts of GRBs will lead to a greater scientific

understanding of the underlying physics by constraining models describing

the them. The scientific questions posed in the previous section can be

ultimately answered by first implementing a low-computational resource GW

inspiral search pipeline that can send out GW triggers with low-latency.

1.7. GOALS OF THESIS 17

1.7 Goals of thesis

This thesis will focus on developing and implementing a new low-latency

time-domain detection pipeline capable of detecting inspiral GW signals from

the coalescence of NS-NS binaries. The construction and implementation

of this pipeline will enable the rapid response robotic telescopes to make

complete observations of the transient EM counterparts. This will ultimately

lead to scientific questions being answered about the nature of such highly

energetic events.

The primary goals of this thesis are to;

1. demonstrate the need for a low-latency gravitational wave search pipeline,

2. develop a new low-latency search pipeline with low computational cost,

3. demonstrate capabilities of this pipeline,

4. present results of the pipeline as applied to realistic GW detector data,

5. achieve sub-minute GW triggers for EM follow-up observations.

1.8 Outline of thesis

The layout out the remaining parts of this thesis are as follows,

• Chapter 2 will define the common mathematical conventions and con-

cepts that will contribute to the development of a new low-latency

search pipeline. This chapter serves as a reference for the rest of the

thesis. First, taking inspiration from Chapter 2 of [1], a rigorous deriva-

tion of the inspiral gravitational waveform is shown. This derivation

is not original, but is presented here in order to make clear the prop-

erties of the inspiral gravitational waveform. The notion of intrinsic

and extrinsic parameters of the compact binary coalescence event will

be discussed, which paves the way to understand how such signals are

searched for. As a reference, the traditional (and optimal) strategy for

searching for the inspiral waveforms in Gaussian detector data will be


shown. Finally, there will be a short introduction on digital time do-

main filters, so that the reader has a better understanding of the basic

components used for the development of the new low-latency pipeline.

• Chapter 3 will introduce the theoretical strategy my collaborators and

I have developed to address the issues for a new low-latency pipeline.

Once I have demonstrated the proof of concept of the strategy, the new

method is tested by searching for a single canonical inspiral waveform

in mock Gaussian detector data. The introduction of the method and

the tests were published in [119]. Hence the major part of this chapter

will be [119] in its entirety.

• Chapter 4 is a short chapter detailing some changes made to the method

introduced in Chapter 3. It will show the significant computational cost

improvements that can be made by modifying the basic implementation

of the method.

• Chapter 5 will focus on the realisation of the pipeline through imple-

mentation to the common software environment available to the wider

GW scientific community. Attention will be given to the details of

this implementation. The newly realised pipeline will be tested by

searching for inspiral GW signals with a variety of parameters in both

mock Gaussian detector data, and previously recorded LIGO science

run data. The latter is a necessary step in addressing the capabilities

of the pipeline in non-Gaussian noise. Both tests are performed offline.

This will allow the pipeline to be tested by injecting many simulated

inspiral signals with different parameters. From this, a discussion will

be made about the detection efficiency of the pipeline and the ability

to estimate signal parameters will follow.

• Chapter 6 will demonstrate the ability of the new pipeline to search for

inspiral GW signals in online detector data. The detector data here are

part of LIGO/Virgo’s second engineering run, which began on July 18

2012 and ended on August 8 2012. This run provided an opportunity

to test the online and low-latency capability of the pipeline. Results

1.8. OUTLINE OF THESIS 19

from the run will be presented and a discussion of its the ability and

shortcomings will ensure. The ultimate aim of this thesis, achieving

low-latency GW triggers for further EM follow-up, will be addressed in

this chapter.

• Chapter 7 will review and discuss the implementation of the new pipeline,

and the performance on both offline and online detector data. In this

chapter there will also be a discussion on what can be done to improve

the pipeline. There have also been several side studies that have used

this pipeline that will be discussed.


Chapter 2

Gravitational Waves

The aim of this chapter is to introduce all the physical and mathematical

concepts that will be used to create a new low-latency inspiral search pipeline

derived and tested in later chapters. The rigorous derivation of the inspiral

gravitational waveform here is inspired by Chapter 2 of [1]. Although the

derivation is not original, it is important to understand the concepts of the

inspiral gravitational waveform in order to design a new time-domain low-

latency algorithm. Readers familiar with inspiral GW data analysis may

wish to skip forward to Chapter 3, where the new low-latency method is first

introduced. Indeed, Chapter 3 was published as a self-contained article in

Physical Review D [119].

Firstly, in Section 2.1 the foundations on which GWs are based, linearised

gravity, will be discussed. For a more complete background on the funda-

mentals of GWs see [3], on which Section 2.1 closely follows. This will lead

to an understanding of how GWs can be detected directly using laser inter-

ferometers in Section 2.2. An explicit derivation of the inspiral gravitational

waveform will follow in Section 2.3. The way the waveform manifests in the

GW detector’s strain signal will be shown explicitly in Section 2.4. A dis-

cussion on how inspiral GWs are traditionally (and optimally) searched for

in detector strain data is covered in Section 2.5, as well as covering more

general signal processing concepts which will be useful in later chapters.

21

22 CHAPTER 2. GRAVITATIONAL WAVES

2.1 Linearised gravity

By introducing some small perturbation hµν to an otherwise flat space (the

Minkowski metric) ηµν , one has the metric tensor,

gµν = ηµν + hµν , (2.1)

where |hµν | 1. In the weak-field limit1, one can expand the Einstein field

equations in powers of hµν keeping only linear terms without much loss of

accuracy [3].

The Einstein field equations are the equations of general relativity that

form the link between the curvature of space-time and matter contained

within it:

Gµν =8πG

c4Tµν (2.2)

where Gµν is the Einstein tensor given by,

Gµν = Rµν − 12Rgµν (2.3)

and Rµν = Rγµνγ is the Ricci tensor and Tµν is the stress-energy tensor. By

introducing the tensor,

hµν ≡ hµν − 12ηµνh (2.4)

where h ≡ ηµνhµν , the linearised Einstein field equations can be expressed

as,

−h α

µν,α − ηµνhαβ

µν, + hα

µα, ν + hα

να, µ =8πG

c4Tµν . (2.5)

Without loss of generality, one can impose the gauge condition hα

µα, = 0

which reduces the above equation outside the source of the waves (i.e. where

1here we are describing the state of the space-time outside the source of GWs

2.1. LINEARISED GRAVITY 23

Tµν = 0) to

hα

µν,α = 0. (2.6)

2.1.1 Plane wave solution

An obvious solution to linearised field equations (2.6) is that of a monochro-

matic plane wave:

hµν = <[Aµνe

ikαxα]

(2.7)

where < [...] denotes the real part, A denotes the amplitude and kα denotes

the wave 4-vector satisfying

kαkα = 0 (2.8a)

Aµαkα = 0. (2.8b)

The first constraint states that kα is a null vector (and hence the wave travels

at the speed of light c), and the second constraint gives A as orthogonal to k.

There are six independent components ofA. However four degrees of freedom

can be removed by choosing a specific gauge, the transverse traceless gauge.

2.1.2 Transverse traceless gauge

Introducing the two gauge transformations,

Aµνuν = 0 (2.9a)

Aµµ = 0, (2.9b)

for any 4-velocity u that is the same throughout all space-time, results in

eight constraints onA in total. This can be seen by choosing a Lorentz frame


in a form where,

hµ0 = 0 only spatial components present (2.10a)

hkj,j = 0 spatial components divergence free (2.10b)

hkk = 0 spatial components are trace free. (2.10c)

Note there is no difference between hµν and hµν in this gauge. This choice

of gauge leaves any symmetric tensor satisfying the gauge conditions (2.10)

transverse, since it is purely spatial, and if described as a plane wave trans-

verse in the direction of propagation and traceless because the trace van-

ishes. Therefore any symmetric tensor satisfying the gauge conditions (2.10)

is called transverse traceless. In such a gauge form, the amplitude A takes

the form,

Aµν =

0 0 0 0

0 h+ h× 0

0 h× −h+ 0

0 0 0 0

(2.11)

where h+ and h× are the called plus and cross polarisation of the plane wave.

They take their name from the effect of a wave passing through a set of test

particles.

2.2 Detection of gravitational waves

As mentioned in Section 1.4, current generation GW detectors (such as the

LIGO/Virgo detectors) are based on kilometre scale Michelson interferome-

ters. A simple Michelson interferometer uses a laser pointed at beam splitter,

which in splits the light into two orthogonal directions. The light runs down

the two “arms” of the interferometer. At the end of the arms are suspended

test mass mirrors, which reflect the light back down each respective arm.

When the light comes back together, it will pass through the beam splitter

back to either the laser (the symmetric port) or towards a photo-detector

2.2. DETECTION OF GRAVITATIONAL WAVES 25

(the asymmetric port). See Figure 2.1 for a schematic of a simple Michelson

interferometer.

test mass

test mass

test mass

test mass

light storage arm

photodetector

laser

beamsplitter

light storage arm

Figure 2.1: A schematic diagram of the laser interferometer designed tomeasure GWs. A laser (rectangular box) in has its beam pointing toward abeam splitter. The beam splitter reflects the light down the one arm, andtransmits down the other arm. Each arm, of length L, is terminated by asuspended test mass mirror, which reflects the light back towards the beamsplitter. The light comes together in the beam splitter, going toward thelaser (symmetric port) and the photo-detector through the asymmetric port.Additional mirrors form a Fabry-Perot cavity, where the power of the lasercan increase. Credit [120].

The mirrors are test masses that are suspended in a free falling frame (for

Earth bound interferometers, the mirrors are under the gravitational effect

of the Earth, however it can be shown that the horizontal motion of the test

masses would be the same as if it were in a free falling frame). Choosing a

coordinate system for our interferometer, let us place mirror X on the x axis

initially a length L from the beam splitter, and mirror Y on the y axis also

initially a length L from the beam splitter. Let us define a polar coordinate

frame (θ, ϕ, ψ) to denote the direction of the incoming GW. Here θ it the

azimuthal angle from the x axis, ϕ the inclination from the normal to the


plane the arms lie in, and ψ is the polarisation angle which completes the

three Euler angles (see Figure 2.2).

x

y

j

Θ

Ψ

eΙ

eΦ

ej

eΘ

N

Figure 2.2: The three Euler angles (θ, ϕ, ψ) used to convert the frame of theGW radiation ( eι, eφ) to the x-y plane of the detector arms.

For the case of a purely + polarised GW with its source directly above

the plane of the detector (ϕ = 0) and its radiation frame aligned to the arms

of the detector (ψ = 0), the space-time interval between the beam splitter

and each test mass is

0 = gµνdxµdxν (2.12)

⇒ c dt2 = (1 + h+(t)) dx2 + (1− h+(t)) dy2 + dz2 (2.13)

where ds = 0 since the laser light is travelling along a null geodesic.

The distance LX,Y between the origin (beam splitter) and the test masses


X and Y as a plus polarised GW passes through will be,

LX(t) =

∫ L

0

√1 + h+(t) dx

≈ L

[1 +

1

2h+(t)

](2.14a)

LY (t) ≈ L

[1− 1

2h+(t)

](2.14b)

where the approximation is made assuming L to be much shorter than the

wavelength λ of the GW.

The phase Φ of the laser light accumulated along the length of the arm

on its return journey will be,

ΦX =

∫ L′

0

2π

λdx−

∫ 0

L′

2π

λdx

≈ 4πL

λ

(1 +

1

2h+

), (2.15a)

ΦY ≈4πL

λ

(1− 1

2h+

). (2.15b)

The phase difference between the two arms is,

∆Φ =4πL

λh+(t). (2.16)

Hence the detector can measure the strain,

h(t) =λ

4πL∆Φ ≈ ∆LX −∆LY

L(2.17)

The actual quantity measured by the photo-detector will depend on the pre-

cise configuration of the experiment. However for our purposes, we will as-

sume that the detector outputs a time series data h(t) which is the observed

strain at any point in time.

In order to calculate the gravitational-wave strain h incident to the plane

of the interferometer, one must calculate the set of geometric transformation

rules which transform the + and × polarisations of the radiation frame to the


frame of the interferometer. This has been previously worked out in great

detail (e.g. [121]). The gravitational strain observed in the detector will have

the form,

h(t) = F+(θ, ϕ, ψ)h+(t) + F×(θ, ϕ, ψ)h×(t) (2.18)

where the detector antenna response functions F+ and F× are functions of

(θ, ϕ, ψ); these can be found in [121].

2.2.1 Noises in interferometer

Like all instruments, the laser interferometer will have sources of noise and

these will corrupt the measured strain h. The main sources of noise are:

Seismic noise This occurs at the low frequency end, mostly at less than

40 Hz. Seismic vibrations due to passing vehicles and disturbances in

the Earth will induce vibrations in the suspended mirrors. Improve-

ments in test mass isolation systems aim to reduce this noise level below

10 Hz for Advanced LIGO/Virgo.

Thermal noise Thermal vibration of the mirrors and the suspension sys-

tem can alter the length L. This occurs around the ∼40-200 Hz band.

Improvements such as monolithic suspension wires and better optical

coatings are leading to increased sensitivity in this bandwidth.

Shot noise This source of noise arises from the quantum nature of the laser

light itself. Since the photo-detector is a photon counter, there is a

Poisson process, with error√N where N is the number of photons per

unit time. To increase the sensitivity, the laser power can be increased.

However simply increasing the power can lead to effects that can in-

crease the noise, such as increased thermal noise of the mirrors. One

way experimentalists plan to decrease the shot noise with increasing

the power of the laser is to use a power recycling mirror located at

the symmetric port. This reflects light back into the cavity, thereby

increasing the power.


These sources of noise are essentially stationary Gaussian processes. This

property can be characterised by use of a (one-sided) power spectral density

Sn(f) (PSD) defined by,

〈n(f)n(f ′)〉 ≡ 12Sn(|f |)δ(f − f ′) (2.19)

where the angled brackets 〈.〉 denote the ensemble average and δ(f − f ′) is

the Dirac delta function, with the property∫ ∞−∞

δ(f) df = 1. (2.20)

Since the above quantity is unit-less, δ(f − f ′) must have units of time. The

Fourier transform of n(t) has units of strain times time:

n(f) =

∫ ∞−∞

n(t)e−2πiftdt. (2.21)

Hence the (one-sided) noise PSD must have units of strain squared by time

(or time, since strain is unit-less). Generally the noise in the detector is

measured as an amplitude spectral density (ASD). This is simply the square

root of the PSD, hence it has units of root time. The common unit of

measuring ASDs is Hz−1/2. Figure 2.3 shows the measured ASDs for the

previous LIGO science runs, and the design sensitivity goal.


Figure 2.3: The best strain sensitivities (ASDs) of previous LIGO scienceruns are shown here. The curve can be broken down into roughly three majorsources of noise. The seismic “wall” below 40 Hz, the suspension thermalnoise between 40 and 200 Hz and the photon shot noise above 200 Hz. Thedetectors are most sensitive to sources with gravitational wave frequenciesaround 100 Hz. Credit: [45]

2.3 Inspiral gravitational waves

In this section the gravitational waveform produced during the inspiral phase

of compact binary coalescence will be derived. The section closely follows

the derivation of the inspiral waveform from [1].

The leading-order magnitude of gravitational radiation observed at a time

t and distance D from the source is given by the quadrupole formula (i.e.

2.3. INSPIRAL GRAVITATIONAL WAVES 31

Equation 36.45a of [3]),

hTTjk (t,x) =

2G

c4D

d2ITTjk (t−D/c)dt2

(2.22)

where ITT is the transverse traceless part of the second moment of mass

distribution I:

Ijk =

∫ρ(x)xjxk d

3x. (2.23)

2.3.1 Geometry of binary system

Consider a binary system with masses m1 and m2, total mass M = m1 +

m2, separated by a distance a in a circular orbit. If a 2GM/c2, then

Newtonian gravity can be used to give a description of the binary dynamics.

For simplicity, let us assume that the bodies behave as point mass objects

with no spin. To describe this system, let us define a Cartesian coordinate

system (x, y, z) where the centre of mass of the system is at the origin, and

the bodies rotate anti-clockwise (as seen from the positive z-axis) in the x-y

plane. The orbital phase φorb(t) is defined as the angle between m1 and the

positive x-axis. See Figure 2.4 for reference. The locations of the masses m1

and m2 at some time t are

(x1, y1, z1) =

(µ

m1

a cosφorb(t),µ

m1

a sinφorb(t)), 0

)(2.24a)

(x2, y2, z2) =

(− µ

m2

a cosφorb(t), − µ

m2

a sinφorb(t), 0

)(2.24b)

respectively. For convenience we have introduced the reduced mass

µ =m1m2

M. (2.25)


xy

z

m1

m2

a

ΦHtL

Figure 2.4: Coordinate system of binary system (x, y, z). The two bodies,m1 and m2 orbit about the centre of mass, which is chosen as the origin ofthe coordinate system. At some time t the phase φorb(t) describes the anglebetween m1 and the x axis. In this example, the masses are approximatelyequal, with a separation a.

The mass distribution of the binary is therefore

ρ(x) = m1

[δ

(x− µ

m1

a cosφorb(t)

)δ

(y − µ

m1

a sinφorb(t)

)δ(z)

]+m2

[δ

(x+

µ

m2

a cosφorb(t)

)δ

(y +

µ

m2

a sinφorb(t)

)δ(z)

]. (2.26)

Substituting the mass distribution (2.26) and the location of the bodies (2.24)

into the quadrupole moment equation (2.23), we find the non-zero compo-


nents of Ijk are Ixx, Ixy = Iyx and Iyy. Using the identity∫δ(x− x0)g(x) dx = g(x0), (2.27)

Ixx can be evaluated as

Ixx =

∫ (m1

[δ

(x− µ

m1

a cosφorb(t)

)δ

(y − µ

m1

a sinφorb(t)

)δ(z)

]+m2

[δ

(x+

µ

m2

a cosφorb(t)

)δ

(y +

µ

m2

a sinφorb(t)

)δ(z)

])x2 dx3

= m1

(µ

m1

a cosφorb(t)

)2

+m2

(µ

m2

a cosφorb(t)

)2

=

[m1

(µ

m1

)2

+m2

(µ

m2

)2]a2 cos2 φorb(t)

= µa2 cos2 φorb(t)

=1

2µa2 (1 + cos 2φorb(t)) (2.28)

Similarly, we can find Iyy and Ixy:

Iyy =1

2µa2 (1− cos 2φorb(t)) Ixy =

1

2µa2 sin 2φorb(t). (2.29)

The second derivatives with respect to time can then be easily worked

out:

Ixx = −Iyy = −2µa2Ω2(t) cos 2φorb(t) (2.30a)

Ixy = Iyx = −2µa2Ω2(t) sin 2φorb(t) (2.30b)

where we have introduced the orbital angular frequency Ω(t), which is the

first time derivative of the orbital phase. At this point, we assume that the

second time derivative of the orbital phase is negligible, i.e Ω(t) Ω2(t)


2.3.2 Orientation of the binary relative to an observer

Before we can insert the second derivatives of the moment into the quadrupole

formula (2.22) for hTTjk (t,x), we need to describe Iij as it would be seen along

a particular direction. Let us then choose a spherical polar coordinate system

(D, ι, φ) which has the centre of mass of the binary at the origin, and the

observer at the coordinates (D, ι, φ) — see 2.5. Here ι is the angle between

the line of sight N and the vector that is perpendicular to the plane of the

binary, i.e the angular momentum direction, L:

N · L = cos ι. (2.31)

For example, if ι = 0 the binary system is orientated “head-on” relative to

the observer, and if ι = ±π/2 the binary is “edge-on”. The orientation of

the x axis to the observer is completely arbitrary within the orbital plane.

For simplicity of transforming Ijk to the line of sight, let us fix the x axis

such that the line of sight lies along the z-x plane (see Figure 2.5).

The unit vectors ( eι, eφ) relate to the Cartesian unit vectors ( ex, ey, ez)

by

eι = cos ι cosφ ex + cos ι sinφ ey − sin ι ez, (2.32a)

eφc = − sinφ ex + cosφ ey. (2.32b)

To transform the second derivative of the quadrupole moment Iij from

Cartesian coordinates to spherical polar coordinates, we use the standard

transformation rule

A′ij =∂xk∂x′i

∂xl∂x′j

Akl. (2.33)

The partial derivatives can be worked out by using

e′i =∂xj∂x′i

ej (2.34)


xy

z

m1

m2

a

ΦHtL

Ι

eΙ

eΦ

N

Figure 2.5: Coordinate system of binary system (x, y, z) with respect to anobserver at (D, ι, 0). Here we have fixed the x axis such that the line of sightto the observer lies in the x-z plane (yellow plane).

and (2.32), yielding

∂x

∂ι= cos ι cosφ = cos ι

∂y

∂ι= cos ι sinφ = 0 (2.35)

∂x

∂φ= − sinφ = 0

∂y

∂φ= cosφ = 1, (2.36)

where we have explicitly chosen the z-x plane to be perpendicular to the

observer (i.e. φ = 0). This greatly simplifies the calculation of the moments


Iij, resulting in

Iιι =

(∂x

∂ι

)2

Ixx +

(∂y

∂ι

)2

Iyy + 2

(∂x

∂ι

∂y

∂ι

)Ixy

= −2µa2Ω(t)2 cos2 ι cos 2φorb(t), (2.37)

Iφφ =

(∂x

∂φ

)2

Ixx +

(∂y

∂φ

)2

Iyy + 2

(∂x

∂φ

∂y

∂φ

)Ixy

= 2µa2Ω(t)2 cos 2φorb(t). (2.38)

and

Iιφ = Iφι =

(∂x

∂ι

∂x

∂φ

)Ixx +

(∂y

∂ι

∂y

∂φ

)Iyy +

(∂x

∂ι

∂y

∂φ+∂x

∂φ

∂y

∂ι

)Ixy

= −2µa2Ω(t)2 cos ι sin 2φorb(t). (2.39)

Since the components of Iij derived are already transverse, we simply

need to remove the trace to get ITTij :

ITTιι = −ITT

φφ = Iιι −1

2

(Iιι + Iφφ

)=

1

2

(Iιι − Iφφ

)= −µa2Ω(t)2(1 + cos2 ι) cos 2φorb(t), (2.40)

ITTιφ = ITT

φι = −2µa2Ω(t)2 cos ι sin 2φorb(t). (2.41)

Substituting these results into the quadrupole formula (2.22) to gives us the

two independent polarisations :

hTTιι = h+(t) = −4Gµa2Ω(t)2

c4D

(1 + cos2 ι

2

)cos 2φorb(t), (2.42a)

hTTιφ = h×(t) = −4Gµa2Ω(t)2

c4D(cos ι) sin 2φorb(t). (2.42b)


2.3.3 Orbital frequency as a function of time

Although the binary system is evolving through a series of quasi-stationary

circular orbits with orbital energy

E = −1

2

GµM

a, (2.43)

there is a loss of energy due to gravitational waves caused by the quadrupole

moment:

dE

dt= − G

5c5〈...I TT

ij

...I TTij 〉 = −32G4

5c5

M3µ2

a5. (2.44)

Because of the energy leak, the orbital separation a will reduce at a rate

da

dt=dE

dt

(dE

da

)−1

= −64G3

5c5

µM2

a3. (2.45)

By integrating this rate, we can obtain the separation as a function of time:∫a3 da =

∫−64G3

5c5µM2dt

⇒ a(t) =

(256G3

5c5µM2

)1/4

(tc − t)1/4, (2.46)

where have integrated up to a “coalescence” time tc. In practise there is a lo-

cal minimum of the effective potential of the system. In the Schwarzschild po-

tential, the minimum distance for the innermost stable circular orbit (ISCO)

is the three times the Schwarzschild radius 2GM/c2. This is reached at a

time

(tc − t) =405

16η

(GM

c3

). (2.47)

We can relate the separation a to the orbital angular frequency Ω by using


Kepler’s third law,

a =3

√GM

Ω2. (2.48)

Hence the orbital angular frequency Ω as a function of time can be obtained

by inserting the expression for the separation a (2.46),

Ω(t) =(GM)1/2

8

((GM)3

5c5η

)−3/8

(tc − t)−3/8

=c3

8GM

(c3η

5GM

)−3/8

(tc − t)−3/8

=c3

8GM[Θ(t)]−3/8, (2.49)

where in the second equality we have introduced the symmetric mass ratio

η = µ/M . In the final step, we have introduced the convenient dimensionless

time parameter Θ,

Θ(t) =

(c3η

5GM

)(tc − t). (2.50)

The orbital phase of the waveform at some time t before coalescence is the

time integral of the orbital angular frequency:

φorb(t) =

∫Ω(t)dt (2.51)

= φorb,c −1

ηΘ(t)5/8 (2.52)

where φorb,c is the orbital phase at coalescence tc.

2.3.4 Higher order multipole corrections

We have derived the gravitational waveform assuming only the lowest or-

der multipole moment, the quadrupole moment. However in reality, there

are higher order multipolar moments that contribute to energy loss, as well

as relativistic corrections to the quadrupolar formula that can change the


phase evolution. Ultimately, the goal is to identify the general form of the

GW to be searched for in the detector strain data. Studies have shown that

the so called restricted post-Newtonian (PN) waveform is sufficient to use in

matched filtering (discussed in the next section) [9]. So far we have evalu-

ated the zero-th order “Newtonian” waveform. The restricted PN waveform

has the amplitude factor still with a Newtonian order, but includes higher

v/c corrections to the phase evolution. The orbital phase to second post-

Newtonian order, generally used in LIGO inspiral searches, has the orbital

phase

φorb(t) = φorb,c −1

η

[Θ(t)5/8 +

(3715

8064+

55

96η

)Θ(t)3/8 − 3π

4Θ(t)−1/4

+

(9275495

14450688+

284875

258048η +

1855

2048η2

)Θ(t)1/8

],

(2.53)

where Θ is defined in equation (2.50).


Figure 2.6: Schematic showing trajectories of two equal mass bodies in abinary system near coalescence. The black circle represents the boundaryof the innermost stable circular orbit (ISCO). Beyond this limit, the bodiesplunge toward each other. This will be the time at which we will terminatethe inspiral waveform. As the bodies inspiral, the separation gets smaller asa 1/4 power law in time, and the orbital angular frequency increases as a-3/8 power law (in the Newtonian limit).

2.4. INSPIRAL WAVEFORM 41

2.4 Inspiral gravitational waveform as seen in

the detector strain

In this section, we turn our attention to describing the inspiral gravitational

waveform as it will be observed in the strain of the detector.

Let us now define the GW + and × polarisations (2.42) in terms of the

“cosine” and “sine” components hc, hs which will be useful later. From

(2.42),

h+(t) =

(1 + cos2 ι

2

)A(t) cos 2φorb(t) (2.54a)

h×(t) = (cos ι)A(t) sin 2φorb(t), (2.54b)

where the amplitude factor A(t) is defined as,

A(t) = −4Gµ

c4Da2Ω2

= −4Gµ

c4D(GMΩ(t))2/3 substituting (2.48)

= −GMη

c2D

(c3η

5GM(tc − t)

)−1/4

substituting (2.49) and µ = Mη

=GMDc2

(tc − t

5GM/c3

)−1/4

. (2.55)

Here we have introduced the chirp mass M = Mη3/5. Let us also choose to

describe the phase of the gravitational waveform as

2φorb(t) = φ(t) + φc (2.56)

where we define the gravitational phase φ(t) (to second post-Newtonian order

as in (2.53)) to be

φ(t) = −2

η

[Θ(t)5/8 +

(3715

8064+

55

96η

)Θ(t)3/8 − 3π

4Θ(t)1/4

+

(9275495

14450688+

284875

258048η +

1855

2048η2

)Θ(t)1/8

].

(2.57)


and φc is the gravitational phase at time of coalescence, tc. The form (2.56)

enables the + and × polarisations of the waveform to be written as

h+(t) =

(1 + cos2 ι

2

)A(t) cos (φ(t) + φc) , (2.58)

h×(t) = (cos ι)A(t) sin (φ(t) + φc) . (2.59)

Substituting these into the strain measured at the detector (2.18) we get

h(t) = F+

(1 + cos2 ι

2

)A(t) cos (φ(t) + φc)

+F× (cos ι)A(t) sin (φ(t) + φc)

(2.60)

where for convenience we have dropped the parameters (θ, ϕ, ψ) from the an-

tenna pattern functions F+, F×. One can use the linear combination trigono-

metric identity

a cos θ + b sin θ =√a2 + b2 cos(θ − α) (2.61)

where tanα = b/a to re-express the strain (2.60) as

h(t) =D

Deff

A(t) cos(φ(t)− φ0) (2.62)

where the “effective distance” factor Deff obtained from the trigonometric

identity (2.61) is

Deff =D√

F 2+ (1 + cos2 ι)2 /4 + F 2

× (cos ι)2(2.63)

and φ0 is related to the antenna pattern functions F+, F× and coalescence

phase φc by

φ0 = arctan

(F× (2 cos ι)

F+ (1 + cos2 ι)

)− φc. (2.64)

The advantage of expressing the strain as (2.62) is that, by the trigonometric

2.4. INSPIRAL WAVEFORM 43

addition identity,

h(t) =D

Deff

[A(t) cosφ(t) cosφ0 + A(t) sinφ(t) sinφ0]

=(1 Mpc)

Deff

[A1 Mpc(t) cosφ(t) cosφ0 + A(t)1 Mpc sinφ(t) sinφ0]

=(1 Mpc)

Deff

[hc(t) cosφ0 + hs(t) sinφ0] , (2.65)

where in the second equality we have explicitly set D in the numerator and

D in the amplitude function A(t) to be 1 Mpc. This allows us to define the

cosine and sine components of the waveform as

hc(t) = A1 Mpc(t) cosφ(t), (2.66)

hs(t) = A1 Mpc(t) sinφ(t). (2.67)

2.4.1 Intrinsic and extrinsic parameters

We see that the inspiral signal h(t) given by (2.65) will take its form from

nine different parameters ϑµ, shown in table 2.1. Most of the parameters

Table 2.1: Table of intrinsic and extrinsic inspiral parameters

Parameter ϑµ Symbol Unit Unit of measurecomponent mass 1 m1 mass Mcomponent mass 2 m1 mass Mtime of coalescence tc time sphase at coalescence φc angle radiansdistance to source D length Mpcinclination ι angle radianssky coordinates (θ, ϕ) angle radianspolarisation angle ψ angle radians

depend on the orientation of the binary with respect to the observer, and

can be combined into the effective distance amplitude Deff given by (2.63).

Of the remaining parameters, tc can searched for explicitly in time, shown

in Section 2.5.4. The unknown phase term φ0 can also be maximised over

using the procedure outlined in Section 2.5.5.


The only remaining parameters are the intrinsic parameters, the compo-

nent masses m1 and m2, which must be explicitly searched for, as explained

in Section 2.5.3.

2.5 Signal processing

The goal of a GW data analyst is to determine whether a GW signal is

present in the GW detector strain data at some time. For those looking

for inspiral signals, this usually means determining when an inspiral signal

(with any number of parameters) just finished. In this respect the time of

coalescence tc, is searched for explicitly. As discussed in the previous section,

the inspiral gravitational waveform as a function of time is well known (for

given intrinsic parameters) and can be scaled by effective distance, given by

(2.63). In order to detect whether an inspiral waveform just finished in the

detector strain data, one must construct a filter K that is cross-correlated

(in time) with the detector data.

2.5.1 Matched Filter

It has long been known [122] that the best way to search for a known signal

in stationary Gaussian noise is to use a matched filter . This is not the name

of a class of filter, such as a FIR or IIR or the specific name of a type of filter

such as Chebyshev or Butterworth, but rather a theoretical framework. In

this section we will derive the form of optimal filter K that will maximise

the signal searched for with respect to the background noise (signal to noise

ratio, SNR). From Section 2.2.1 we know that the input signal (in our case

strain) s, contains stationary Gaussian noise n(t) and possibly a signal h(t)

with a set of parameters ϑµ.

s(t) =

n(t) if signal is absent

n(t) + h(t) if signal is present.(2.68)

2.5. SIGNAL PROCESSING 45

The input to the optimal filter will be s(t) = h(t) + n(t) and the output

will be something that scales with SNR. We want to find the filter K, that

produces the filtered output z:

z =

∫ ∞−∞

s(t)K∗(t) dt =

∫ ∞−∞

s(f)K∗(f) df.

=

∫ ∞−∞

h(f)K∗(f) df +

∫ ∞−∞

n(f)K∗(f) df

= H +N. (2.69)

This is a cross-correlation of the filter K(t) and the input s(t). The first

term H is well-defined, but the second term is a random process. Using the

properties of the noise given in Section 2.2.1, the ensemble average N2 is

found to be

〈N2〉 =

∫ ∞−∞

∫ ∞−∞

K∗(f)K∗(f ′)〈n(f)n(f ′)〉 dfdf ′

=

∫ ∞−∞

∫ ∞−∞

K∗(f)K∗(f ′)12Sn(|f |)δ(f − f ′) dfdf ′

=1

2

∫ ∞−∞|K(f)|2Sn(|f |) df. (2.70)

We are looking for the filter K that maximises the ratio of H2 to 〈N2〉, i.e.

ξ =H2

〈N2〉 =2∣∣∣∫∞−∞ h(f)K∗(f)e2πift df

∣∣∣2∫∞−∞ |K(f)|2Sn(|f |) df

=2∣∣∣∫∞−∞ [h(f)/

√Sn(f |)

] [K∗(f)

√Sn(f |)

]df∣∣∣2∫∞

−∞ |K(f)|2Sn(|f |) df. (2.71)

The Cauchy-Schwarz inequality tells us that for two functions A(f) and B(f),∣∣∣∣∫ ∞−∞

A(f)B(f)df

∣∣∣∣2 ≤ ∫ ∞−∞|A(f)df |2

∫ ∞−∞|B(f)df |2 (2.72)


if and only if A and B are linearly dependent, i.e.

A(f) = C ·B(f) (2.73)

where C is a constant (see proof in [122]). By setting A(f) = K∗(f)√Sn(f)

and B(f) = h(f)/√Sn(f), the numerator in (2.71) is maximised, yielding

ξ =

2C

[∫∞−∞

∣∣∣h(f)∣∣∣2 /Sn(f) df

] [∫∞−∞

∣∣∣K(f)∣∣∣2 Sn(f) df

]∫∞−∞ |K(f)|2Sn(|f |) df

(2.74)

= 2C

∫ ∞−∞

∣∣∣h(f)∣∣∣2

Sn(f)df. (2.75)

Inserting the chosen A and B into (2.73) gives,

K∗(f)√Sn(f) = h∗(f)/

√Sn(f) (2.76)

→ K∗(f) = 2h∗(f)

Sn(f), (2.77)

where we have set the arbitrary constant C = 1. Hence the optimal filter

scales with,

z ≡ 2

∫ ∞−∞

s(f)h∗(f)

Sn(|f |) df. (2.78)

2.5.2 Inner product

The definition of the optimal filter (2.78) naturally leads us to the idea of an

inner product 〈· |·〉. Let us define the inner product of two real vectors s and

h to be

〈s |h〉 ≡ 2

∫ ∞−∞

s(f)h∗(f)

Sn(|f |) df. (2.79)


This definition can be applied to any two real vectors, i.e. two waveforms

with slightly different intrinsic parameters, h and h′:

〈h |h′ 〉 = 2

∫ ∞−∞

h(f)h′∗(f)

Sn(|f |) df. (2.80)

These can be normalised such that

~h =h√〈h |h〉

, (2.81)

leading to the definition of the normalised inner product,

⟨~h∣∣∣~h′⟩ =

⟨h√〈h |h〉

∣∣∣∣∣ h′√〈h′ |h′ 〉

⟩=

〈h |h′ 〉√〈h |h〉

√〈h′ |h′ 〉

. (2.82)

2.5.3 Template bank

Clearly the matched filter can only search for a single “template” — a wave-

form with a specific set of parameters. As we have already discussed, the

extrinsic parameters D, ι, ψ, θ, ϕ simply scale the waveform in the effective

distance Deff . In order to find a waveform with any distribution of intrinsic

parameters ϑµ using the matched filter, we must first define the continuous

manifold on which all waveforms reside. The geometry of this parameter

space manifold will allow us to find the minimum spacing between templates

such that for any waveform with unknown intrinsic parameters, at least one

matched filtered template will produce a high SNR.

In this section, we will only present a short derivation on template bank

construction. For a full discussion on template bank construction, see [123]

(for example).

Using the normalised inner product (2.82), we define the “match” M

between two normalised waveforms ~h(ϑµ) and ~h(ϑν) as

M = max⟨~h(ϑµ)

∣∣∣~h(ϑν)⟩

(2.83)

≈ 1 +1

2

(∂2M

∂∆ϑµ∂∆ϑν

)∆ϑα=0

∆ϑµ∆ϑν . (2.84)


The concept of a match leads to the idea of a “mismatch” MM,

MM = 1−M ≡ γµνdϑµdϑν (2.85)

where the metric γµν defines the differential geometry of the intrinsic param-

eter space. Each point on this space corresponds to intrinsic parameter ϑµ.

The distance between two points on the manifold can be found using the met-

ric, which will tell us the mismatch. From this metric (or rather a coordinate

transformation of it), a discrete “template bank” can be constructed.

2.5.4 Matched filter as a function of unknown time of

coalescence

The chosen filter K (2.77) depends on the unknown time of coalescence tc.

This parameter can be “pulled” out via the following rearrangement of h(f),

h(f) =

∫h(t)e−2πift dt

=

[∫h(t)e−2πif(t+tc) dt

]e2πiftc

let t = t′ − tc, dt = dt′

=

[∫h(t′ − tc)e−2πift′ dt′

]e2πiftc

= F [h(t− tc)] · e2πiftc (2.86)

From now on we explicitly use h(f) to represent the Fourier transform of

h(t − tc) (which does not explicitly depend on tc). This way, the unknown

time of coalescence tc can now be searched over by changing the value of tc

in the matched filter,

z(tc) = 2

∫ ∞−∞

s(f)h∗(f)

Sn(|f |) e2πiftc df. (2.87)


By introducing

x(f) =s(f)

Sn(|f |) (2.88)

where x represents the “over-whitened” detector output s(t), weighted by the

inverse noise power spectral density Sn(f), we can use the cross-correlation

theorem to define the matched filter (2.87) in the time domain:

z(tc) =

∫ ∞−∞

x(f)h∗(f)e2πiftc df = 2

∫ tc

−∞x(t)h(t− tc) dt, (2.89)

recalling that we have now defined h(t) to explicitly have tc = 0.

2.5.5 Matched filter of unknown phase

The unknown phase constant φ0 can be maximised over by filtering both

components hc and hs (which correspond to orthogonal phases φ0 = 0 and

φ0 = π/2) separately and then combining them to form a complex signal

z(tc) = 2

∫ ∞−∞

s(f)h∗c(f)

Sn(|f |) e2πiftcdf + i2

∫ ∞−∞

s(f)h∗s(f)

Sn(|f |) e2πiftcdf. (2.90)

The real part corresponds to the matched filter output for the (real) template

hc, and the imaginary part corresponds to the matched filter output for the

(real) template hs.

The cross-correlation theorem gives the complex time domain two-phase

matched filter (2.90):

z(t) = 2

∫ t

−∞x(t′)hc(t

′ − t)dt′ + i2

∫ t

−∞x(t′)hs(t

′ − t)dt′ (2.91)

= 2

∫ t

−∞x(t′)h(t′ − t)dt′ (2.92)

where, for simplicity of later equations, we define

h(t) = hc(t) + ihs(t) = A1 Mpc(t)eiφ(t) (2.93)


(recalling again that amplitude A(t) and phase φ(t) implicitly have tc = 0).

2.5.6 Signal to noise ratio

Since the matched filter output z maximises the signal to noise ratio, we

could simply use this as detection statistic by declaring detection if it has a

value above some threshold. However first we normalise by the variance σ2

of the real part of the real filter:

σ2 = VAR (〈n |hc 〉)⟨(2

∫ ∞−∞

n(f)h∗c(f)

Sn(f)df + 〈2

∫ ∞−∞

n(f)h∗c(f)

Sn(f)df〉)2⟩

= 〈(

2

∫ ∞−∞

n(f)h∗c(f)

Sn(f)df + 2

∫ ∞−∞〈 n(f)√

Sn(f)〉 h∗c(f)√

Sn(f)df

)2

〉

= 〈(

2

∫ ∞−∞

n(f)h∗c(f)

Sn(f)df

)2

〉

= 2

∫ ∞−∞

hc(f)h∗c(f)

Sn(f)〈 n(f)n∗(f)

Sn(f)〉df

= 2

∫ ∞−∞

hc(f)h∗c(f)

Sn(f)df

= 〈hc |hc 〉 = 2

∫ ∞−∞

∣∣∣hc(f)∣∣∣2

Sn(|f |) df. (2.94)

We could define the SNR as,

SNR(t) =z(t)

σ. (2.95)

However this clearly means SNR could be complex, which does not really

make sense. Following convention (c.f. [124]), let us define the amplitude

SNR, ρ, as the absolute value of the two-phase matched filter z divided by


the standard deviation of its real part,

ρ(t) =|z(t)|σ

. (2.96)

By normalising the matched filter like this, in the absence of a GW, the SNR

squared ρ2 is Chi-square distributed with two degrees of freedom (one for

each real and imaginary component). Hence the probability of finding an

SNR value greater than ρ∗ in the absence of a waveform is [125]

P (ρ2 > ρ2∗) = e−ρ

2∗/2. (2.97)

Conversely, in the presence of an inspiral waveform (3.12) the expected ρ2 is

〈ρ2〉 =〈|z|2〉σ2

=

(1 Mpc

Deff

)2

σ2. (2.98)

Hence the amplitude SNR ρ forms a very useful detection statistic, as values

well above unity are unlikely to caused by Gaussian noise alone. The relation

(2.98) can also determine the effective distance Deff for a recovered amplitude

SNR ρ.

2.5.7 Discrete time domain filtering

In practise the detector data s(t) is discretely sampled at intervals of ∆t

with a sampling rate of fs ≡ 1/∆t. The discretisation of the continuous time

domain matched filter (2.92) is

zk = 2k∑

j=−∞xjhj−k∆t (2.99)

where the index k denotes discretely sampled times of t, namely tk = k∆t. In

practise, the inspiral waveform template hi is bounded in both the time and

frequency domains because the detector is only sensitive over a finite band-

width, fmin–fmax. Hence the discretisation of the matched filter conforms to


the standard form of a finite impulse response (FIR) filter,

yn =M∑k=0

bkxn−k (2.100)

where standard signal processing terminology denotes the feed-forward coef-

ficients bk, which in our example relate the time domain template hj. The

term impulse response comes from the filter’s output (response) when there

is only one non-zero unity input sample (impulse). In this case, M samples

after the unit impulse the output is zero. Hence the term finite impulse

response.

The approximate duration of the template in the time domain can be

worked out by inverting the orbital frequency (2.49) and noting that the inspi-

ral gravitational frequency, f , is twice the orbital frequency (f = 2(Ω/2π) =

Ω/π); thus

tc − tf =5GM

ηc3

(8GMπf

c3

)−8/3

(2.101)

where tf is the time when the template has a gravitational frequency f .

As discussed in Section 2.3.3, we define the end of the inspiral waveform

to be the time of ISCO. Substituting the time of ISCO (2.47) into the above

equation, we can see that for a typical NS-NS system (where the total mass

is less than 10M), the time between coalescence and ISCO is negligible

(< 1 second). Hence the actual duration T of the waveform will depend on

the lower frequency boundary of the detector’s sensitive bandwidth, which

is primarily dependent on seismic noise (see 2.2.1). For LIGO’s fifth science

run, this was around 40 Hz. For Advanced LIGO, this is expected to go down

to 10-15 Hz. Although one could place a coefficient bj for every sample hj,

a more efficient method would be to place coefficients at each quarter wave

cycle Ncyc (for example). Hence M could be proportional to the number of

cycles rather than the template duration. Table 2.2 shows the duration T

and number of cycles Ncyc of a canonical 1.4-1.4 M NS-NS template as a

function of fmin.


Table 2.2: Table of approximate template durations T and number of cyclesNcyc of a 1.4-1.4M template based on starting frequency fmin

fmin Duration T (seconds) Number of cycles Ncyc

40 2.5× 101 1.6× 103

15 3.4× 102 8.2× 104

10 1.0× 103 1.6× 104

3 2.5× 104 1.2× 105

2.5.8 Infinite Impulse Response Filter

Infinite impulse response (IIR) filters differ from FIR filters in that the filter

output depends on previous filter inputs (feed-forward) as well as previous

filter outputs (feed-back). The name infinite comes from their characteris-

tic of having a non-zero response any time after an impulse. The generic

difference equation of an IIR filter is,

yn =N∑k=1

akyn−k +M∑k=0

bkxn−k. (2.102)

Hence IIR filters have two sets of coefficients, ak’s (feed-back coefficients)

and bk’s (feed-forward coefficients).

Analogs of the coefficients are capacitance, resistance, inductance (if the

system is electrical) or mass, coefficient of damping, and coefficient of re-

silience (if the system is mechanical), or thermal capacitance, thermal con-

ductance (if the system is thermal).

IIR filter design

When designing FIR filters, the coefficients are generally chosen by first find-

ing the desired frequency response and then simply using the inverse Fourier

transform to find the time domain coefficients. For IIR filters, choosing co-

efficients ak and bk is not as easy. In fact it is well known that there is no

direct method for designing IIR filter coefficients. However standard IIR fil-

ter design falls into three classes; impulse invariance, bi-linear transform, and

optimisation methods, all of which use the z-transform. The z transform of


(2.102) is

Y (z) = Y (z)N∑k=1

akz−k +X(z)

M∑k=0

bkz−k (2.103)

which leads to the transfer function

H(z) =Y (z)

X(z)=

∑Mk=0 bkz

−k

1 +∑N

k=1 akz−k. (2.104)

One key aspect of IIR filter design is stability. A system is defined as

stable if, given any bounded input, the output will always be bounded. This

condition is satisfied when all poles of the transfer function (2.104) lie within

the unit circle.

There are many different types of forms for IIR filters. Examples include

Direct Form I and II, modified Direct Form I and transposed Direct Form

II. All different forms return the same output, but are constructed in such

a way as to optimise performance (i.e. minimise memory requirements, or

maximise computational efficiency).

A number of IIR filters can also be used in combination to give a single

output. In this way, IIR filters can either be cascaded (one filter’s output is

the next filter’s input), or operate in a parallel configuration (e.g several filters

take the same input, and then add their outputs to give a single output). The

low-latency inspiral search pipeline introduced in the next chapter is based

on the latter, parallel IIR filters. The transfer response function of parallel

IIR filters is the addition of each IIR filter’s transfer response function. As

will be seen in the next chapter, a summation of IIR filters can approximate

the transfer response of an inspiral waveform.

2.6 Summary

In this chapter we have provided all the physical and mathematical concepts

necessary to design a search method for GWs from the inspiral phase of a

CBC source. In Section 2.1 we have shown how a small perturbation to an

2.6. SUMMARY 55

otherwise flat space can allow for plane wave solutions to the Einstein field

equations. A description of how this perturbation causes a differential frac-

tional change in arm length of large-scale Michelson interferometers was given

in Section 2.2. With some calibration, the induced strain can be measured as

a fluctuation of photons at the asymmetric port of the interferometer. The

strain from a true GW is subject to many noise sources all corrupting the

purity of any GW signal. The gravitational waveform produced by the inspi-

ral phase of a CBC is given in detail in Section 2.3. The way an inspiral GW

manifests in the strain data signal of ground-based GW detectors was shown

Section 2.3. Due the noise level of the signal, the concept of calculating a

signal to noise ratio (SNR) was introduced in Section 2.5, as well as other

signal processing techniques which will enable the detector strain data to be

analysed.

We now have all the fundamental knowledge of inspiral gravitational

waves and signal processing techniques with which to design a new inspi-

ral search pipeline.


Chapter 3

Low-Latency Gravitational

Wave Detection Method

In this chapter, a new time-domain low-latency pipeline will be presented.

Our method aims to achieve low latency by filtering the over-whitened strain

x in the time domain using a bank of parallel IIR filters. The idea of using

IIR filters to search for inspiral GW signals in detector data was first explored

in [126]. However our method is independent of theirs, as ours depends on

a bank of fixed single pole IIR filters, and theirs depends on adaptive line

enhancer filters.

The remainder of this chapter consists of an article we published in Phys-

ical Review D [119]. Sections 3.0 to 3.6 is our paper verbatim.

3.0 Paper abstract

With the upgrade of current gravitational wave detectors, the first detection

of gravitational wave signals is expected to occur in the next decade. Low-

latency gravitational wave triggers will be necessary to make fast follow-up

electromagnetic observations of events related to their source, e.g., prompt

optical emission associated with short gamma-ray bursts. In this paper we

present a new time-domain low-latency algorithm for identifying the presence

of gravitational waves produced by compact binary coalescence events in

57

58 CHAPTER 3. LOW-LATENCY GW DETECTION METHOD

noisy detector data. Our method calculates the signal to noise ratio from

the summation of a bank of parallel infinite impulse response (IIR) filters.

We show that our summed parallel infinite impulse response (SPIIR) method

can retrieve the signal to noise ratio to greater than 99% of that produced

from the optimal matched filter.

3.1 Introduction

The interferometric gravitational wave (GW) detectors LIGO [35], and Virgo

[37] have reached a sensitivity at which the detection of GWs is possible.

The LIGO detectors are currently undergoing a major upgrade to Advanced

LIGO, for which the sensitivity will be improved ten fold relative to Initial

LIGO [48]. Hence in the era of advanced detectors GWs produced from

inspiralling compact binaries will be detectable within a volume of space one

thousand times larger than that of initial LIGO, out to ∼200-300 Mpc [49].

The emission of GWs produced by compact binary coalescence (CBC)

can be modelled with a high degree accuracy [127]. When two compact

bodies, such as neutron stars or black holes are in orbit, Einstein’s equations

predict the generation of GWs. As the bodies spiral towards each other a

GW is created that increases in frequency over time until the bodies merge,

following what is known as the inspiral waveform. Ground based detectors

have frequency passbands that allow them to be sensitive to the final stages

of such events up to total system masses of several hundred M.

Neutron star binary mergers are widely thought to be the progenitors of

short hard gamma-ray bursts (short GRBs) [63, 128]. The delay between

the final GW emission and the onset of the GRB is estimated to be as

short as 0.1 seconds or as long as tens to hundreds of seconds [82, 83]. The

electromagnetic emission of the GRB event is not well understood. Related

to the initial GRB there is thought to be a prompt emission in X-ray and

optical wavelengths followed by a delayed afterglow of cascading wavelengths.

Prompt optical emission may occur tens to hundreds of seconds after the

initial burst. The low-latency detection of the GW associated with a neutron

star merger could lead to the localization of a GRB source event on the

3.1. INTRODUCTION 59

sky, enabling fast moving telescopes to observe the prompt optical emission.

Data collected from a multitude of sources — GWs, gamma-rays, X-rays and

optical counterparts of the GRB — will lead to maximum insight into these

highly energetic events.

The standard strategy for searching for the existence of inspiral wave-

forms in the detector data is based on matched filtering [127] (and references

therein). This method, based on Wiener optimal filtering, is a correlation

of an expected inspiral waveform template and the detector data, weighted

by the inverse noise spectral density of the detector [122]. In order to save

computational costs, this correlation is performed in the frequency domain,

via a Fourier transform of a finite segment of detector data. In previous

LIGO searches, the detector data is split up into “science blocks”, which are

further divided into “data segments” chosen to be at least twice the length

of the longest waveform in the template bank [129]. Each data segment is

chosen to overlap the previous one by 50%. Each segment therefore must be

matched filtered in a time that is half the length of the segment for a real-

time analysis; that is, the filter output rate is equal to the data input rate.

In this case, the matched filter process has a minimum latency (from signal

arrival to signal detection) that is proportional to the longest template (see

[4] for more details). Advanced LIGO will have an increased bandwidth over

Initial LIGO, with the lower bound dropping from 40 Hz to 10 Hz [49]. GW

signals from CBC events spend much more time at these lower frequencies.

Hence waveforms used for matched filtering in Advanced LIGO will be much

longer (thousands of seconds). This in turn means the segment length will

be increased, further increasing the latency. The latency of this method to

produce GW triggers is longer than the time to onset of prompt optical emis-

sion after coalescence (10s to 100s of seconds). After this amount of time,

the early electromagnetic counterpart of a GRB event will be significantly

faded, and may be missed by telescopes altogether.

A low-latency GW detection method is required to trigger follow-up elec-

tromagnetic observations of the prompt optical emission. So far two fre-

quency domain methods have been developed to solve this issue. The VIRGO

group has produced a low-latency pipeline based on Multi-Band Template


Analysis (MBTA) [116], and LIGO is also working on a new method, Low-

Latency On-line Inspiral Data analysis (LLOID) method. In MBTA the

matched filtering technique is split over two frequency bands, and the out-

put is coherently added, reducing latency. A latency of less than 3 minutes

until the availability of a trigger using this method has been achieved [116].

Low-latency in the LLOID method is achieved by first down-sampling the

incoming data into multiple streams and then applying frequency domain

finite impulse response (FIR) filters [130]. The computational cost of this

pipeline is reduced by decreasing the number of templates via singular value

decomposition [118].

We introduce a new method to detect CBC signals in the time domain

using infinite impulse response (IIR) filters. Approximating an inspiral wave-

form by a summation of time shifted exponentially increasing sinusoids en-

ables us to construct a bank of parallel single-pole IIR filters. Each IIR filter

acts as a narrow bandpass filter. When each appropriately delayed IIR fil-

ter is added the coherent output approximates the matched filter output of

the exact waveforms. We call this the summed parallel infinite impulse re-

sponse (SPIIR) method. Figure 3.1 visually demonstrates the idea of using

a bank of IIR filters as narrow bandpass filters. For a full explanation of the

mathematical principles, see [4].

In this follow up paper, we numerically address the issues essential to

the practical use of this method for the upcoming advanced detectors. We

calculate the filter coefficients and demonstrate via numerical simulations

how well our method approximates the optimal matched filter as a function

of the number of filters per bank using a range of parameters. We also show

that the detection rate of the SPIIR method is very similar to that of the

matched filter method. It has been shown theoretically that in order to get

the same latency as the SPIIR method, the frequency domain matched filter

method would require greater computational resources [4].

The structure of this paper is as follows: In section 3.2 we will go through

the formal introduction of the inspiral waveform and matched filtering, and

how to get from the continuous frequency domain matched filter to the time

domain discrete matched filter. This will lead to a demonstration on how

3.2. METHODOLOGY 61

f1

d1

f2

d2

f3

d3

fn

dn

Input Output+

Figure 3.1: A schematic overview of the SPIIR method. The input is splitinto different channels, time delayed by an amount d, then passed through anarrow bandpass IIR filters, each with a different central frequency f . Finallythe output of each individual IIR filter is summed, giving the output of theSPIIR method.

it is possible to approximate an inspiral signal by a sum of exponentially

increasing sinusoids. The methodology is explained in Section 3.3 and will

cover how we set up our simulation to test the efficiency of the SPIIR method

as opposed to the frequency domain matched filter. Section 3.4 will analyze

the results of the simulation and Section 3.5 will discuss the implications of

these results for advanced detectors.

3.2 Methodology

Gravitational wave interferometers output the strain induced by gravitational

waves incident on the detector, as well as inherent noise. In unitless strain,


the detector output will be

s(t) =

n(t) if signal is absent

n(t) + h(t) if signal is present(3.1)

where n(t) is the noise inherent in the detector, assumed to be a stationary

Gaussian process with mean zero. The sensitivity of the instrument can be

characterized by the (one-sided) power spectral density Sn(f) defined by

〈n(f)n∗(f ′)〉 =1

2Sn(|f |)δ(f − f ′) (3.2)

where 〈. . .〉 denotes the ensemble average over detector noise, and the tilde

represents the forward Fourier transform,

q(f) =

∫ ∞−∞

q(t)e−2πiftdt. (3.3)

3.2.1 The Inspiral Waveform

The gravitational-wave strain incident at the interferometer is given by

h(t) = F+(θ, ϕ, ψ)h+(t) + F×(θ, ϕ, ψ)h×(t) (3.4)

where the detector antenna response functions F+ and F× are functions of

(θ, ϕ) — the standard spherical polar co-ordinates measured with respect

to the detector’s frame, and ψ is the polarization angle. The detector an-

tenna response functions, F+ and F×, can be found in [121]. The + and ×polarizations of the waveform are

h+(t) =

(1 + cos2 ι

2

)A(t) cos (φ(t) + φc) (3.5)

and

h×(t) = (cos ι)A(t) sin (φ(t) + φc) . (3.6)

For non-spinning binaries with component masses m1, m2 in the range

3.2. METHODOLOGY 63

of (1 − 3)M — which we will hereafter assume — the waveforms can be

modelled to very high accuracy using the Restricted post-Newtonian (PN)

expansion [9, 41, 131] in the LIGO band (assumed to be 10-1500 Hz for

advanced LIGO). For restricted waveforms, only the leading order of the

amplitude A(t) is taken,

A(t) = −GMη

Dc2

(η

5GM/c3(tc − t)

)−1/4

, (3.7)

where M = m1 +m2 is the total mass, and η = m1m2/M2 is the symmet-

ric mass ratio. The phase of the gravitational waveform φ produced by a

coalescing compact binary system evolves at twice the rate of the instanta-

neous orbital phase. The orbital phase can be approximated via the post-

Newtonian expansion. To second post-Newtonian order the phase of the

inspiral gravitational waveform is [9]

φ(t) = −2

η

[Θ(t)5/8 +

(3715

8064+

55

96η

)Θ(t)3/8

−3π

4Θ(t)1/4

+

(9275495

14450688+

284875

258048η +

1855

2048η2

)Θ(t)1/8

].

(3.8)

Where we have used the convenient dimensionless time parameter Θ,

Θ(t) =

(c3η

5GM

)(tc − t). (3.9)

In addition to the component masses m1,m2, there are several unknown

parameters: the time of coalescence tc, the phase of the gravitational wave-

form at coalescence φc, distance from observer to source D, the inclination

angle of the binary’s orbital plane relative the line of sight ι, and the polar-

ization angle ψ. One can use the linear combination trigonometric identity

a cos θ + b sin θ =√a2 + b2 cos(θ − α) (3.10)


where tanα = b/a to re-express the strain (3.4) as

h(t) =D

Deff

A(t) cos(φ(t)− φ0) (3.11)

=(1 Mpc)

Deff

[hc(t) cosφ0 + hs(t) sinφ0] , (3.12)

where the factor Deff obtained from the trigonometric identity (3.10) is

Deff =D√

F 2+ (1 + cos2 ι)2 /4 + F 2

× (cos ι)2(3.13)

and φ0, an unknown phase term is

φ0 = arctanF× (2 cos ι)

F+ (1 + cos2 ι)− φc. (3.14)

Here we define hc and hs as the “cosine” and “sine” components of the gravi-

tational waveform. They are equivalent to the polarizations of a gravitational

waveform that would be produced by an optimally orientated inspiralling bi-

nary. An optimal orientated binary system is one that has its orbital plane

perpendicular to and has its orbit centered on the detector’s z-axis (i.e. where

the unknown phase term would be φ0 = 0 and φ0 = π/2 respectively). Al-

though not strictly necessary, the usual convention is to scale the cosine and

sine components to an optimally orientated template at a distance of 1 Mpc,

which we have done by setting D in equation (3.12). Hence the cosine and

sine components are defined as

hc(t) = A1 Mpc(t) cosφ(t), (3.15)

hs(t) = A1 Mpc(t) sinφ(t) (3.16)

where A1 Mpc is simply (3.7) with D = 1 Mpc.

3.2. METHODOLOGY 65

3.2.2 Two-Phase Matched Filter

The matched filter is the optimal linear filter for detecting known signals in

noisy data [122]. In this paper, we will follow the derivation of the matched

filter as it appears in section III of [132], which itself is based on classical

signal analysis methods. We define the output of the matched filter as a

correlation of the detector data s and the filter Q, weighted by the noise-

spectral density Sn(|f |)

z = 2

∫ ∞−∞

s(f)Q∗(f)

Sn(|f |) df. (3.17)

Note that z may be a complex value depending on the choice of Q. In the

case that the detector data contains Gaussian noise n(t) only, the expectation

value of the matched filter output z is zero. In such a case, the variance of

the output of the matched filter z is

σ2Q = 〈z · z∗〉 = 2

∫ ∞−∞

∣∣∣Q(f)∣∣∣2

Sn(|f |) df. (3.18)

Let us assume the gravitational waveform is present in the detector data,

ending at some time tc not known a priori. There is also a constant phase

term φ0 that is not known ahead of time. A common way [41, 127, 132] to

search for the unknown time of coalescence tc and phase term φ0 is to filter

both components hc and hs (which correspond to orthogonal phases φ0 = 0

and φ0 = π/2) separately and then combine them to form a complex signal.

This can be done by using the complex filter

Q∗(f) =[h∗c(f) + ih∗s(f)

]e2πiftc , (3.19)

where we now explicitly use h(f) to represent the Fourier transform of h(t)

when tc = 0. This convention shall be used throughout the remainder of the

paper. This way, the unknown time of coalescence tc can now be searched

over as an extrinsic parameter by changing the value of tc in the complex


matched filter

z(tc) = 2

∫ ∞−∞

s(f)h∗c(f)

Sn(|f |) e2πiftcdf

+i2

∫ ∞−∞

s(f)h∗s(f)

Sn(|f |) e2πiftcdf.

(3.20)

This is a cross correlation of the components hc and hs with detector data

s, weighted by inverse noise spectral density. The real part corresponds to

the matched filter output for the (real) template hc, and the imaginary part

corresponds to the matched filter output for the (real) template hs. In the

stationary phase approximation [133] the components hc and hs are exactly

orthogonal. It then follows that hc(f) = −ihs(f) for f > 0. When this

property is applied to equation (3.20), we have the form of the two-phase

matched filter as

z(tc) = 4

∫ ∞0

s(f)h∗c(f)

Sn(|f |) e2πiftcdf, (3.21)

commonly found in inspiral search papers [41, 127]. In this paper we prefer

to maintain the form of the two-phase filter in equation (3.20). Following

convention (c.f. [124]), the amplitude signal to noise ratio (SNR) ρ is de-

fined as the absolute value of the two-phase matched filter z divided by the

standard deviation of its real part:

ρ(t) =|z(t)|σ

(3.22)

where

σ2 = 2

∫ ∞−∞

∣∣∣hc(f)∣∣∣2

Sn(|f |) df. (3.23)

In the absence of a waveform, the SNR squared ρ2 is Chi-square distributed

with two degrees of freedom (one for each of the components). Hence the

probability of finding an SNR value greater than ρ∗ in the absence of a

3.2. METHODOLOGY 67

waveform is [125]

P (ρ2 > ρ2∗) = e−ρ

2∗/2. (3.24)

3.2.3 Discrete Time Domain Filtering

The two-phase matched filter (3.20) is a cross correlation of each component

hc,s(t) and the detector output s(t), weighted by the inverse noise spectral

density Sn(f). By defining the quantity x as the over -whitened strain data,

x(t) =

∫ ∞−∞

s(f)

Sn(|f |)e2πiftdf, (3.25)

we can use the cross-correlation theorem to define the two-phase matched

filter (3.20) in the time domain:

z(t) = 2

∫ t

−∞x(t′)hc(t

′ − t)dt′ + i2

∫ t

−∞x(t′)hs(t

′ − t)dt′ (3.26)

= 2

∫ t

−∞x(t′)h(t′ − t)dt′ (3.27)

where we have redefined hc(t) and hs(t) to represent the cosine and sine

components (3.15) and (3.16) when tc = 0. For simplicity of later equations,

we define h(t) = hc(t) + ihs(t) = A1 Mpc(t)eiφ(t) where the amplitude and

phase terms have tc = 0.

In practice the detector data s(t) is sampled at intervals of ∆t. The

discretized form of the continuous time domain matched filter (3.27) is

zk = 2k∑

j=−∞xjhj−k∆t (3.28)

where the index k denotes discretely sampled times of t, namely tk = k∆t.

In practice, the inspiral waveform template hi is bounded (because the de-

tector is only sensitive over a bandwidth) and the summation becomes finite,

making this a finite impulse response (FIR) filter.


3.2.4 Infinite Impulse Response Filter

Now let us introduce an alternative digital filter, the infinite impulse response

(IIR) filter. The difference equation of a general IIR filter is,

yk =N∑n=1

anyk−n +M∑m=0

bmxk−m (3.29)

where yk is the filter output at time step k (tk = k∆t), xk is the filter input,

and a’s and b’s are complex coefficients.

Examples of IIR filters in common usage are Chebyshev, Butterworth and

elliptic filters [134, 135]. IIR filters use much less computational resources

than an equivalent FIR filter. This is because they have “memory” — the

previous outputs are fed back into the filter. However digital IIR filter design

is a more complex process than FIR design. Obtaining the coefficients is

usually done by first constructing an equivalent analog filter and applying

well-known methods, such as the bi-linear transform or impulse invariance.

Multiple IIR filters used together have different forms, such as direct form

I & II, cascade (series) and parallel. In a series configuration, the overall

transfer function is the multiplication of each IIR filter transfer function.

In a parallel bank of IIR filters, where the output is summed together, the

overall transfer function is the summation of the different transfer functions.

First, let’s analyze the simplest single-pole IIR filter. The difference equa-

tion of this filter is

yk = a1yk−1 + b0xk. (3.30)

A solution to this first-order linear inhomogeneous difference equation is

yk =k∑

j=−∞xjb0a

k−j1 . (3.31)

By defining the complex coefficient a1 in the form

a1 = e−(γ+iω)∆t (3.32)

3.2. METHODOLOGY 69

+

×

×

Figure 3.2: A signal processing schematic showing the flow of data througha digital single-pole IIR filter. The input, xk is multiplied by a complexconstant b0, then added to the previous output that has been multiplied byanother complex constant a1, resulting in the current output yk. It shouldbe noted that this filter, in principle, should be have been run forever.

and comparing (3.28) and (3.31), it is easy to see that the output of the

simple filter (3.30) is the cross-correlation of xk and a complex sinusoid un

with frequency ω and a magnitude that increases with an exponent factor γ

for n < 0:

un = b0e(γ+iω)n∆tΘ(−n) (3.33)

where Θ(−n) is the Heaviside function.

3.2.5 Approximation to an inspiral waveform

Since φ(t) is not linear in time, a complex sinusoid (3.33) cannot approximate

the hc,s components of the inspiral waveform h(t) = A1 Mpc(t)eiφ(t). However

we can easily linearize the components by a first-order Taylor expansion

about the time t∗l :

A1 Mpc(t)eiφ(t) ' A1 Mpc(t

∗l )e

iφ(t∗l )+iφ(t∗l )(t−t∗l ); (3.34)

since the amplitude A1 Mpc(t) does not increase at the same rate as φ(t),

only a linear expansion of φ(t) is required. Multiplying by the window func-

tion eγl(t−tl)Θ(tl − t) makes this approximation an exponentially increasing


constant frequency complex sinusoid with cutoff time tl:

ul(t) = A1 Mpc(t∗l )e

i(φ(t∗l )+φ(t∗l )(tl−t∗l ))

× e(γl+iφ(t∗l ))(t−tl)Θ(tl − t). (3.35)

The expansion point t∗l is chosen to be near the cutoff time t∗l = tl − αTl,

where α is a tunable parameter and the interval Tl is the duration in which

the approximation is valid:

|12φ(tl)T

2l | = ε < 1 (3.36)

with ε a tunable parameter chosen to be to small. Equation (3.35) implies

that the coefficient b0 for the lth complex sinusoid is

b0,l = A1 Mpc(t∗l )e

i(φ(t∗l )+φ(t∗l )(tl−t∗l )) (3.37)

and the frequency ωl = φ(t∗l ).

In this paper, we chose the cutoff time tl of the first sinusoid to correspond

to the time at which the waveform has the highest frequency detectable by

the LIGO detector band. The next sinusoid is chosen by moving to an earlier

time, tl+1 = tl − Tl. Since we want the lth sinusoid to be mostly present on

the interval tl − Tl < t < tl, we choose the damping factor to be γl = β/Tl,

where β is a tunable parameter. This procedure is repeated until the time tl

corresponds to a time in the waveform that has frequency below the LIGO

detector band. Hence the number of sinusoids is dependent on the value

of ε, the rate of frequency change φ(t), which is dependent on the masses

of the system, and the detector bandwidth. For more information on this

procedure, see [4].

We can now approximate the components h(t) = A1 Mpc(t)eiφ(t) by an ad-

3.2. METHODOLOGY 71

dition of a series of damped sinusoids u(t) with cutoff times tl:

A1 Mpc(t)eiφ(t) ' U(t) =

∑l

ul(t)

=∑l

b0,le(γl+iωl)(t−tl)Θ(tl − t). (3.38)

Figure 3.3 shows an illustration of how damped constant-frequency sinusoids

can add to give an inspiral like waveform.

(e)

(d)

...

(c)

...

(b)

(a)

Figure 3.3: An illustrative diagram demonstrating the ability to linearlysum exponentially increasing constant-frequency sinusoids to approximatean inspiral like waveform. The top three panels (a-c) show three examplesinusoids with different damping, frequency and cutoff time factors. Panel(d) shows the linear addition of all the sinusoids (at different scales). Panel(e) shows the exact inspiral-like waveform. Note that this figure is only forillustrative purposes.


3.2.6 Summed Parallel IIR filtering

Each complex sinusoid ul(t) in equation (3.38) can be searched for in the data

x using the single pole IIR filter (3.30). Here the cutoff time is incorporated

by running each filter on a delay, dl = tl/∆t. The output of the lth filter at

time k is

yk,l = a1,lyk−1,l + b0,lxk−dl . (3.39)

The linear summation of the output of all filters is the cross-correlation of

the data x and the approximate waveform U(t) in (3.38):

zk ' 2∆t∑l

yk,l. (3.40)

Here z is equivalent to the value computed by the discrete time domain two

phase filter (3.28) when using a template h(t) = U(t). From equation (3.22),

it follows that the absolute value of the summation (3.40) divided by σU is

the SNR, which we term the output of the Summed Parallel Infinite Impulse

Response (SPIIR). The normalization factor σU is defined as

σ2U = 4

∫ ∞0

∣∣∣Uc(f)∣∣∣2

Sn(|f |) df, (3.41)

where Uc(f) is the Fourier transform of the real part of U(t), which approx-

imates hc(t). The similarity of the SPIIR output and the matched filter

output will depend on how well U(t) approximates the given template.

3.3 Implementation for Performance Testing

3.3.1 IIR bank construction

To confirm the ability of the SPIIR method to recover a good SNR, it is first

necessary to show that the approximate inspiral waveform (3.38) is a good

“match” to the theoretical inspiral waveform (3.12). We define the overlap

3.3. IMPLEMENTATION FOR PERFORMANCE TESTING 73

∆ as the inner product of the normalized approximate waveform U and the

template h:

∆ =1

σ · σU

√√√√(2

∫ ∞−∞

hc(f)Uc∗(f)

Sn(|f |) df

)2

+

(2

∫ ∞−∞

hs(f)Us∗(f)

Sn(|f |) df

)2

(3.42)

where Us(t) approximates hs(t). We initially approximate a canonical 2PN

1.4-1.4 M inspiral waveform band-limited to 10-1500 Hz using the value of

the tunable parameters ε, α and β to be consistent with the high overlap

results of [4]. With some minor variation of their values, we aim to recover

the highest overlap possible. Once a good choice of α and β is found for the

2PN 1.4-1.4 M template, we use the same values for other templates, but

vary the value ε (and consequently the number of IIR filters in each bank)

to see the effect on overlap.

3.3.2 Detector Data Simulation

To test the detection efficiency of the SPIIR method compared to the fre-

quency domain matched filter, we will filter two mock signals, one for which

the input data is just LIGO-like noise, and the other with the same noise

plus an inspiral waveform injection scaled to represent a source at a chosen

effective distance Deff .

For this test, we need to construct a finite segment of detector data to

filter. Being infinite impulse response filters, in principle the filters should be

run for an infinite length of the input data before the output has stablized.

In order to approximate this behavior, we need to run the IIR bank for a

finite “warm-up” period before the output is consistent with that of an IIR

filter that has been running for an infinite amount of time. We choose to

run each filter for 2 e-foldings of time before we accept the output as being

identical to one which has run for an infinite amount of time. Additionally,

since each IIR filter in the bank runs on a delay, the summed output of all

the IIR filters will not be produced until after the longest delay time (dmax)

has passed. The filter that has the longest delay (dmax) is also the one that


has the longest decay rate γmax. In total, the input data must be at least

dmax + 2γ−1max in length before any output is produced. Hence the length of

the input data is

Ninput = dmax + 2γ−1max +Nanalysis (3.43)

where Nanalysis is the length of analysis period, which we choose to be 4

seconds. Hence the 4 s SPIIR output will tell us whether there is an injection

that ended somewhere within those 4 seconds. At a sample rate of 4096 Hz,

the analysis period is Nanalysis = 16834 data points long. In our simulation,

we find dmax = 4081683 and 2γ−1max = 149432, resulting in Ninput = 4247499.

Noise generation

The LIGO-like noise data is produced by creating a normally distributed

white noise time series of length Ninput, then colouring it by the theoretical

advanced LIGO noise spectrum Sn(f) (3.46). We then over-whiten this time

series using equation (3.25) to produce the waveform-free noise input data x:

xnoise(t) = now(t). (3.44)

Waveform injection

We create our waveform injections by first producing an inspiral waveform

band-limited between 10 and 1500 Hz. The injection is padded with zeros so

that it has the length Ninput. The end of the waveform is chosen so that it

finishes somewhere after dm + 2γ−1m data points. The injection signal is then

over whitened using equation (3.25). The over-whitened injection can then

be placed in the over-whitened noise signal:

xnoise+injection(t) = xnoise(t) + how(t). (3.45)

Matched filter comparison

As a comparison, we will also perform a frequency domain correlation matched

filter. For this process, since the input data is already over-whitened, it only

3.3. IMPLEMENTATION FOR PERFORMANCE TESTING 75

needs to be cross-correlated with the waveform. Section 3.2.2 outlines how

this is done. The cosine component hc(t) gets pre-padded with enough zeros

to get to length Ninput. This ensures that hc(f) has the same spectral res-

olution as s(f). The matched filter (3.21) produces a time series of Ninput

length. However the first Ninput − Nanalysis data points are erroneous wrap-

around caused by the FFT. Only the interval [Ninput −Nanalysis + 1, Nanalysis]

is used to determine if a waveform is present.

3.3.3 Detection Efficiency

To test the detection efficiency of the SPIIR method compared to the tra-

ditional matched filter method we will construct several receiver operating

characteristic (ROC) curves for 2PN 1.4-1.4 M waveforms injected for dif-

ferent effective distances Deff . To create each ROC curve, we first find the

false alarm rate. The false alarm rate is found by realizing an Ninput length

LIGO-like noise time series, filtering this input data, and analyzing the out-

put of the 4 s analysis period (the SNR). We will count this realization as a

false positive if at any point within the 4 seconds the SNR goes over a given

SNR threshold. Several thresholds will be chosen, giving the false positive as

a function of threshold. After > 106 noise realizations, the false alarm rate is

simply the ratio of total number of false positives to number of noise realiza-

tions. Likewise, to see if the IIR filter doesn’t miss too many true positives,

we inject a 2PN 1.4-1.4 M waveform using the prescribed method in 3.3.2

for a given Deff into LIGO-like noise. After filtering, if at any point within

the analysis period the SNR is above a given threshold, this realization is

counted as a true positive. Again, after > 106 noise realizations, we calculate

the detection rate as a ratio of the total number of true positives to number

of realizations. The plot of false alarm rate versus detection rate gives the

ROC curve.


3.4 Results

3.4.1 Inspiral Waveform Overlap

Starting with the canonical 1.4-1.4 M second order post-Newtonian binary

waveform band limited to be between 10 and 1500 Hz we found, using the

parameters ε = 0.04, α = 0.99, β = 0.25 in the procedure outlined in Section

3.2.5, that we can recover an overlap of 99% using 687 IIR filters.

We find that increasing the value of ε will in general increase the overlap,

as the frequency space is more finely sampled. However there seems to be

a limit, as the damping factor γ causes the adjacent IIR filters to run into

each other.

With this choice of α and β we are able to recover a high overlap for

different mass pairs as well. Figure 3.4 shows the overlap as a function of

number of IIR filters for six different mass pairs.

200 400 600 800 1000 1200 1400

0.976

0.978

0.98

0.982

0.984

0.986

0.988

0.99

0.992

0.994

Number of sinusoids per waveform

Over

lap

1.4+1.4 M⊙1.0+1.0 M⊙1.0+3.0 M⊙2.0+2.0 M⊙2.0+3.0 M⊙3.0+3.0 M⊙

Figure 3.4: The overlap between the exact inspiral waveform and the ap-proximate inspiral waveform as a function of number of damped sinusoids.In general the greater the number of sinusoids per waveform, the greater theoverlap. However the choice of γls greatly affects the overlap.

3.4. RESULTS 77SN

R

t − τc (ms)−15 −10 −5 0 5 10 15

0

1

2

3

4

5

6

7

8

9

10

IIR filter output

Matched filter output

−1 0 17.5

8

8.5

Figure 3.5: The SNR output of both the SPIIR method and a traditionalmatched filter method. The plot is centered on t − τc where τc is the timeat which the injection ends. From the two curves, it is clear that the SPIIRmethod can return a very similar SNR to that from the optimal filter. Theinset shows a close-up of the time around t = τc to show how similar the twomethods are.

3.4.2 Ability to Recover SNR

Figure 3.5 shows the SNR produced from both the matched filter technique

and the SPIIR method. The input time series is constructed following Section

3.3.2. The injection of a 2PN 1.4-1.4 M waveform scaled for an effective

distance of 500 Mpc is added to LIGO-like noise. The x-axis of the plot is

centered about the end of the injection (t = τc), which is directly in the

middle of the analysis period. Around this time, the SNR peaks to 8, which

is near the expected value of 7.9 for an injection at this distance. This plot

shows that the SPIIR method is capable of recovering a very similar SNR to

the matched filter at all times.


3.4.3 Detection Efficiency

We analyzed over 106 independent noise realizations, for which the waveform

had been injected at Deff of 500, 600, 700, 800 Mpc. We performed both

IIR filtering and traditional matched filtering. Figure 3.6 shows that the

SPIIR method recovers most of the same events as the traditional matched

filter method. At false alarm rates of greater than 10−5, the SPIIR method

recovers greater than 99% of the injections recovered by the matched filter

when searching for injections at an effective distance of 500 Mpc (SNR∼8).

Even in the worst case, at a false alarm rate of 10−6, the SPIIR method

catches 4.5% of injections scaled at an extreme 800 Mpc (SNR∼5), whereas

the matched filter catches 5% of injections at this scale.

3.4. RESULTS 79

False Alarm Rate

De

tectio

n R

ate

SNR ~8 (500 Mpc)

SNR ~6.6 (600 M

pc)

SNR ~

5.7(

700

Mpc

)

SNR

~5

(800

Mpc

)

10−6

10−5

10−4

10−3

10−2

10−1

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SPIIR Method

Matched Filter

Figure 3.6: The receiver operating characteristics (ROC) of both the IIR filtermethod and the traditional matched filter method. The x-axis shows the falsealarm rate, and the y-axis the detection rate. A one-to one relationship,which is the worst case scenario, is shown by the boundary of the shadedarea. We show four different ROC curves, where each curve represents thedetection rate as a function of false alarm rate for waveforms injected ateffective distances of 500, 600, 700 and 800 Mpc (SNR ∼ 8, 6.6, 5.7 and 5respectively).


3.5 Summary and Discussion

We have demonstrated that the through the use of a parallel bank of sin-

gle pole IIR filters, it is possible to approximate the SNR derived from the

matched filter with greater than 99% overlap. The main advantage of our

SPIIR method is that it operates completely in the time domain, and in

principle it has zero latency (not taking into account whitening or computa-

tional time). The SPIIR method recovers most of the injections the optimal

matched filter recovers.

The use of a bank of simple IIR filters for each template as opposed to

the matched filter method enables us get two extra processes for minimal

additional cost. The first is that the individual IIR filter outputs can be

arranged into groups, such that their total summed output is roughly in-

dependent and orthogonal to each other. This enables, with minimal extra

overhead, the calculation of a χ2 distributed statistic, giving a secondary

method of verification. We will demonstrate this in an upcoming paper. The

second natural advantage of using a parallel bank of single-pole IIR filters

is that they can easily be executed in parallel using multi-threaded proces-

sors, such as graphics processing units (GPUs). Indeed, a side study has

shown that this is possible [136]. This leads to the future possibility that a

single personal computer may be able to process the detection of GWs from

inspiralling compact binaries.

A further way to reduce the computation of the IIR calculation is to split

the incoming data into differently down-sampled channels. The output of

each IIR filter in the bank is the correlation of a fixed frequency sinusoid and

the incoming data. For the sinusoids that have frequencies <124 Hz, the in-

coming data need only be sampled at 256 Hz. The current pipeline of LLOID

uses a similar multi-channel down-sampling in their detection pipeline. Their

pipeline consists of the integration of the open-source real-time multimedia

handling software gstreamer and the LIGO Algorithm Library (LAL) [130].

This software library is an ideal platform to integrate the SPIIR method.

The total computation can also be further reduced by sharing IIR filters (via

interpolation) between different templates [4].

3.6. ACKNOWLEDGEMENTS 81

Although the design of the IIR filter so far only applies to chirping, post-

Newtonian approximation inspirals, we have performed preliminary tests us-

ing more complicated combinations of single-pole IIR filters to replicate the

waveform of an inspiral with spin. If the amplitude/frequency beating of a

spinning inspiral waveform can be simulated by the linear addition of two

different non-spinning inspirals with different masses, then it can be approx-

imated by a linear addition of damped sinusoids. In this case, the SPIIR

method can produce the SNR for the beating waveform. There is also the

possibility of using higher order IIR filters, although designing the coefficients

can be very difficult.

We foresee that the use of IIR filters for time domain filtering of Advanced

LIGO will be ideal, as the waveforms will be much longer. The frequency

domain matched filter will take more time to calculate GW triggers, essen-

tially ruling out the possibility of triggering the detection of prompt optical

emission related to neutron star mergers (GRBs). We have shown that the

use of a parallel bank of IIR filters requires less computational cost, with

minimal detection rate loss, and most importantly can be calculated in the

time-domain with near zero latency.

3.6 Acknowledgements

We would like to thank Kipp Cannon, Drew Keppel and Chad Hanna for de-

tailed discussion on the design and implementation of low-latency detection

algorithms. This work was done in part during the LIGO Visiting Student

Researcher program, which was partially funded by the 2009 UWA Research

Collaboration Award. This research was supported by the Australian Re-

search Council. SH gratefully acknowledges the support of an Australian

Postgraduate Award. LW acknowledges the support of the Australian Re-

search Council Discovery Grants and Future Fellow program.


3.7 Noise Spectral Density

The noise spectral density (in units of strain/√

Hz) we use is based on an

algebraic expression prediction of the Advanced LIGO noise curve given in

the LAL suite reference manual [137] defined by,

Sh(f) = S0

(f

f0

)−4.14

− 5

(f0

f

)2

+

111

1−(ff0

)2

+ 0.5(ff0

)4

1.+ 0.5(ff0

)2

;

(3.46)

where, f0 = 215Hz and S0 = 1049 strain/√

Hz.

Chapter 4

Multi-rate SPIIR method

The last chapter has shown that the SPIIR method can, in theory, recover

inspiral GW signals from Gaussian noise. In this short chapter, we will

discuss modifications to the method designed to improve performance when

using it as part of a real search pipeline for multiple inspiral templates. A key

aspect to this is the introduction of the multi-rate SPIIR method. Significant

computational gains can be made by employing this method. The theoretical

benefits of a multi-rate design were first discussed in the initial study of using

IIR filters [4].

Section 4.1 will introduce the notion of operating the SPIIR method at

multiple sample rates. The next section (4.2) will expand the use of the

multi-rate SPIIR method to accommodate filtering a template bank with

many templates. These modifications are important to discuss before moving

on to a full scale implementation of the SPIIR method in a realistic setting.

4.1 Multi-rate SPIIR filtering

The computational cost of the SPIIR method as it has been described is

proportional to the number of IIR filters NIIR per template, and the sample

rate fs at which the IIR filters operate. Each IIR filter (3.39) has 12 real

number floating operations (FLOP) per sample point. Hence to run the

SPIIR method in real-time (e.g. filter one second of data in one second of

83

84 CHAPTER 4. MULTI-RATE SPIIR METHOD

clock time) for data sampled at fs Hz requires

Cs = 12fsNIIR (4.1)

FLOPS (FLOP’s per second). For a given set of tunable parameters α, β and

ε listed in section 3.4.1 the total number of IIR filters for the 2PN 1.4−1.4M

canonical template is NIIR = 687. With a sample rate of 4096 Hz, the total

computational cost for the SPIIR method to search for this template in real-

time is ∼ 33.8 MFLOPS (1 MFLOPS = 106 FLOPS).

Given that an inspiral waveform has a broad frequency range (for Ad-

vanced LIGO from 10 Hz to fISCO), it is natural to assume that not all IIR

filters need to be filtering data sampled at the original sample rate of 4096 Hz.

It can be shown that the individual IIR filters can filter data sampled at only

twice of the IIR characteristic frequency f , which can be significantly lower

than 4096 Hz, without a significant loss of signal. This was first considered

in [4].

The factor at which the input strain data can be decimated can be de-

termined by each filter’s characteristic frequency f , the native sample rate

f0 = 1/∆t, and a small > 1 padding factor p. In practise, it is generally

easier to choose a decimation factor M (the ratio of native sample rate f0 to

new sample rate fs) to be a power of two,

M = 2−dlog2(2pf/f0)e = f0/fs. (4.2)

Hence the total computational cost of the multi-rate SPIIR method to search

for a template in real-time scales as

Ctot =∑s

12fsNIIR,s. (4.3)

Where the summation is over sub-groups s for each sample rate fs and

NIIR,s is the number of IIR filters in the s sub-group (NIIR =∑

sNIIR,s).

For the canonical template, with a padding factor of p = 1.1 and a min-

imum cutoff frequency of 10 Hz, we expect eight sub-group sample rates:

4.1. MULTI-RATE SPIIR FILTERING 85

32, 64, 128, 256, 512, 1024, 2048, 4096 Hz.

Before the input strain data can be decimated and filtered it must first

have a low-pass filter applied. The computational cost of the down-sampling

process is trivial and can be ignored, as it is only performed once for all

templates (as will be shown in the next section).

In order to run the IIR filter sub-groups s at the reduced rate fs = f0/M,

first the coefficients a1, b0 and delay d must be modified. As can be seen

from the definition of a1 in equation 3.32, the new feed-back coefficient a′1needs to be exponentiated by M. The new delay d′ is just the old delay

divided by M to the nearest integer. The new feed-forward coefficient b′0needs to have a phase shift applied because the new delay d′ is in a new

position, possibly up to M/2 original samples away from where the original

d was. Without correction, this could introduce a phase shift of up to πp, as

well as an amplitude change by eγ(M/2). The following equations define the

transformations to the new coefficients and delay;

a′1 = aM1 , (4.4)

b′0 = b0aMd′−d, (4.5)

d′ = d(d+ 1)/Me. (4.6)

Sub-groups of IIR filters can now filter data that has first been down-

sampled to a sample rate fs. The output from each IIR filter group can be

added together, up-sampled by a power of two and added to the next highest

sample rate. See figure 4.1 for a visual explanation of the data flow in the

multi-rate SPIIR method.

The cost of up-sampling the output of each filter sub-group scales as the

number of templates increases, and must be considered to show that the

reduction in computational cost of using the multi-rate scheme is beneficial.

For each successive up-sample, a sample point with zero value is inserted

between adjacent samples, and a low-pass filter with N↑ coefficients must

be applied. Hence the computational cost of each up-sampler scales with

the sample rate fs and the number of low-pass filter coefficients N↑. The

number of low-pass filter coefficients N↑ will depend on the type and quality


x zFilter sub-group 0 +

2048 HzFilter sub-group 1 +

Filter sub-group s

... ...

4096 Hz

+

...

Figure 4.1: A schematic of the multi-rate SPIIR pipeline. In this design, theincoming data x is down-sampled simultaneously to different sample ratesfs. Each sample rate stream is then filtering by using the SPIIR method.The output of the each SPIIR block is up-sampled by a power of two andadded to the next highest SPIIR block’s output.

of up-sampler used, but it will be typically be between 16 and 192.

Table 4.1 shows the division of IIR filter sub-groups for the canonical

1.4-1.4 M template with the α, β and ε values as used in section 3.4.1.

With multi-rate SPIIR filtering and up-sampling, the total cost to search for

a single template in real-time is between 1.633 and 3.057 MFLOPS, a saving

of 90-95%.

4.2. MULTIPLE TEMPLATES 87

Table 4.1: The relationship between the characteristic frequency f (in unitsof Hz) of a filter and the decimation factor M in order to run it at a samplerate f s is shown. On the right hand side the number of IIR filters pergroup (NIIR,s) is shown, along with real-time computational cost in units ofMFLOPS of the SPIIR method both with and without the multi-rate applied(CIIR and C∗IIR respectively). The scale value of the computational cost of theup-sampler is also shown.

M fs( Hz) f ( Hz) NIIR,s C∗IIR CIIR C↑ (×N↑)1 4096 1

p[1024, 2048) 0 0 0 4096

2 2048 1p[512, 1024) 13 0.6390 0.3195 2048

4 1024 1p[256, 512) 22 1.081 0.2703 1024

8 512 1p[128, 256) 41 2.015 0.2519 512

16 256 1p[64, 128) 72 3.539 0.2212 256

32 128 1p[32, 64) 127 6.242 0.1951 128

64 64 1p[16, 32) 225 11.06 0.1728 64

128 32 1p[8, 16) 187 9.191 0.0718 32

Total 687 33.77 1.503 8160

4.2 Multiple templates

So far we have described the SPIIR method for a single canonical 1.4−1.4M

template. As stated in section 2.5.3, a real inspiral search pipeline will need

to search for many templates over a large parameter space. The multi-rate

SPIIR method described in the previous section can be easily expanded to

search for multiple templates. The down-sampled detector strain can provide

the input for a number of IIR filter sub-groups. See figure 4.2.

For each template, a different set of IIR filters is needed for the SPIIR

method. Using the Taylor-expansion method prescribed in section 3.2.5, the

total number of IIR filter coefficients NIIR scales as,

NIIR ∝f−5/6min − f−5/6

max

ε1/2M5/6(4.7)

Hence each template will have a different number of IIR filters NIIR. For

convenience in the actual implementation of the SPIIR method, we have

chosen to organise the IIR coefficients as follows:


• Sort template bank by chirp mass

• For each template place IIR filters into S sub-groups using the method

prescribed in section 4.1

• For each template place the sub-group coefficients a′, b′ and d′ and

delays s as rows into the matrices As, Bs and Ds.

• Pad each row of each matrix with zeros so that all matrices are regular.

Hence each matrix will have N1IIR,s columns and M rows (one for each

template). For example a given A matrix is arranged as

A =

a1[1, 1] a1[1, 2] · · · a1[1, N1

IIR]

a1[2, 1] a1[2, 2] · · · a1[2, N2IIR] 0 · · · 0

... a1[3, 2]...

a1[M, 1] a1[4, 2] · · · a1[M,NMIIR] 0 · · · 0

, (4.8)

where there will be S As, Bs and Ds matrices (one for each sample rate).

Packing the matrices this way implies that there will be some computer

cycles wasted by using IIR filters with coefficients that are zero. However, in

practise, the number of templates per instance of the pipeline is limited (to

100 templates). As the templates are sorted by chirp mass, there will not

be a significant number of zeros in any given row and the matrix will be far

from sparse.

4.2. MULTIPLE TEMPLATES 89

Filter sub-group 0 +




32 Hz

Filter sub-group s +

Filter sub-group s +

...

...

... ...

......

...

...

...

...

...

z

z

2

M

1

2048 HzFilter sub-group 1 +

Filter sub-group s

... ...

+

...

x z+4096 HzFilter sub-group 0

1

2

M

1

2

M

1

2

M

Figure 4.2: A schematic of the multi-rate SPIIR pipeline with many tem-plates. This is an extension to the multi-rate SPIIR method discussed insection 4.1. The advantage of this implementation is that there is only oneset of down-samplers for all templates.


4.3 Discussion

In this chapter, we have introduced two modifications to the SPIIR method.

We have shown there are significant computational cost savings (up to 90-

95% per template) to be made simply by operating the IIR filters at reduced

sample rates. The extension of the “multi-rate” SPIIR method to search

for multiple templates has also been shown. Of specific interest for later

chapters is the introduction of the As, Bs and Ds matrices. With these

concepts introduced, the following chapters can focus on the application of

the SPIIR method in a realistic inspiral search environment.

Chapter 5

Offline SPIIR pipeline

This chapter will describe the implementation of the SPIIR design in the

form of an executable computer application available to the greater GW

community, and its use in a full scale inspiral search pipeline. The pipeline

includes template bank construction, interferometric data collection, filter-

ing the data into a series of GW triggers, trigger coincidence tests between

detectors, vetoes based on instrument characterisation, and the calculation

of false alarm rates for coincident triggers into candidate events.

As a preliminary test of the new implementation, a search for low-mass

NS-NS inspiral signals in existing initial LIGO detector data was performed

(approximately two weeks of data from LIGO’s fifth science run — S5). Re-

sults from other inspiral searches from these science run data were published

in [138]. The aim of using these data was not to find any signals missed by

other searches, but rather to test the SPIIR method on real detector data.

This was a necessary step to confirm that the SPIIR method behaves as

expected in real detector data. Both the SPIIR method and the matched

filter method are optimised to maximise the SNR of a signal present in sta-

tionary Gaussian noise. However it is well known that to date GW detector

data are non-stationary and non-Gaussian [139]. Although non-stationarity

occurs on many timescales, we are mostly concerned with how the SPIIR

method will respond to short durations of non-stationarity, generally called

glitches. Glitches can cause the SNR to peak simply due to noise transients.

91

92 CHAPTER 5. OFFLINE SPIIR PIPELINE

To compare how the SPIIR method behaves in non-Gaussian data, we also

performed an identical analysis in Gaussian noise generated with the initial

LIGO design sensitivity.

Although the SPIIR method was designed for low-latency, and therefore is

best used as part of an online search pipeline, in this chapter we demonstrate

its ability to search for inspiral signals in an offline pipeline. Since we are

demonstrating its ability on initial LIGO data, the templates searched for

began at 40 Hz, and not the Advanced LIGO 10 Hz which the SPIIR method

is computationally optimised for. Since we were operating in an offline mode,

low latency was not considered.

The main reason to run in an offline configuration was to show the

pipeline’s detection efficiency and sensitivity. The detection efficiency of the

pipeline is a measure of how many signals the pipeline is expected to recover

as a function of distance. The volume of space that the pipeline is sensitive

to can be measured by the efficiency-weighted volume integral. We were able

to measure this by injecting many simulated inspiral signals into the data

with randomly chosen parameters drawn from a uniform distribution. If it is

assumed that the noise in the detector is Gaussian, the expected SNR of an

injected signal is a function of known intrinsic and extrinsic parameters, and

inversely proportional to the effective distance (which can be measured). A

theoretical Gaussian-noise detection efficiency can then be obtained indepen-

dently of our SPIIR pipeline. This provides an ideal method of characterising

the detection efficiency for comparing our pipeline with other low-mass in-

spiral pipelines, which may have vastly different search conditions (such as

duration of search, number of injections, injection parameter space, etc).

By recovering the injected signals, we were able to test how well the SPIIR

method can estimate the measured parameters of the signals. Although

parameter estimation can be done better using other offline inspiral search

pipelines, it was particularly important to show that the time of arrival of the

signal in each detector was measured accurately. Since the SPIIR pipeline

was designed to provide triggers for EM follow-up, accurate arrival time

information is required for sky localisation.

The first section of this chapter will give a description of the offline SPIIR

5.1. THE SPIIR APPLICATION 93

application itself, and how it produces a database of GW triggers. Changes

made to this pipeline to make it an online pipeline will be discussed in chap-

ter 6. Section 5.2 will show how we divided up roughly two weeks of S5

data for analysis. Section 5.3 describes how the IIR template banks were

generated for the pipeline. To test the detection efficiency of the pipeline,

we performed the analysis both with and without injecting simulated inspiral

signals. Section 5.4 describes the parameters of the injected signals. We com-

pare the output of the SPIIR method in Gaussian noise to real non-Gaussian

S5 data in Section 5.5. In order to make confident detections, we introduce

the ranking statistic, false alarm rate, in Section 5.6. Section 5.9 shows how

well the pipeline recovered the injected signals, and Section 5.10 shows how

well the injected signal parameters (including the timing accuracy) were re-

covered. Finally, we discuss how the pipeline performed overall, and what

its limitations are.

5.1 The SPIIR application

The SPIIR method has been realised through an executable computer ap-

plication called as gstlal iir inspiral. This application1 is intercon-

nected with the greater GW analysis software library, LIGO Algorithm Li-

brary (LAL)[140]. The gstlal iir inspiral application resides within the

gstlal project [130], which wraps, amongst other things, components of

the LAL library into GStreamer elements. GStreamer [141] is a multimedia

framework consisting of a suite of pre-written signal processing tools. It pro-

vides a framework for programmers to build custom media processing appli-

cations. Typical examples include applications that can record and playback

audio/video sources from either disk or streaming sources. Programmers can

build a network of interconnected media-handling components, called ele-

ments, that apply transformations to packets of flowing data. GStreamer’s

plug-in environment allows programmers to build their own elements to pro-

cess data packets in a variety of ways. This framework allows GW data

1written by my collaborators and myself


analysts to concentrate on writing GW-analysis specific elements whilst hav-

ing access to common digital signal processing tools (such as re-samplers

and amplifiers) and without having to be concerned with the management of

real-time data processing. For instructions on how to install gstlal please

see the project page [130].

The gstlal iir inspiral application searches for templates in multiple

detector data streams over a fixed period of time. The end result is a database

of triggers and related information. At this stage, the triggers have not been

ranked by significance. That comes at a later stage of the overall offline

SPIIR pipeline (discussed in section 5.7).

5.1.1 Internal structure of gstlal iir inspiral

The gstlal iir inspiral application consists of a collection of intercon-

nected GStreamer elements that operate on discrete packets of data known

as buffers. Each buffer contains a finite duration of discretely sampled time

series data, with particular attributes, e.g. timestamp, duration, sample rate,

data format, etc. Each buffer can also have a number of parallel channels.

Buffers are passed between elements, where each element is responsible

for some transformation of the input buffer into an output buffer. Figure 5.1

schematically describes the flow of data from raw gravitational frame files

(the strain data), through several processing stages, ultimately producing a

database of triggers.

At execution, the gstlal iir inspiral application is provided with a

start and end time with which to search for inspiral signals. The first stage

of gstlal iir inspiral is to read interferometric strain data from disk, and

package it into buffers. Within each instance of gstlal iir inspiral there

is one source of data for each detector i. The GW strain data is read from

gravitational wave frame (gwf) files stored on disk. The gstlal iir inspiral

application is provided with a path to the gwf files and a list of time segments

that correspond to “science mode”. Science mode refers to segments of time

when each detector was in lock (i.e the laser in the interferometer was res-

onant in the arm cavity), there was no other experimental work performed


Template M

yes no

Start time, end time,Science segments list.

Path to frame files

Flag as gap buffer

Veto segments list

Injection file

Apply SPIIR methodfor each template

...

Does SNR goabove threshold?

IIR templatebank file(s)

Calculate statistic

Detector 1

Template 1

yes

Whiten data

Store trigger information in database

*

Initial PSD

Create gap buffer

Inject simulation

Is thisanalysis segment in

science mode?

Injectsimulation?

yes no

Should this buffer be vetoed?

yes no

Create buffer

Is trigger coincident between

detectors?yes no

Mark as coincident Mark non-coincident


Calculate statistic

Detector n

yes

...

...

...

...

...

yes no

Flag as gap buffer


Whiten data

*

Create gap buffer

Inject simulation

Is thisanalysis segment in

science mode?

Injectsimulation?

yes no

Should this buffer be vetoed?

yes no

Create buffer

...

Figure 5.1: The flow of data through gstlal iir inspiral. Given a startand end time, data follows synchronously through a series of stages, explainedin the text. After conditioning the data for each detector (up to the ∗), theSPIIR method as described in Section 4.1 and schematically shown in figure4.2 is applied to the data. At least one template bank file is required; howeverit is possible to have multiple template bank files that branch from the ∗ stage(not shown here due to complexity). At the application termination, a singledatabase containing trigger information and coincidence tables is written todisk.


at the site, and the segment quality was confirmed by a human monitor. If

these conditions are met the strain data from the gwf files are packaged into

discrete buffers. If these conditions are not met, buffers are still created, but

have no data and are flagged as “gap” buffers. A gap buffer is a regular

buffer with a timestamp and a duration which has been flagged as having no

valid data. In this way, there is a continuous flow of buffers, regardless of the

status of the detectors.

At this point simulated inspiral signals can be injected into the buffers if

a list of injection parameters is provided. If no list is provided, no injections

will be made. The specific set of injections for this study will be discussed

in Section 5.4.

The next stage whitens the spectral properties of the data in the non-gap

buffers. The whitener element is provided with an initial PSD frequency

series which is an estimate of the average noise spectral properties of the

detector. Every 4 seconds, the Fourier transform of the last 8 seconds of

data is taken, and divided by the square root of the stored averaged PSD (the

ASD). The inverse Fourier transform of the resulting time series should have

approximately equal power in all frequency bins (i.e. white), and is scaled

so that the time series has (on average) mean zero, unit variance. Hence

the whitener element pushes out a single channel buffer every 4 seconds.

The stored PSD is a running estimate of the arithmetic mean square of each

frequency bin, which is updated up new incoming data.

Before searching for inspiral signals, a final veto stage is performed. A

veto segment list is provided to gstlal iir inspiral. A veto segment is

any time that was flagged as being not worthy of analysis for some other

reason. In our analysis, we only veto times of hardware injected signals. If

the buffer’s timestamp occurs within a veto segment, it is flagged as a gap

buffer.

The next stage is where the multi-rate SPIIR method as described in

Section 4.2 is applied to the single channel buffer containing whitened strain

data (with approximately zero mean, unit variance). In order to form a

continuous stream, we have taken the position of filtering gap buffers, but

only after setting the data within them to zero for all samples. The multi-rate


design as shown in figure 4.1 is applied to this buffer, ultimately producing

an output buffer for each detector with 2 × M channels, where M is the

number of templates searched for. There is one channel each for the real and

imaginary part of the complex SNR time series z/σ, for each template.

A peak finding algorithm is then applied to the complex time series from

each template in each detector. The triggering element finds the highest

value of the absolute SNR (= |z/σ|) for each template over a prescribed

time window. If this peak is over a given threshold, a consistency check

is performed to confirm that the shape of the SNR time series around the

peak conforms to one that would be expected if the SNR peak was caused

by a true signal. This has the same purpose as the χ2 consistency check

performed in other inspiral search pipelines based on matched filters [132] –

to distinguish SNR peaks due to true signals from those due to non-stationary

noise transients (glitches). However our method of calculating a χ2 statistic

differs greatly. In the presence of Gaussian noise, the expected SNR time

series is simply the auto-correlation of the whitened inspiral signal. Thus

if an inspiral signal is truly responsible for the SNR peak, it should look

something like the auto-correlation time series. Summing the squares of the

residuals of the measured and expected time series provides a χ2-like statistic

that acts as an indicator of the goodness of fit. Although in this study we call

that statistic the χ2 value, it is not strictly χ2 distributed in the presence of a

true signal or noise. However, as will be shown in Section 5.5, the χ2 statistic

we calculate can help to distinguish SNR peaks caused by true signals and

those caused by glitches, since the latter will not significantly match the

auto-correlation of the whitened inspiral signal. In this study, we chose to

use a time window of 4 seconds for the peak finding and SNR threshold of

4. Hence the triggering element passes buffers that at most have one trigger

per 4 seconds for each template. The intrinsic parameters relating to the

template responsible for the trigger, as well as the measured Deff and time of

arrival, are also recorded in the buffer. This defines a single detector trigger.

The final stage of gstlal iir inspiral is to perform a coincidence test

between detectors. A trigger is considered coincident between detectors if

it occurs within roughly the same time (considering the light-travel time


between detectors), for exactly the same template. The trigger information

and a coincidence table is stored in memory and written to disk in the form

of a standard database when the application finishes.

5.2 Data for offline run

In order to test the gstlal iir inspiral application, we chose to analyse

data from approximately two weeks of LIGO’s S5 data from 28 July, 2007

01:46:35 UTC to 11 August, 2007 01:45:35 UTC. This time period was chosen

as the detectors were operating normally, and is a typical reflection of S5

data. Because we are only interested in triggers coincident between non-co-

located detectors, we only analysed data from detectors Livingston 1 (L1),

and Hanford 1 (H1).

Since we are testing the pipeline in an offline mode we can achieve faster

than real-time analysis (i.e. analyse approximately two weeks of data in less

than two weeks) by running many instances gstlal iir inspiral in parallel

across machines, each analysing short segments of time. Because coincident

triggers could only occur during times when both detectors were in science

mode, only the times when both H1 and L1 detectors were in science mode

were analysed. We chose to analyse segments in the following way;

• Take intersection of H1 and L1 science segments,

• For each segment, protract by 1024 seconds,

• Take union of the protracted segments,

• Break up segments longer than 30000 seconds into segments not greater

than 30000 seconds,

• Protract segments so that they overlap by 1024 seconds.

The final step overlaps each broken up segment by 1024 seconds so that no

signal is missed due to breaking up. Figure 5.2 shows the actual distribution

of analysed segments for this two week run. In total there were 63 time

segments to be analysed.

5.2. DATA FOR OFFLINE RUN 99

28Jul

00:00

29Jul

00:00

30Jul

00:00

31Jul

00:00

01Aug

00:00

02Aug

00:00

03Aug

00:00

04Aug

00:00

05Aug

00:00

06Aug

00:00

07Aug

00:00

08Aug

00:00

09Aug

00:00

10Aug

00:00

11Aug

00:00

Time (UTC)

H1 data

L1 data

H1 ∩ L1data

ProtractedH1 ∩ L1

Analysedsegments

(a)

01Aug

00:00

01Aug

02:00

01Aug

04:00

01Aug

06:00

01Aug

08:00

01Aug

10:00

01Aug

12:00

01Aug

14:00

01Aug

16:00

01Aug

18:00

01Aug

20:00

01Aug

22:00

Time (UTC)

H1 data

L1 data

H1 ∩ L1data

ProtractedH1 ∩ L1

Analysedsegments

(b)

Figure 5.2: Here we show how we chose to break up the two weeks of S5 datainto 63 analysed segments. Figure 5.2a shows the full analysis period fromthe beginning of 28 July, 2007 UTC to the end of 10 August, 2007 UTC. Fora visual reference of how the segmentation process was performed, Figure5.2b shows the break up on day five, 1 August, 2007.


5.3 IIR Bank generation

The SPIIR element in each instance of gstlal iir inspiral requires the

As, Bs and Ds matrices introduced in Section 4.2 to filter the data. This

was done by providing gstlal iir inspiral access to a standard xml file

containing the matrices for each detector. The xml file also contains the

corresponding template bank intrinsic parameters (i.e. component masses).

The procedure to produce this xml file was a multi-step process as outlined

in figure 5.3.

Generate base

template bank

lalapps_tmpltbank

Split bank

by chirp mass

gstlal_bank_splitterGenerate

IIR template banks

gstlal_iir_bank

Measure PSD

gstlal_reference_psd

gstlal_iir_bank

gstlal_iir_bank

0000-H1_split_bank.xml



iir_0000-H1_split_bank.xml



reference_psd.xml.gz

bank.xml

... ......

Figure 5.3: A flow chart describing how IIR template banks were created.First we generated a list of mass pairs using lalapps tmpltbank in the formof a standard LAL xml document, e.g. bank.xml. The full list of templateswas divided into several subset lists, each containing at most 100 templates,sorted by chirp mass. Each subset list was recorded into a file, and along witha PSD frequency series was then passed to the gstlal iir bank applicationwhich resulted in a final IIR template bank xml file containing the As, Bs

and Ds matrices for that subset list.

First, a large template bank was produced using the LAL application

lalapps tmpltbank. In practise, this study was limited by available com-

puter resources, so the chosen parameter space was somewhat small. It was

designed to cover a component mass range of 1.1M–1.7M. By requiring

5.3. IIR BANK GENERATION 101

a minimal mismatch of 3%, lalapps_tmpltbank returned a template bank

with 1248 templates. The PSD used by lalapps_tmpltbank to place the

templates over the intrinsic parameter space was the initial LIGO S5 design

curve. Although the measured PSD changed over the two week analysis pe-

riod, the design PSD used to place the templates was conservative, meaning

that templates were over-sampled in parameter space. This meant that there

was no expectation of a drop in efficiency due to a poorly covered parameter

space.

A property of lalapps_tmpltbank is that for small parameter spaces, it

tends to place templates outside of the desired mass space, and also over-

populate the equal mass (η = 1/4) curve. Figure 5.4 shows distribution

of the 1248 templates generated in both component mass space and total

mass – symmetric mass ratio space. Many of the templates are actually out-

side the desired parameter space. We chose to include these templates in

our search, since they still correspond to NS-NS signals, albeit with slightly

higher masses.

In order to make the best use of computational resources, we chose to

run multiple instances of gstlal iir inspiral for different parts of the pa-

rameter space. We divided the full template bank into 13 subset template

lists each with 100 templates (except for the last subset, which contains

the remaining 48 templates). Each subset bank was then passed to the

gstlal_iir_bank application, along with a reference PSD, which used the

IIR coefficient generation algorithm described in Section 3.2.5 to create an

xml file containing the As, Bs and Ds matrices for each detector. The refer-

ence PSD used was determined by measuring the time segment data ahead

of time. During this procedure, we weighted the feed-forward b0 coefficients

by the inverse square root of the reference PSD, since the input data to the

filter was weighted by the inverse square root of the measured PSD (i.e. the

data was whitened). This application also calculated the auto-correlation

time series of each template and stored 100 samples either side of the peak

of each template’s auto-correlation in the xml file.

Using the IIR tuning parameters of α = 0.99, β = 0.25 and ε = 0.02, we

found that each IIR template had an overlap of at least 98.6% compared to


1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Component mass 1, m1 (M)

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Com

pone

ntm

ass

2,m

2(M

)

(a)

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

Total mass, M (M)

0.21

0.22

0.23

0.24

0.25

Sym

met

ricm

ass

ratio

,η(b)

Figure 5.4: The distribution of the 1248 templates is shown in both compo-nent mass (m1,m2) space (5.4a) and total mass – symmetric mass ratio space(5.4b). The target parameter space is shaded in grey. A known feature oflalapps tmpltbank is that it places many templates outside the target areafor small parameter spaces. Also, it seems to overpopulate the equal masscurve (m1 = m2, η = 1/4). The colours represent the 13 different subsettemplate lists that make up the full parameter space.

the 2PN waveform it was trying to approximate. Figure 5.5 shows the SNR

overlap as a function of total number of IIR filters and also of chirp mass.

This figure also shows how the total number of IIR filters per template (NIIR)

depends on chirp mass. Because the overlap is not 100%, we expect some

loss in detection efficiency, as was seen in the previous chapter 3.

The gstlal iir inspiral application is capable of processing multiple

template banks simultaneously by branching the data flow after whitening

the data (denoted by an ∗ in figure 5.1). This provides multiple threads

to better manage the execution of the application on multi-core CPU ma-

chines. Through trial and error, we found that a good combination of number

of template banks and template bank size for this study was five banks of

100 templates each. Since we had twelve template banks of 100 templates

each and one template bank of the remaining 48 templates, there were three

5.4. SIMULATED INSPIRAL SIGNALS 103

250 275 300 325 350 375Number of IIR filters

0.986

0.987

0.988

0.989

0.990

SN

Rov

erla

p

0.90 1.05 1.20 1.35 1.50Chirp massM (M)

250

275

300

325

350

375

Num

bero

fIIR

filte

rs

NIIR ∝M−5/6c

Figure 5.5: This series of plots show the SNR overlap of an IIR responsewith respect to its 2PN waveform, the total number of IIR filters in a giventemplate, and the chirp mass.

“bank sets” (template banks 1–5, 6–10, and 11–13 respectively). In total,

that meant each run required 189 (3 bank sets ×63 time segments) different

instances gstlal iir inspiral.

5.4 Simulated inspiral signals

In order to test the sensitivity of our pipeline to recover GW signals we

performed separate runs both with and without injecting simulated inspi-

ral signals into the detector data. By running the pipeline with injected

signals and searching for them in the list of candidates, we are able to eval-


uate the detection efficiency of the pipeline. As discussed in Section 5.1,

when gstlal iir inspiral is supplied with a list intrinsic and extrinsic pa-

rameters, simulated inspiral waveforms are injected into the detector data

before the whitening stage. To be as realistic as possible, we chose to inject

waveforms with Newtonian order in amplitude and 3.5PN order in phase.

The current implementation of the IIR filter coefficient design (as described

in Section 3.2.5) has some problems generating 3.5PN based templates (be-

cause the 3.5PN phase is generally defined in the Fourier domain). However

in reality, a true signal could have any order of post-Newtonian amplitude

and phase. We were interested to see how well the SPIIR method could

recover 3.5PN signals from 2PN templates. We expected this to result in

some loss in detection efficiency; however since we chose to randomly draw

injections with component masses from a rather narrow mass range close to

the canonical 1.4M–1.4Mwaveform, we did not believe that using 2PN

templates to search for 3.5PN simulations would be a problem. Indeed, we

chose the component masses of the injections to be restricted to be between

1.39M and 1.41M. The reason for the narrow mass range was so that we

can build a smooth detection efficiency curve for a canonical 1.4M–1.4M

system.

The extrinsic parameters of the injected inspiral signals were randomly

drawn from the following distributions:

1. The injections were uniformly distributed across the sky (cos θ, ϕ),

2. The cosine of the inclination angle ι was uniformly distributed between

0 and 1 (no preference of orientation between the observer’s line of sight

and the binary direction),

3. The polarization angle ψ was distributed uniformly between 0 and 2π,

4. Physical distance D was distributed log uniformly. We chose to start

at a minimum distance of 770 kpc and go out to 40 Mpc,

5. In our injections, no spin was given to the simulations.

5.5. BEHAVIOUR IN NON-GAUSSIAN DATA 105

The time distribution of the injections must be limited in order to avoid

placing them too close together in time, which would cause them to overlap

and produce unexpected results. We chose the geocentric end time tc of the

injections to be uniformly distributed over a 20 second interval every 100

seconds. In theory this limited the number of injections we could do over the

two week run period. However we can simply do additional runs, each with a

different injection parameter list (drawn with a different random seed). The

number of injections could be arbitrarily high without affecting the response

of the SPIIR output. In practise, we chose six different injection lists.

As injections were uniformly distributed over time, we did not expect to

recover injections during times when at least one detector was not in science

mode, or one detector had a veto segment (see Section 5.2). Over the six

injection runs, there were 54424 injections over valid segments with a uniform

distribution of parameters.

5.5 Behaviour in non-Gaussian data

As discussed in Section 2.5.6, the SNR ρ is a well defined statistic to declare

detection in stationary Gaussian noise. As stated, it is well known that to

date GW detector data is non-stationary and non-Gaussian [139]. For this

reason, we compared the output of the SPIIR pipeline in real, non-Gaussian

S5 data to an identical run in Gaussian noise.

Our method of calculating the χ2 statistic (described in Section 5.1) differs

from the standard matched filter way [132]. However we find that the distri-

bution of χ2 values is sufficient to distinguish SNR peaks due to glitches from

those due to genuine signals. Figure 5.6 shows the distribution of recorded

single-detector ρ and χ2 values for triggers from both the non-injection run

(black crosses) and coincident triggers from the injection runs that were asso-

ciated with injections (red crosses). The top plot show the ρ-χ2 distribution

for the non-Gaussian S5 runs, and the bottom plot the Gaussian runs.

In the non-Gaussian run, glitches can for the most part be distinguished

from genuine noise or simulation events as they have larger χ2 values. How-

ever there is a small population of triggers that have large χ2 values for


relatively small ρ values. These odd triggers were discovered to be misiden-

tified templates. Our pipeline requires triggers to be found in coincidence,

and for the ρ value to be above the threshold (of 4). For these odd triggers,

the L1 counterpart to the L1-H1 pair had a very low SNR, below threshold.

Consequently, a coincidence near the time of the injection was found, but

the incorrect template was identified (causing the large χ2 value).

Calculating both a ρ and χ2 value for each trigger will enable the ranking

of each trigger to determine it’s significance.

5.5. BEHAVIOUR IN NON-GAUSSIAN DATA 107

(a)

(b)

Figure 5.6: The distribution of χ2 values as a function of SNR ρ for detectorH1 (the distribution for L1, not shown, is similar). The black crosses repre-sent triggers from the non-injection run. The red crosses represent triggersfrom the injections runs that occurred within a 9 second window of an injec-tion, and were coincident in with a trigger from L1. The upper plot 5.6a isthe distribution of triggers in S5 (non-Gaussian) data, and the bottom plot5.6b is for Gaussian noise.


5.6 Ranking triggers

Each of the 189 instances of gstlal iir inspiral produced a separate co-

incident trigger database. For each trigger in each database, we need to de-

termine their significance by using a “ranking” statistic. By histogramming

the likelihood ratios of individual triggers that occurred in signal-free data,

and modelling the coincidence procedure, the pipeline is able to transform

the likelihood ratio for a multi-detector coincidence into a false-alarm proba-

bility (FAP). Knowing the livetime of the analysis (roughly two weeks), and

assuming that the triggers due to noise are the result of a Poisson process,

the FAP was converted to a false alarm rate (FAR, in units of Hz). The

FAR is the primary quantity used in determining the significance of each

trigger. Ultimately, we will only consider a coincident trigger a candidate for

follow-up if it has a FAR less than a given threshold FAR∗.

The background FAR distribution in other inspiral search pipelines is

typically estimated by repeating the coincidence test after shifting in time

single detector triggers relative to each other. Typically 100 time slides are

applied, resulting in 100 independent trials. Triggers from these trials are

called time-slide coincident triggers, as opposed to zero-lag coincident trig-

gers where no time slide has been applied. If the time shift is longer than

the light travel time between detectors, any coincident triggers found must

necessarily be a false alarms. Due to the computational overhead, this pro-

cedure has traditionally been performed offline. In an online configuration,

this would require excess computation that could be used elsewhere. In or-

der to calculate the FAR without performing many time slides, we use the

procedure outlined in [142], which was specifically developed for low-latency

online inspiral searches. This method relies on first determining the likeli-

hood that a trigger from detector i with a measured ρi and χ2i values and

intrinsic parameters θ occurred due to noise alone (i.e. the definition of a

false alarm), which is given by the standard likelihood ratio,

L(ρ1, χ21, . . . ρD, χ

2D, θ) =

P (ρ1, χ21, . . . ρD, χ

2D, θ|s)

P (ρ1, χ21, . . . ρD, χ

2D, θ|n)

≈D∏i

Li(ρi, χ2i , θ). (5.1)

5.6. RANKING TRIGGERS 109

Here P (. . . |s) is the probability of observing (. . . ) given a signal, and P (. . . |n)

is the probability of observing (. . . ) given noise. In the second equality the

simplification that the likelihood ratio is a product of the individual detector

likelihood ratios has been introduced. This is valid in the case that the signal

and noise distribution in each detector are independent.

The probability of a coincident trigger having a likelihood value L in the

presence of noise only is the false alarm probability, P (L|n). However over

the course of the analysis, there are likely to be many independent coincident

triggers with a certain likelihood value. Ultimately, we are interested in the

probability of getting at least one coincident trigger with L > L∗ (P (L∗|n) =

P (L > L∗|n)) after M independent coincident triggers are found. This is

given by the complement of the binomial distribution,

P (L∗|n1, . . . , nM) = 1− (1− P (L∗|n))M . (5.2)

The FAP (5.2) can be converted to a FAR by assuming that false alarms

occur as a Poisson process. Strictly speaking, the triggers produced by our

pipeline are limited to one trigger per 4 seconds for each template (as de-

scribed in Section 5.1). At an even shorter timescale, the frequency response

of each template filter limits the minimum interval between false alarms.

However the timescale of dead-time around each event multiplied by the

event rate is so small that it will have an insignificant effect on the event

distribution. Hence the background process is well modelled as a Poison pro-

cess. For a Poisson process with mean λ, the probability of observing N or

more events is,

P (N |λ) = 1− e−λN−1∑i=0

λi

i!. (5.3)

Setting N = 1 and equating this with the FAP (5.2), we see the mean number

of false alarms is,

λ(L∗) = − ln [1− P (L∗|n1, . . . , nM)] . (5.4)


To convert this to a FAR we simply need to divide by the livetime of the

run T ,

FAR = λ/T. (5.5)

The numerator of the likelihood ratio (5.1) is evaluated by assuming that

signals follow their theoretical distribution in Gaussian noise. This is a rea-

sonable assumption since detections of real signals are likely to come from

times of relatively stationary Gaussian noise. The denominator of (5.1) is

found by histogramming single-detector triggers that were not found in coin-

cidence. In practise, this means that a catalogue of non-coincident triggers,

sufficient to accurately estimate the likelihood ratio, must be recorded before

any FAP or FAR values can be assigned. As mentioned, running instances

of gstlal iir inspiral is only one part of the offline SPIIR pipeline.

5.7 The offline SPIIR pipeline

In Section 5.2 we described that for each run there were 63 time segments,

and 3 separate bank sets (Section 5.3). Hence in total, there were 189 in-

stances of gstlal iir inspiral for each run, each producing a database of

triggers. To calculate the FAR for each trigger as described in the previous

section a series of operations must be performed on the 189 databases. Figure

5.7 schematically outlines the post-gstlal iir inspiral procedure, which

calculates the likelihood ratio, FAP and FAR of each trigger, and reduces

the 189 databases to a single database of triggers. This constitutes the entire

offline inspiral search pipeline.

For every instance of gstlal iir inspiral the non-coincident trigger

SNR ρi and χ2i values were recorded. As described in Section 5.6, this infor-

mation was used to build a background distribution of probabilities which

gives the denominator of (5.1). The numerator of (5.1) was found by assum-

ing signals follow a theoretical distribution in Gaussian noise — which we

believe to be valid as we expect signals to only be observed when the noise

is relatively stationary and Gaussian. The likelihood ratio of each trigger for

5.7. THE OFFLINE SPIIR PIPELINE 111

Bank set 1(template banks 1-5)

Triggers from time segment 1

...

Calculate likelihoods

Cluster database

Merge databases

CalculateFAR

Write triggersto database


Cluster database

...

...

...

...

Marginalise likelihoodover parameter space

Cluster database

Merge databases

Cluster database



...


Cluster database

Merge databases


Cluster database

...

...

...

...

Cluster database



...


Cluster database

Merge databases


Cluster database

...

...

...

...

Cluster database

Figure 5.7: The post-gstlal iir inspiral procedure. The 189 triggerdatabases from gstlal iir inspiral are distilled down to a single trig-ger database that contains triggers ranked by FAR. See text for detailedexplanation.

each time segment and bank set is then assigned.

For each of the 189 trigger databases, it is highly likely that many coinci-

dent triggers are correlated across both time and intrinsic parameter space.

In order to reduce the amount of data, a cut is made on coincident trig-

gers. This process is known as “clustering”. Coincident triggers that are

within 4 seconds of other coincident triggers with higher SNR values are

deleted from the database. This means at most there is one trigger every

four seconds. This significantly reduces the amount of triggers in the origi-

nal databases. Databases from different time segments within the same bank


set are then merged together, and the clustering applied again. The three

resulting databases from each bank set are then merged to a single database

and the clustering cut is performed a final time.

The final trigger database, along with the marginalised likelihood infor-

mation are sent to the FAR estimation code that assigns a FAP and FAR to

each coincident trigger.

At the end of the pipeline there is a single database that has a list of

coincident triggers which each contain a FAR. This will be the primary

detection statistic that will determine if a trigger is worthy of follow-up. We

perform the entire search pipeline both with and without injections.

5.8 Confirmation of false alarm rate estima-

tion

For the non-injection run, we confirmed that the false alarm rate estimation

code was working as expected by plotting the cumulative number of false

alarms as a function of inverse FAR (figure 5.8). Assuming that false alarms

are a Poisson process, one would expect the total number of false alarms less

than a given FAR (or greater than a given inverse FAR) to be the product

of the experiment duration (the livetime) and the FAR (see equation 5.5).

For example, for the two week (14 day) run we expect to have on average 14

false alarms with a FAR less than one per day.

In addition to the zero-lag coincident triggers, we also applied a single

5 second time slide to the L1 individual detector triggers before coincidence

with H1 individual detector triggers was made. This time period was longer

than the light travel time between the two detectors meaning that any trig-

gers found in coincidence must necessarily be due to noise alone and not gen-

uine GW events. Both the zero-lag and time-slide distributions are shown

in figure 5.8). Both follow a Poisson distribution to within 1σ variation. We

believe this confirms the ability of the FAR estimation code.

5.9. SENSITIVITY OF SEARCH 113

105 106 107 108

Inverse False-Alarm Rate (s)

100

101

102

Num

bero

fCoi

ncid

entT

rigge

rs

Time-slideZero-lag〈N〉±√N

±2√N

±3√N

±4√N

±5√N

Figure 5.8: The number of coincident triggers less than a given false alarmrate, or greater than a given inverse false alarm rate is shown. The bluetrace is for the non-injection run without any time slide applied (zero-lag),and the black trace is for the non-injection run with a 5 second time slideapplied. The expected number of events assuming a Poisson distribution offalse alarms is given as the product of the livetime (∼ two weeks) and theFAR. This is what the dashed line shows. The shaded regions show the 1, 2,3, 4, and 5σ variance given by a Poisson distribution of false alarms.

5.9 Sensitivity of search

We quantify the sensitivity of our pipeline to recover injections by measuring

the detection efficiency ε. The detection efficiency is defined as the ratio of

found injections (Nf ) to total injections (Nt) at a given distance D.

Over the period of the two week injection runs, our pipeline produced

many coincident triggers that were within valid time segments. In order to as-

sociate these possible recoveries with the injected simulations, we ran the co-

incident trigger database through the LAL application ligolw inspinjfind.

This application associates coincident triggers to injected simulations that

happened within a 9 second time window. No other coincidence test was

performed (such as a mass coincidence or amplitude consistency check). If a


coincident trigger that was associated with an injection had a FAR less than

a given threshold FAR∗, we declared that the pipeline found it. Hence the

detection efficiency ε of our pipeline is a function of distance D and FAR∗,

ε(FAR∗, D) =Nf (FAR∗, D)

Nt(D). (5.6)

We can compare our measured detection efficiency to a detection efficiency

that would be obtained using the optimal matched filter and assuming that

the noise in the data is Gaussian. An optimal matched filter would recover all

injections with an SNR above a given threshold ρ∗. The strength of a signal

in Gaussian noise can be obtained from equation 2.98. This value can be ob-

tained by inserting the known injection signal parameters θ, ϕ, ψ, ι,D,m1,m2

and the detector’s noise PSD Sn(f) at the time of the injection tc,

ρ =1

D

√√√√2(F 2

+ (1 + cos2 ι)2 /4 + F 2× (cos ι)2) ∫ ∞

−∞

∣∣∣hc(f ;m1,m2)∣∣∣2

Sn(|f |) df, (5.7)

where the detector antenna response functions F+ and F× are functions of sky

location (θ, ϕ) and polarisation angle ψ. An injection in Gaussian noise would

be recovered from an optimal matched filter if it has an SNR (5.7) above a

given threshold ρ∗ in both detectors. Since all of our injections have masses

close to the canonical 1.4M–1.4M waveform, and the extrinsic parameters

are known, the number of injections recovered above ρ∗ in each detector

assuming Gaussian noise can be determined completely independently of the

SPIIR pipeline by measuring the PSD at the time of the injection.

We have plotted the SPIIR pipeline’s detection efficiency and the de-

tection efficiency obtained from a matched filter in Gaussian noise in figure

5.9. The measured detection efficiency will depend on the chosen FAR∗, and

Gaussian noise detection efficiency on ρ∗. For realistic searches, a high level

of confidence will be required to declare that a trigger was due to a real

inspiral GW event and not noise. The five sigma confidence level generally

relates to a FAR threshold of 3×10−13 Hz (∼ one per 105 years). A typically

quoted SNR value associated with this confidence level is 8 or above. Hence

5.9. SENSITIVITY OF SEARCH 115

as a comparison, we show the Gaussian noise detection efficiency by requiring

injected simulations to have an SNR ρ greater than 8 in each detector.

0 5 10 15 20 25 30 35 40

Distance (Mpc)

0.0

0.2

0.4

0.6

0.8

1.0

Effi

cien

cyε

FAR < 3× 10−13 HzGaussian noise (ρi > 8)

Figure 5.9: The detection efficiency of SPIIR pipeline for 54424 inspiralinjections with randomly chosen parameters. The solid blue curve representsthe SPIIR pipeline’s detection efficiency for trigger with a FAR < 3×10−13 Hz(∼ one per 105 years). Shown for comparison is the detection efficiencyobtained from an optimal matched filter assuming the signals were presentin Gaussian noise and recovered with a SNR of 8 or greater in each detector.

A useful quantity to compare the sensitivity of our pipeline to others is

the “sensitive volume”. The sensitive volume Vs is defined as the efficiency

weighted volume integral,

Vs(FAR∗) =

∫ ∞0

4πD2ε(FAR∗, D) dD. (5.8)

We show the sensitive volume of our pipeline in figure 5.10, along with the

volume obtained by the Gaussian noise detection efficiency (for an SNR >8 in

each detector). Multiplying the sensitive volume by the true coalescence rate

R (in units of events per unit volume per unit time) will give the expected

detection rate N (events per unit time) of the pipeline.


10−12 10−11 10−10 10−9 10−8 10−7

FAR∗ (Hz)

5000

6000

7000

8000

9000

10000

11000

12000

13000S

ensi

tive

Volu

meV

s(M

pc3)

measuredGaussian noise (ρi > 8)

∼ 105 yrs ∼ 104 yrs ∼ 103 yrs ∼ 102 yrs ∼ 10 yrs ∼ 1 yrs ∼ 1 mth

Figure 5.10: The sensitive volume of the SPIIR pipeline is shown as a functionof FAR∗ (and inverse FAR along the top axis). The volume is the efficiencyweighted distance integral given in Eq (5.8). The red bar is the sensitivevolume obtained by a detection efficiency assuming signals were present inGaussian noise and the optimal matched filter gave an SNR greater than 8.

Another way to compare the sensitivity of our pipeline to other NS-NS

inspiral searches is to measure the “sensitive range” rs. The sensitive range

is the radius which would give a sphere with volume equal to the “sensitive

volume” (Vs = 4πr3s/3). The sensitive range can be directly compared to

the “SenseMon” range commonly found in GW detector sensitivity papers

[143, 144, 145]. The SenseMon range is the estimated distance at which an

interferometer is sensitive to a volume of homogeneously distributed 1.4M–

1.4M inspiral sources averaged over all possible sky positions and orienta-

tions, with a minimum SNR of 8. The published SenseMon range for both

H1 and L1 over the entire S5 run was around 15 Mpc. However this value

is highly dependent on the properties of the noise PSD at the time of the

injection. We estimate our own SenseMon range by inverting the theoretical

detection efficiency weighted volume integral for signals with an SNR greater

than 8 (in each detector). This gives an expected range of ∼ 11.7 Mpc, which

5.10. PARAMETER ESTIMATION 117

is plotted in figure 5.11. Our pipeline obtains an equivalent range for a FAR∗

of 2× 10−11 Hz (∼ one per 1600 years).

10−12 10−11 10−10 10−9 10−8 10−7

FAR∗ (Hz)

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

14.5

Sen

sitiv

eR

ange

r s(M

pc)

measuredGaussian noise (ρi > 8)

∼ 105 yrs ∼ 104 yrs ∼ 103 yrs ∼ 102 yrs ∼ 10 yrs ∼ 1 yrs ∼ 1 mth

Figure 5.11: The sensitive range of the SPIIR pipeline is shown as a functionof FAR∗(and inverse FAR along the top axis). The range is the distance whichwould give a sphere of volume equal to the sensitive volume (fig 5.10). Thered bar is the sensitive volume obtained by a detection efficiency assumingsignals were present in Gaussian noise and the optimal matched filter gavean SNR greater than 8.

5.10 Parameter Estimation

A final step in validating the SPIIR pipeline is to make sure that the recovered

signals estimated the injected simulation parameters within an acceptable

error bound. We are only interested in showing that the SPIIR pipeline can

reasonably estimate the parameters, as the design goal of the pipeline is to

send triggers for EM follow-up with low-latency.

When using a discrete template bank to search for real GW signals that

may have any parameters, the template that gives the highest SNR will be

chosen. For a given signal with fixed parameters, different noise realisations


may cause the SNR to change, and hence a different template to be chosen.

If the SNR is high enough, the error in the measured parameters will have a

Gaussian distribution centred around the actual value [146]. This error can

be measured from the covariance matrix, which is the inverse of the Fisher

information matrix of the parameters θ. In practise, the actual error is also

related to the strength of the signal, however to confidently claim detection

of a signal would require a sufficiently high SNR, so this is a reasonable

estimate [146].

The only parameters we are concerned with in this low-latency pipeline

are the intrinsic mass parameters and time of arrival. For the former, we

considered the fractional accuracy in chirp mass defined as,

∆MM =

Mrecovered −Minjected

Minjected

. (5.9)

In [146], the authors estimate the error in ∆M/M to be 0.0383% for 3.5PN

templates for signals with an SNR of 10. They arrive at this value using

a theoretical initial LIGO noise curve, and assume signals have an SNR of

10. Figure 5.12 shows the distribution of measured fractional accuracy of

recovered injections that have a FAR less than 3 × 10−7 Hz (approximately

one per month, since we would not expect any false alarm trigger with a FAR

less than this value within the two week run), and a SNR in both detectors

of around 10 for the Gaussian injection runs. We observed that there was

a slight bias of approximately 1.5%, but the distribution otherwise mostly

matched. This bias is most likely caused by a mismatch in templates searched

for. The templates searched for were 2PN, whilst the injections were 3.5PN.

Although we are not too concerned with parameter estimation for this

pipeline, we must provide astronomers with accurate information for sky

localisation of the GW signal’s source. Generally sky localisation is based

on timing differences obtained from n different non-co-located detectors. For

two non-co-located detectors, the timing differences yield an annulus on the

sky of possible origins. Additional information, such as signal amplitude,

can reduce the annulus to localised regions, however the errors can remain

5.10. PARAMETER ESTIMATION 119

−0.001 0.000 0.001 0.002 0.003 0.004 0.005

Fractional Accuracy (∆M/M)

0

200

400

600

800

1000

1200

Nor

mal

ised

pdf

ExpectedMeasured

Figure 5.12: The fractional chirp mass accuracy ∆M/M distribution ofrecovered injections with a FAR < 3×10−7 Hz and an SNR of around 10. Thered dashed line shows the expected distribution (see text). Our distributionhas a similar uncertainty, with a bias of ∼1.5%. This is likely due to theincorrect PN order templates searched for.

as large as hundreds to thousands of square degrees. For more precise sky

localisations, more non-co-located detectors are needed.

The most important quantity to be accurately estimated in this respect

is the difference between the time of arrival of the signal ti at each detector i.

Since the error on the sky is a function of arrival time differences, we measure

the timing accuracy as,

δt = (trecoveredL1 − trecovered

H1 )− (tinjectedL1 − tinjected

H1 ). (5.10)

In [146], the expected error in measuring the time of arrival for each detector

is estimated to be 0.476 ms for 3.5PN templates with an SNR of 10. Since

we expect the noise from the two detectors to be independent, the expected

timing accuracy error (5.10) is this value added in quadrature, ∼0.673 ms.

Figure 5.13 shows the distribution of δt from recovered injections with a


FAR < 3× 10−7 Hz and an SNR of around 10 in both detectors. We see that

it follows the expected distribution.

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0

Timing Accuracy δt (ms)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nor

mal

ised

pdf

Expected σ = 0.673Measured

Figure 5.13: Timing accuracy δt distribution of SPIIR pipeline for injectionswith a FAR < 3 × 10−7 Hz and an SNR of around 10 in both detectors.This follows the expected distribution (see text for explanation) closely. Theroughness of the normalised pdf is most likely due to the finite number ofinjections.

5.11 Discussion

In this chapter we have realised the SPIIR method in the form of an exe-

cutable application, gstlal iir inspiral. This application was built into

part of the larger GW data analysis software project, LAL, and is avail-

able to other scientists in the community. We have tested the ability of the

SPIIR method to operate in real, non-Gaussian detector data by running this

application as part of a search for low-mass NS-NS inspiral signals in approx-

imately two weeks of historically recorded S5 data. As we were limited by

available computer resources, we chose to search over a rather small param-

eter space, 1.1M–1.7M component masses, which gave us 1248 templates

5.11. DISCUSSION 121

with a minimal mismatch of 3%. Using the IIR filter coefficient design de-

scribed in 3.2.5, we created a bank of IIR templates that approximate the

2PN templates to greater than 98.6%.

We ran the SPIIR pipeline in both non-Gaussian S5 data and Gaussian

noise generated from the initial LIGO design curve. Figure 5.6 shows the ρ-χ2

distribution for the single detector triggers. Single detector triggers that are

due to noise transients, known as glitches, are immediately distinguishable

from those caused by injections. This proves that the SPIIR pipeline is

capable of running on non-Gaussian data.

We have chosen to determine the significance of each coincident trigger

produced by the SPIIR pipeline by false alarm rate (FAR). Our method of

estimating the FAR differs from other search inspiral pipelines that take many

time slides. We have purposely chosen not to do this, as taking multiple time

slides won’t be feasible in an online environment. The new FAR estimation

code has been verified by showing that coincident triggers obtained from a

single time slide follow a Poisson distribution (figure 5.8).

The SPIIR pipeline’s sensitivity was determined by calculating the de-

tection efficiency, the sensitive volume and the sensitive range. By injecting

many simulations into the data with a uniform distribution of signal param-

eters we were able to count the number of injections found as a function of

distance and FAR∗. The detection efficiency curve (Fig 5.9) is shown for

a minimum FAR∗ of 3 × 10−13 Hz (∼ one per 105 years). In lieu of com-

paring this sensitivity directly with other inspiral pipelines, which can be

difficult due to different search conditions, we have chosen to compare it to

an efficiency curve obtained by requiring the SNR (Eq 5.7) found by an op-

timal matched filter to be greater than 8 in each detector. This method is

completely independent of the SPIIR method, as it depends solely on the

injection parameters and the noise PSD at the time of each injection. As

Figure 5.9 shows, at an efficiency of 50%, the distance is reduced by about

10% compared to the same theoretical maximum efficiency assuming Gaus-

sian noise. The sensitive volume (Fig 5.10) for detection requiring a five

sigma confidence level is about 25% of the Gaussian noise sensitive volume.

This is consistent with other low-mass inspiral search pipelines, such as [147],


although the conditions of their search were different from ours. However it

is also important to point out that the sensitivity in non-stationary noise is

influenced by techniques independent to the choice of filtering engine, such

as χ2-tests, other vetoes, coincidence tests, ranking statistics, etc.

By taking the efficiency weighted volume integral, we found the sensitive

volume as a function of FAR∗. By multiplying this volume by the true

coalescence rate, expected to be 10−8 − 10−6 events per Mpc3 per year [29],

we can calculate a detection rate (in units of events per year) of our pipeline.

The radius at which gives a sphere equal to the sensitive volume is known as

the sensitive range. This range can be compared to the “SenseMon” range

reported in GW detector sensitivity papers. Since the sensitivity depends

strongly on the non-stationary noise in the interferometer, we calculated

our own SenseMon range by inverting the sensitive volume (5.8) based on

a minimum SNR of 8 in each detector. This range was ∼11.7 Mpc. To

obtain an equivalent sensitive range in our pipeline, we would be looking for

coincident triggers with a FAR less than 2×10−11 Hz (∼ one per 1600 years).

It was important to show that the SPIIR pipeline could provide reasonable

parameter estimation of GW triggers for follow-up. Although our search

mass range was very small, our pipeline recovered the chirp mass parameter

to within the theoretical limit 5.12. Perhaps more importantly, the time of

arrival difference between detectors was considered. This too seems to follow

an expected distribution 5.13. This implies that the localisation error of this

method will be no worse than any other inspiral method.

The SPIIR method has been implemented and tested in an offline pipeline.

The pipeline can handle non-Gaussian detector data, can recover injections

with a detection efficiency close to the ideal amount, and can estimate the

template parameters within a reasonable amount. We believe we are in good

shape to scale up to the next stage, a test of the SPIIR pipeline on online,

live detector data.

Chapter 6

Online SPIIR pipeline

In this chapter we will discuss the implementation and execution of the SPIIR

pipeline in an online environment, as opposed to the offline environment dis-

cussed in the previous chapter. The very core of the SPIIR pipeline remains

the same — only the way the input to the pipeline (the data source) and the

output of the pipeline (the candidate coincident triggers for follow-up) are

handled have changed.

The Advanced detectors are expected go online between 2015 and 2020

with ever-increasing sensitivity [148]. As both the LIGO and Virgo detectors

are currently in the process of being upgraded to their advanced configuration

state, there is currently no online strain data to analyse. For this reason,

the LIGO scientific collaboration has organised a series of engineering runs,

designed to provide the collaboration with a realistic environment in which to

test and develop infrastructure that will be used in the advanced detector era.

LIGO’s second engineering run, dubbed ER2, occurred over three weeks from

18 July, 2012 to 8 August, 2012. During this run, re-coloured S6 data was

broadcast from the detector sites in real-time as though the detectors were

online. This gave us the opportunity to use real S6 like data in a real-time,

online environment. No S6 data by itself was used in this thesis. Simulated

gravitational wave signal injections were made at a realistic astrophysical

rate, with parameters unknown to data analysts ahead of time. This provided

an ideal situation for us to test the SPIIR pipeline on live, streaming detector

123

124 CHAPTER 6. ONLINE SPIIR PIPELINE

data. Our pipeline found many coincident triggers with low false alarm rates

and automatically submitted event information to the online listening service

called Gravitational-wave Candidate Event Database (GraCEDb) for further

follow-up (sky localisation, astronomical alerts, etc).

We participated in the ER2 experiment alongside other low-mass inspiral

search pipelines. However it is worth pointing out that our goal of operating

the SPIIR pipeline in ER2 was to implement the pipeline in a realistic setting

and identify possible future improvements, and not to directly compete with

other pipelines. Hence making direct comparisons with other inspiral search

pipelines can be difficult, as different pipelines may have different motiva-

tions. Additionally, different configurations of the same pipeline can alter

the design goals; level of latency, total computational resources, etc. Our

result in this chapter is not a final determination of the SPIIR pipeline ca-

pabilities, but rather a proof that the SPIIR method can work in a online

environment.

We begin this chapter by describing the relatively minor changes made

to the offline program gstlal iir inspiral to transform it into

gstlal iir ll inspiral— an implementation of the SPIIR method capa-

ble of reading online strain data, calculating the FAR of coincident triggers,

and sending low FAR triggers to the GraCEDb service in real-time and with

low-latency. Details of GraCEDb will be described in 6.1.1. The purpose

and goals of LIGO’s second engineering run, ER2, as well as the real-time

data distribution plan will be outlined in Section 6.2. We describe the oper-

ating procedure of the SPIIR pipeline we have designed to run on ER2 data

in Section 6.3, and the chosen parameter space for this run in Section 6.4.

Results of our ER2 search will be given in Section 6.5, and details of the

simulated inspiral injections placed in the strain data throughout ER2, in-

cluding whether or not our pipeline found them, will be discussed in Section

6.6. Finally, in Section 6.7 we will discuss the weak and strong points learnt

from this analysis on live engineering run data, and suggest how the pipeline

can be improved in future analyses.

6.1. SPIIR ONLINE PIPELINE 125

6.1 SPIIR online pipeline

In this section we will describe the application gstlal iir ll inspiral,

which is a modification of the gstlal iir inspiral application described

in the previous chapter (c.f. Section 5.1), designed to submit triggers from

live, online data. The core of the gstlal iir ll inspiral where the whiten-

ing, SPIIR filtering, triggering and coincident tests take place, is identical

to gstlal iir inspiral — only the way the input and output are handled is

different. A key difference is that upon execution, the gstlal iir ll inspiral

application looks for the latest available gravitational wave frame data from

a shared memory location on the machine running the application, and will

continue running indefinitely. The other major difference is that the FAR

of coincident triggers must be calculated immediately after it is found. This

section will describe in detail the flow of data through the application, as is

schematically shown in Fig 6.1.

After execution, the gstlal iir ll inspiral application reads the most

recent gravitational wave frame file from a shared memory location. Both the

strain data and a data quality vector are de-multiplexed from the frame data,

and packaged into regular GStreamer buffers (each buffer has a timestamp,

duration, sample rate, etc). The data quality vector contains information

about the state of detectors and other environmental factors that may oth-

erwise affect the veracity of signals found in the data. For our analysis, we

required that the data quality have the following conditions:

• The data must be flagged as being in science mode1, the interferometer

must be in lock, and the strain data itself, h(t) must be available.

• There must not be any burst injections and there must not be any

other reasons to believe that the data is not analysable (such as other

experimental work on site or environmental factors impacting on the

operation of interferometers).

If these conditions are met, the strain data buffer is passed onwards. Oth-

erwise it is flagged as a gap buffer and passed on. A gap buffer is a regular

1discussed in Section 5.1


buffer with a timestamp and a duration, but it has been flagged as having no

valid data. This is done to preserve a continuous flow of buffers throughout

the application.

The strain data buffer is then passed to the whitener element, as described

in Section 5.1. During the execution of gstlal iir ll inspiral, no initial

PSD is provided to the whitener element for which to initially estimate the

noise PSD. Hence some time is required after program execution for the

internal PSD to be correctly estimated.

The buffer is then passed on to the SPIIR element which operates as

described in Section 5.1. If the SPIIR element receives a gap buffer, it con-

siders the data in the buffer as zeros, and filters it. In this respect, there

were wasted computer cycles, but there was also a synchronous flow of data.

As in the offline run, our pipeline employs multiple SPIIR elements to pro-

cess separate template banks covering different parts of the parameter space

simultaneously (the specific banks used for the online run will be discussed

in section 6.4).

After the SPIIR element, the filtered buffer has 2×M channels (where M

is the number of templates). This buffer is sent to the triggering element. As

described in Section 5.1, the triggering element looks for peaks in absolute

SNR above a given threshold within a fixed time window. If there is a peak

above this threshold, then the χ2 value is calculated, and the ρ and χ2 values,

along with the trigger information (masses, time, etc), are passed along. For

this online study, we have chosen a SNR threshold of 4, and a 4 second time

window.

A time coincidence test is then performed for triggers between detectors.

A trigger is considered coincident between detectors if it occurs within 20

milliseconds for exactly the same template. If coincidence between two or

more detectors does not occur, the non-coincident trigger ρi and χ2i values for

the ith detector are recorded to memory for use in the background calculation

(see Section 5.6). The background probability distributions P (ρi, χ2i |n) (for

each detector i) and a table of independent trials are periodically (every hour

or so) written to memory. If there is a coincident trigger, the FAP (Eq (5.2))

is estimated from the likelihood probability P (L∗|n) where the likelihood L

6.1. SPIIR ONLINE PIPELINE 127

has been marginalised over all detectors and intrinsic parameter space θ.

This probability distribution is supplied to gstlal iir ll inspiral via an

external file (how this works will be described in Section 6.3). The FAR

of the coincident trigger is calculated as the quotient of the FAP and the

livetime of the experiment to date (see Eq (5.5)).

If the FAR of the coincident trigger is less than the a given threshold FAR∗

it is considered a candidate worthy of follow-up. The trigger information

(component masses, state of detector PSDs, time of arrival, etc) is passed on

to GraCEDb for further follow-up.

6.1.1 GraCEDb

The Gravitational-wave Candidate Event Database, GraCEDb, is a proto-

type online service designed to coordinate and centralise candidate events

from GW events (not just those from inspiral sources). This environment

provides a central platform from which to receive GW data analysis prod-

ucts and digest the information. Localisation routines and alerts for the most

interesting (and likely) GW candidates can be automatically generated. In

essence, it is a prototype interface between internal GW scientists and exter-

nal astronomers and other interested parties.

Within gstlal iir ll inspiral a client process is started whenever a

trigger below the provided FAR threshold is found. The event information

(template parameters, time of signal arrival, noise PSD of the detectors, etc)

is sent via the client to the GraCEDb server. When GraCEDb receives the

event information, a unique identification code is assigned in the database,

and a webpage storing the event information is made (the format of which

is dependent on the type of analysis).

Submitting events to the GraCEDb server formed the final stage of our

pipeline. The trigger latencies, discussed in Section 6.5, are defined as the

difference between the GraCEDb submission time, and the measured time of

arrival of the signal.


Template M

yes no


...


IIR templatebank file(s)

Calculate statistic

Detector 1

Template 1

yes

Whiten data

Store ,

*

Create gap buffer

Is thissegment of good

quality?

Create buffer

Is trigger coincident between

detectors?yes no

Mark as coincident Mark non-coincident


Calculate statistic

yes

...

...

...

...

... ...

Read frame filefrom memory

De-multiplex

DQ vector Strain

Marginalisedlikelihood file

Compute FAR

Is trigger FARbelow threshold?

yes

Send event informationto graCEDb

yes no


Detector n

Whiten data

*

Create gap buffer

Is thissegment of good

quality?

Create buffer

Read frame filefrom memory

De-multiplex

DQ vector Strain

Figure 6.1: Flow chart describing the flow of data from top to bottom throughthe (online) SPIIR pipeline program gstlal iir ll inspiral.

6.2. LIGO’S SECOND ENGINEERING RUN 129

6.2 LIGO’s second engineering run

Whilst LIGO and Virgo are undergoing a major upgrade to the advanced

configuration state, there has been a planned series of engineering runs de-

signed to prepare the collaboration for the era of advanced detectors. The

engineering runs provide a realistic environment from which to test GW data

analysis pipelines designed for the advanced detector era. The goals are to:

• Assist the development during construction and commissioning of the

instrument,

• Determine how to integrate GW data analysis with detector character-

isation and instruments during project planning,

• Provide an environment to develop the actual infrastructure of data

acquisition, online characterisation, instrument calibration and low-

latency transfer,

• Give an opportunity for data analysts to provide feedback to the in-

strument designers.

The run had a soft start beginning 11 July, 2012, but did not officially

start until 18 July, 2012, and went until 8 August, 2012. Mock frames con-

taining strain data were created ahead of time by re-colouring S6 strain data

for the LIGO detectors and VSR3 strain data for the Virgo detector. The

re-colouring process used the noise PSD curve based on the zero-detuning of

the signal recycling mirror and high laser power configuration [149] for the

Advanced LIGO detectors, and the design curve [47] for the Advanced Virgo

detector. However, half way through ER2 the mock strain data from L1 was

switched over to re-coloured Power Stabilised Laser (PSL) output. This was

designed to incorporate online subsystems and facilitate feedback between

data analysts and engineers.

Throughout the run, the data was broadcast live and in real-time from

the two LIGO detector sites: Livingston, Louisiana (L1) and Hanford, Wash-

ington (H1), and from the Virgo detector site located in Cascina, Italy (V1).


The broadcast was received by computers at the California Institute of Tech-

nology (CIT) located in Pasadena, California and the Virgo site in Cascina.

The head node of the CIT computer cluster transferred the data to the shared

memory of all the nodes in the cluster. This provided the optimal environ-

ment in which to test the SPIIR pipeline in an online configuration. Figure

6.2 shows the data distribution network.

6.2. LIGO’S SECOND ENGINEERING RUN 131

CIT

Pasadena,

California

Head node

(switch)

Node 1(shared memory)

TCP/IP

...

LHO

Hanford,

Washington

Re-coloured

S6 H1 Strain

Data

DMT

process

(package

gwf file)

H1 V1L1

Node 2(shared memory)

H1 V1L1

Node n(shared memory)

H1 V1L1

TCP/IP

TC

P/IP

LHO

Livingston,

Louisiana

Re-coloured

S6 L1 Strain

Data

DMT

process

(package

gwf file)

Virgo

Cascina,

Italy

Re-coloured

S6 V1 Strain

Data

DMT

process

(package

gwf file)

Broadcast in realtime

Figure 6.2: Data distribution topology of the ER2 run. S6 strain data wasre-coloured at the sites LIGO Hanford observatory (LHO), LIGO Livingstonobservatory (LLO) and Cascina for H1, L1 and V1 respectively. The datawas sent to local Data Monitoring Tool (DMT) processes, which packagedthe data into 4 second gravitational frame files and broadcast the data overTCP/IP in real-time. This data was received at a head node at CIT inPasadena (as well as computers at Cascina), and distributed over the localcluster network to the shared memory of multiple nodes.


6.3 Analysis setup

During the ER2, we were allocated 25 dedicated nodes on the CIT com-

puter cluster from which to run our online SPIIR pipeline. This enabled

us to run many instances of gstlal iir ll inspiral, each searching for

inspiral signals from different parts of the parameter space. Each instance

of gstlal iir ll inspiral across the cluster was scheduled and managed

by the Condor High Throughput Computing [150] workload management

system.

In the offline analysis we estimated the likelihood, FAP and FAR of each

trigger after collecting all triggers from the analysis period. The background

of many independent non-coincident triggers collected over the course of the

two-week offline run was used to to estimate the background probability

P (ρi, χ2i , θ|n) (the denominator of the likelihood ratio 5.1). However in our

online environment, the FAR estimation of each trigger must be made im-

mediately after a coincident trigger is found.

As discussed in Section 6.1, the FAP (5.2) of coincident triggers is esti-

mated within gstlal iir ll inspiral, immediately after the trigger has

been found. This is done by providing gstlal iir ll inspiral with a

marginalised likelihood file containing the probability P (L∗|n) where the

likelihood L has been marginalised over all detectors and intrinsic parameter

space θ. The marginalised likelihood file is updated periodically from the

most recent background probabilities by a separate application. We simul-

taneously had a script running on a separate node that would wait one hour

in between marginalising the likelihood from the 25 nodes running different

instances of gstlal iir ll inspiral (i.e. marginalise across intrinsic pa-

rameter space θ). This file was updated in place, meaning that each instance

of gstlal iir ll inspiral always had access to the latest marginalised like-

lihood file from which to estimate the FAP and FAR of each trigger. In this

respect the pipeline needed to be run for some time to collect a background

of independent trials from which to create the likelihood statistics.

Figure 6.3 shows the dependence of the multiple applications across the

nodes.

6.4. ER2 SEARCH PARAMETER SPACE 133

Job 1

marginalized

likelihood file

Marginalise

likelihoods over jobs

(runs every hour)

IIR template

banks 1,2SPIIR program

(always running)

Triggers sent

to graCEDb

...

...

...

IIR template

banks 3,4SPIIR program

(always running)

Triggers sent

to graCEDb

IIR template

bank 49SPIIR program

(always running)

Triggers sent

to graCEDb

Job 2

Job 25

Write

Write

2

Write

25

Figure 6.3: The 25 independent nodes on the CIT cluster each executean instance of gstlal iir ll inspiral. Each gstlal iir ll inspiral

reads a different IIR template bank(s) (in practise two banks each) and themarginalised likelihood file. Periodically (approximately once an hour) a sep-arate application on a different node is run which marginalises the likelihoodfrom the background probability P (ρi, χ

2i , θ|n) distributions across detectors

(i) and intrinsic parameter space θ. If a coincident trigger with a FARless than the provided threshold is found, it is sent to GraCEDb for furtherfollow-up.

6.4 ER2 search parameter space

The template banks used for this search were created using the same proce-

dure as outlined in Section 5.3, but using the zero de-tuning, high power nose

curve for Advanced LIGO [149] and the design curve for Advanced Virgo [47].

As we were limited by available computer resources, we restricted the

desired component mass range to be between 1.2Mand 1.475M. This

corresponds to a typical neutron star mass, and was a subset of the expected

injections (discussed in Section 6.6). Since the advanced detectors are sensi-

tive to signals with frequencies as low as 10 Hz, a properly placed template


bank will be more densely populated than the initial detector templates dis-

cussed in the previous chapter. By requiring a minimum mismatch of 3%

between templates and a starting frequency of 10 Hz, lalapps tmpltbank

produced 4865 templates. As in the offline case lalapps tmpltbank placed

some templates outside of the desired parameter space, as shown in Figure

6.4.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Component mass 1, m1 (M)

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

Com

pone

ntm

ass

2,m

2(M

)

(a)

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Total mass, M (M)

0.238

0.240

0.242

0.244

0.246

0.248

0.250

Sym

met

ricm

ass

ratio

,η

(b)

Figure 6.4: The distribution of the 4865 templates for the online analysisis shown in both component mass (m1,m2) space (6.4a) and total mass -symmetric mass ratio space (6.4b). The target parameter space is shadedin grey. Although a smaller space than for the offline run (c.f. Fig 5.4),the parameter space for the online run is more densely sampled. Againlalapps tmpltbank placed some templates outside the target area. Also,it seems to overpopulate templates along the equal mass curve (m1 = m2,η = 1/4). The colours represent the 49 different split template banks thatmake up the full parameter space.

In order to make the best use of available resources (the 25 nodes), we

decided to split 4865 templates into banks of 100 templates each ordered

by chirp mass, giving us 48 banks with 100 templates and one template

bank with 65 templates. This meant that each gstlal iir ll inspiral

application would run at most two split template banks each.

6.4. ER2 SEARCH PARAMETER SPACE 135

Each split bank was then passed to the gstlal_iir_bank application,

along with a file containing each detector’s design PSD. This application

used the IIR coefficient generation algorithm described in Section 3.2.5 to

create an xml file containing the As, Bs and Ds matrices required by the

SPIIR element for each detector. The IIR tuning parameters α = 0.99,

β = 0.25 and ε = 0.03 used meant that each IIR template had a greater than

99% match to the 2PN waveform it was trying to approximate (see Figure

6.5).

750 780 810 840 870 900 930Number of IIR filters

0.990750

0.990775

0.990800

0.990825

0.990850

SN

Rov

erla

p

1.02 1.08 1.14 1.20 1.26Chirp massM (M)

750

780

810

840

870

900

930N

umbe

rofI

IRfil

ters

NIIR ∝M−5/6c

Figure 6.5: This series of plots show the SNR overlap of an IIR responsewith respect to its 2PN waveform, the total number of IIR filters a giventemplate, and the chirp mass. As compared to the offline case which was forinitial LIGO (5.5) we can see that there is a ∼ 3 fold increase in the numberof IIR filters NIIR.

Using a lower cut-off frequency of 10 Hz instead of 40 Hz meant that there


were around three times the number of IIR filters NIIR per template than for

the initial LIGO S5 run (see Eq (4.7)). However this added a small amount

of computational cost (around 10%), since the extra filters were run at low

sample rates of 32 and 64 Hz. Figure 6.6 shows the number of IIR filter as a

function of sample rate NIIR,s for each of the 49 split banks (distinguished by

colour) and the estimated computational cost (obtained from Eq 4.1). As we

can see, only the computational cost of the 32, 64 and some of 128 Hz filter

groups are additional to the offline case (this can be compared with table

4.1).

6.5. RESULTS OF SEARCH 137

32 64 128 256 512 1024 2048 40960

50

100

150

200

250

300

Num

bero

fIIR

filte

rs

32 64 128 256 512 1024 2048 4096Sample rate bin

0

10

20

30

40

50

60

70

80

90

Com

puta

tiona

lCos

t(M

FLO

PS

)

Figure 6.6: Number of IIR filters NIIR,s and computational cost Cs per samplerate, per split template bank. There are 49 banks, each represented by adifferently coloured column.

6.5 Results of search

Over the course of ER2 we were able to run the online SPIIR pipeline inter-

mittently. For each execution of gstlal iir ll inspiral, GraCEDb up-

loads were initially disabled by default so there was time for the background

probabilities to be estimated by collecting non-coincident triggers. The FAR

threshold could be manually changed dynamically by the user at any time

(once enough non-coincident events were found to reasonably estimated the


FAR). However, the time at which FAR threshold was manually changed

and its value was not recorded, making it difficult to tell post-analysis when

the pipeline could have been submitting events to GraCEDb.

Also, there were times when the broadcast of the data from the individual

detector sites to the CIT computer cluster was interrupted. During these

times, each instance of gstlal iir ll inspiral stored some data (several

minutes worth) from each detector in memory, then immediately went into

a paused state waiting for the eventual arrival of the data. Once the missing

data arrived at the CIT node’s shared memory, the application tried to first

clear the data in the stored in memory from all detectors, and then the

missing data. However since we had designed gstlal iir ll inspiral to

search the highest number of templates possible in real-time, there were

times when there were not enough extra computational resources to clear

the backlog (i.e. filter the offline data faster than real time). When this

happened the pipeline would either crash, or be indefinitely stuck processing

with a several minute latency, requiring a manual restart.

Those problems aside, we submitted 1994 events to GraCEDb between

the 20 July, 2012 (UTC) and the end of ER2, 17:00 8 August, 2012. Fig-

ure 6.7 shows the FAR of each event submitted to GraCEDb as a function of

event time (the calculated arrival time of the end of the signal). There are no

triggers below 3×10−13 Hz (one in 105 year event), suggesting that no events

were genuine signals. However there were several events with FAR less than

10−13 Hz, and were often submitted within a short period of time. We believe

that these were due to short-lived periods of non-stationary noise (glitches)

in the data. The whitener element within gstlal iir ll inspiral can ac-

curately track a slowly changing noise spectrum, but short lived glitches are

difficult to remove. Indeed, this is one of the reasons why a data quality

vector was used. Also, recall that the online pipeline had, unlike like the

offline pipeline, no clustering over time and parameter space2. This meant

that glitches could ring up in both mass space and time. For clarity, we

applied a 20 second window post-analysis to the 1994 submissions, retaining

2other than the internal 4 second time clustering per instance ofgstlal iir ll inspiral.


only the lowest FAR events within the 20 second interval. A period of 20

seconds was chosen as it is not expected that glitches occur more often than

this. From this we found 193 clustered events (shown by red crosses in Figure

6.7). From this figure we can tell that the FAR threshold was set manually

to 10−5 Hz, 10−4 Hz or 10−3 Hz at different times.

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Figure 6.7: The FAR of events that were submitted to GraCEDb as shown asa function of time (blue crosses). Although there was no logging mechanismto show what the GraCEDb FAR threshold was at a given time, we cansee that it was most likely 10−4 Hz, 10−3 Hz and at times 10−2 Hz. A largenumber of events were submitted close together in time, but with a FAR lessthan 10−13 Hz, suggesting that they were glitches in the data. We applieda 20 second window post-analysis, which reduced the number of clusteredevents to just 193 (red crosses). Shown in grey are times when there were atleast two detectors in science mode.

From Figure 6.7 we can tell that there were days when the analysis was

either not running, or the FAR threshold had not been set (e.g. 30 July, 2

August). However, without an adequate logging mechanism, it is difficult to

tell if the pipeline was not running at all, or the GraCEDb FAR threshold

was not set to a physical number, or there was an interruption in the data


broadcast, or there were less than two detectors operating.

An observation of the post-analysis clustered events reveals that the FAR

estimation takes time to collect enough statistics to accurately estimate the

FAR. Figure 6.8 show the number of clustered events per day that had a

FAR less than 10−4 Hz. On average, we would expect 8.64 events per day.

Also shown is the cumulative average. Considering that there were some

days when no events were submitted (most likely because the FAR threshold

had not been set), by the end of the run, the trigger rate was around two

thirds of the estimated amount.

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Figure 6.8: The number of clustered events with a FAR below 10−4 Hz per(UTC) calender day. The cumulative daily event rate average is shown inblue. Shown in grey are times when there were at least two detectors inscience mode.

Perhaps most importantly, we were able to submit many of the events

with relatively low-latency, where latency is defined as the difference in time

between the event time and submission to GraCEDb. Figure 6.9 shows the

time distribution of the latencies of both the unclustered (blue crosses) and

clustered (red crosses) events. There were some high latency events, which


mostly occurred during times of highly frequent submissions, which prob-

ably overloaded the available resources. This was most likely due to non-

stationary noise artifacts in the data. At other times when the submissions

were at a more realistic rate (and the noise being relatively stationary), com-

mon values of the latencies were around 30 seconds (see Figure 6.10). This

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Figure 6.9: The latency (time difference between the event time and submis-sion to GraCEDb) is shown as a function of time over ER2. Many of the highlatency events occurred during times of frequent submissions, indicating thatavailable resources were hung up trying to clear the abundance of triggers.A common value of latency for the clustered (red crosses) events was around30 seconds. Shown in grey are times when there were at least two detectorsin science mode.

value is about what is expected from the data flow procedure as described

in Section 6.1. There is an approximately 4 second latency in the collecting

of data and packaging into frame files, the whitener element takes a Fourier

transform every 4 seconds, the filtering elements store up to 10 seconds of

data, the triggering element is over a 4 second window, and the coincidence

test is around 10 seconds. A total of 32 seconds is what was expected, and

on average was seen.


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Figure 6.10: The distribution of the latency time of clustered events through-out the ER2 period is shown. Three modes are seen: a common modewhen the pipeline was running as expected, mid-range latencies when themarginalised likelihood file was being updated, and high latencies when majorglitches occurred, causing too many triggers to overload the computationalresources.

Although there were some GraCEDb submissions with very low false

alarm rates, these happened when the pipeline had only been running for

a few days. We do not believe that the pipeline had built up enough back-

ground statistics to accurately estimate the FAR, so we do not claim these

as detections. Over the rest of the run, we did not submit any events with a

very low FAR, so we claimed no detections of real signals.

6.6 Blind software injections

During ER2 there was a set of simulated inspiral (amongst other) signals

injected in the strain data. The parameter boundaries and distributions were

known to GW data analysts before the run, but not the specific parameters.

Hence ER2 served as a blind analysis.

6.6. BLIND SOFTWARE INJECTIONS 143

The inspiral injections were split into three classes: binary neutron star

(NS-NS), neutron-star-black-hole binary and binary black hole, categorised

by component masses. The NS stars were classified as having masses of

1M–2M and black holes 5M–20M. Our pipeline was designed for NS-

NS signals and given our chosen component mass parameter space (1.2M–

1.475M) we could indeed only aim to pick up signals from a subset of the

NS-NS class. The component masses were uniformly distributed between

1M and 2M, and had no spin. The rate at which the simulations were

injected corresponded to the most up-to-date neutron star binary coalescence

rate estimates of 1 Mpc−3 Myr−1[29]. Assuming that the detectors were in

science mode and had an the average inspiral range of 200 Mpc [149] through-

out the three weeks, approximately 1.9 events would have been expected to

be observed on average. However, as we only covered a small part of the

NS-NS mass space (approximately (1.475−1.2)2/(2−1)2 ∼ 7.5%), there was

only a one in fourteen chance that we would discover at least one injection

for this run.

The specific injection parameters were revealed in September 2012, when

the analysis was un-blinded. From the entire list of all injections, there were

only five NS-NS injections that were within the searched parameter space of

our pipeline and occurred when at least one detector was online. Two of these

occurred when only one detector was online, and none had expected SNRs

above 4 in at least two on detectors, which was the minimum requirement

for our pipeline to report coincident triggers. These potential events are

summarised in table 6.1. Hence we should not have expected to find any

injections.


Table 6.1: Shown in this table are the five injections within our parameterspace (c.f. Section 6.4) that occurred when at least one detector was operat-ing. The expected SNRs in the three detectors are listed; bold indicates thatthe detector was online and in science mode. Our pipeline requires a mini-mum SNR of 4 for a minimum of two detectors to report coincident triggers.There were no such injections that met this criteria, so we should not haveexpected to find anything.

End time (UTC) Total mass (M) L1 SNR H1 SNR V1 SNR26 Jul 19:27:08 2.53 3.64 3.18 3.6020 Jul 04:34:56 2.77 3.80 3.14 0.3604 Aug 07:57:54 2.82 3.45 2.57 4.5721 Jul 19:54:15 2.92 5.24 4.37 2.1320 Jul 04:44:07 2.95 5.78 4.11 3.16

6.7 Discussion

By executing the SPIIR pipeline in the realistic, online environment of ER2,

we were are able to gain valuable insight into how the pipeline can be im-

proved during the advanced detector era.

In general, we found that analysing the event rate after the end of ER2

was difficult as the FAR threshold was not set by default. Although there

was good reason to manually change the threshold (i.e. to avoid flooding the

GraCEDb server with high-FAR events), the time at which it was changed

and its value was not recorded, making it nearly impossible to tell post-

analysis when we would have expected events to be submitted. Adding to this

complexity was that fact that there were occasions when the computer cluster

management program Condor would restart the gstlal iir ll inspiral

processes due to external reasons. When the jobs were restarted on the

nodes, the default FAR threshold was not set. However, no alert of the

restart was sent out, and no automation of the FAR threshold was reset.

This led to large gaps of online data not being searched. Additionally, there

were several times when broadcast of data from the sites was interrupted, and

our pipeline went into a paused state. Again, these times were not recorded

by our pipeline.

The most obvious feature in need of improvement found post-analysis was

6.7. DISCUSSION 145

the ability to log when the pipeline was running and when it was expected

to submit events to GraCEDb. This lack of logging has made it difficult to

tell how accurate the FAR of coincident triggers were. Poor FAR estimation

was to be expected in the early days after the SPIIR pipeline began, as there

were not enough non-coincident triggers collected to accurately estimate the

background. By the end of the run, enough statistics had been collected to

reasonably estimate the FAR (within a factor of two thirds). This is vastly

different from the offline case, where the FAR estimation was done post-

analysis and had the entire two weeks worth of background events (from all

templates) to estimate the likelihood from. Considering that the offline run

correctly estimated the FAR of events after collecting almost two weeks of

triggers from a wider parameter space, we are confident that given more time

to run on the online case, and given a wider parameter space, the FAR rate

would have converged to its expected value. In the future we expect to collect

many more non-coincident events before relying on the accuracy of the FAR

estimation. This could be done by running the pipeline on pre science mode

data for a few weeks.

The quality of the data during this run also seemed to cause, at times,

an over-reporting of candidates. These were most likely caused by short

periods of non-stationary noise (i.e. glitches) in the data where the change

to spectrum was too rapid for the PSD to be correctly estimated and the data

whitened. The flood of events submitted during these times likely confused

the FAR estimation code, as it took some time to learn how to deal with

high event rates such as this. Given more time, and a history of these types

of glitch-caused events, the FAR estimation code would have been able to

better assign FAR values to events like these.

Although we knew the parameter distribution and injection rate of the

blind injections, we did not know the specific parameters. Our searched

parameter space, which was limited by available computer resources, was

really too small to expect to discover anything (a one out of fourteen chance

on average). After the un-blinding of the injection parameters, we found that

there were five injections within our parameter space that occurred during

times that at least one detector was online. Our pipeline requires that a


coincident trigger be present in at least two detectors, and have a SNR above

4. None of the five injections met these criteria, hence we accurately made

no claims at true detections. In future searches, we expect to have access

to more computers with which to search for a larger parameter space. A

wider parameter space would also result in more trials for estimating the

background statistics, leading to a better FAR estimation.

Perhaps the best outcome of our test on real online data was the ability

to send out triggers for follow-up with a very low latency. We uploaded most

event candidates that were not clustered in time with a latency of around

30 seconds. Some did have higher latencies, as can be seen in the latency

tail of figure 6.10, but these mostly occurred during times of glitches, when

resources were overloaded. An improvement in data quality flags may reduce

these kinds of events, and a history of events like these can improve the FAR

estimation. The current pipeline also pauses every four to eight hours to read

in new background statistics. Future versions of the pipeline may be able to

speed the reading up. To put this in context, the latency achieved in S6 by

the MBTA method [116] was three minutes until the availability of a trig-

ger. Although the SPIIR method is a time domain method, and therefore

in principle has a zero (or at least sample time) latency, it’s implementa-

tion introduces some latency overhead, e.g; buffering of time series data into

discrete packets, whitening, coincidence tests, etc. The configuration of the

existing pipeline can be modified to improve this latency, but they may come

at a cost of other important considerations, such as computational efficiency

or parameter space. Although sub 30 second latency may be possible within

the current implementation, we believe this level of latency to be reasonable

for the amount of computational resources consumed and the current infras-

tructure of data delivery. Our measured latencies were consistent with the

other low-mass inspiral search pipelines that operated in ER2.

LIGO’s second engineering run, ER2, provided a unique opportunity to

operate the SPIIR pipeline in an online environment. From this run we were

able to assess and evaluate the strengths and weaknesses of the pipeline, and

suggest how it can be improved. The end goal of submitting GW event can-

didates for further analysis was achieved by enabling access to the GraCEDb

6.7. DISCUSSION 147

service. The interface seems to work well, and most clustered triggers were

uploaded with a latency of around 30 seconds.


Chapter 7

Conclusion

In this chapter, I will re-cap the overall aim of this thesis, and outline how

it was achieved. In the second section, I will then explain what future work

could be done to improve the research.

7.1 Thesis motivation

Numerous tests have all verified general relativity to astonishing accuracy [6].

One of the last remaining predictions of general relativity yet to be directly

observed is that of gravitational waves. Caused by non-axis symmetric bodies

that have a time-changing mass quadrupole moment, these are perturbations

of space-time. The effect of a passing GW on matter is to alternately stretch

and squeeze it. So far there has been no direct detection of this stretching

and squeezing, although there is indirect evidence for its existence in the

form of the Taylor-Hulse binary [17, 18, 19]. A worldwide effort is currently

underway to observe the strain h induced by GWs. Both the LIGO and Virgo

interferometric detectors are undergoing a major upgrade, which will see their

sensitivity improve by a factor of ten from their previous configurations.

With the imminent detection of GWs comes the question of what kind

of EM emissions they have, and what science can be explored by observing

it. Some predicted GW sources are thought to be the progenitors of highly

energetic and poorly understood astrophysical events — gamma-ray bursts.

149

150 CHAPTER 7. CONCLUSION

As transient events, GRBs are known to have a complex EM emission pro-

cess (see Section 1.5 for more information). In particular, short-hard GRBs

are thought to have compact binary coalescence events (e.g. neutron star-

neutron star binaries) as their progenitors. A rapidly fading, often not very

bright, optical prompt emission may happen within tens of seconds after the

GRB event [87]. In order to observe the EM component, one must either have

a telescope pointing in the direction of the GRB, or otherwise have a trigger

to motivate the slewing of rapid response telescopes. Considering that the

γ-ray emissions of GRBs are likely beamed [54], not all coalescence events

may have a GRB component (although their optical and radio afterglows

may be observable off axis). Another possibility is that a trigger could be

provided by detecting the preceding GW emission of the binary coalescence

progenitor. A trigger would be required to be calculated in real time and with

low-latency (of order tens of seconds). This would enable a greater scientific

understanding of the underlying physics by constraining models describing

the GRBs. The scientific questions posed can be ultimately be answered by

first implementing a low-computational resource GW inspiral search pipeline

that can send out GW triggers with low-latency.

7.2 Thesis aim

In order to observe the EM counterpart of inspiral sources, fast moving

robotic telescopes are required to respond to triggers from GW searches.

The triggers need to be provided as close to real-time as possible to max-

imise the amount of EM information recorded, and thereby validate models

of GRBs. This notion of low-latency GW detection is a relatively new direc-

tion in GW data analysis, since typical traditional methods search for GWs

in the Fourier domain, which in most cases are performed offline. The chal-

lenge of low-latency GW detection of inspiral sources will only become more

difficult in the era of advanced detectors, where the bandwidth is wider, the

signals longer, and the number of signals needed to be searched for increases.

There is a clear need for a new algorithm to search for inspiral gravita-

tional waveforms in advanced detector data with the goal of providing GW

7.2. THESIS AIM 151

triggers as close to real-time as possible. This was the aim of this thesis. This

aim was met by breaking the task into three phases: design, implementation

and testing of a new GW search method that is able to provide inspiral GW

triggers from advanced detector data with very low-latency.

Design Chapter 2 provided the background knowledge necessary to under-

stand GW data analysis, specifically for searching for inspiral gravita-

tional waveforms. The information required to build a new low-latency

inspiral search pipeline including an introduction to time-domain digi-

tal infinite impulse response (IIR) filters was given.

In Chapter 3, the design of the new time-domain low-latency algorithm

for identifying the presence of gravitational waves produced by com-

pact binary coalescence events in noisy detector data was described in

detail. This chapter was published in its entirety as an article in Phys-

ical Review D [119]. The new method is similar to matched filters,

but achieves low-latency by searching for approximate inspiral signals

strictly in the time-domain. Computational efficiency (when compared

to time-domain matched filters) is achieved by approximating the in-

spiral waveform as a summation of damped sinusoids, and then using

a summed bank of parallel infinite impulse response (SPIIR) filters to

search for the approximated waveform. Since the IIR filters operate

strictly in the time domain, the signal to noise ratio can be calculated

with in principle sample time latency. Throughout Chapter 3, my col-

laborators and I compared the ability of the SPIIR design to recover a

single canonical 1.4M–1.4M waveform in Gaussian noise to that of

the optimal matched filter approach. It was shown that with a good

choice of tuned parameters in deciding the IIR filter coefficients, the

SPIIR method can retrieve the signal to noise ratio to greater than 99%

of that produced by the optimal matched filter. The test was repeated

with many different noise realisations showing that there was not a sig-

nificant loss in detection efficiency compared to the optimal matched

filter result. This confirmed that the SPIIR method is a feasible alter-

native to the optimal matched filter.


Before entering the implementation phase, Chapter 4 described some

extensions to the basic SPIIR design to optimise the computational

efficiency in a real online search. Through the use of multi-rate filtering,

the theoretical computational cost to search for a single template was

shown to be reduced by 90-95%. Also discussed in this chapter was

a design strategy to efficiently filter many templates, not just a single

template.

Implementation Although Chapter 3 showed the feasibility of the SPIIR

design, the test was not necessarily computationally efficient and exe-

cutable on a scale that would be required by a real inspiral search on

live advanced detector data. Much effort was put in to place to incor-

porate the SPIIR method into the greater GW scientific community’s

software library (LAL). This had two distinct advantages: to draw on

the wide experience of the greater data analysis group, and to make

available our code for testing and use by other GW data analysts.

From an early stage it was decided to implement the SPIIR method

within the gstlal [130] project. This is project wraps, amongst other

things, the LIGO Algorithm Library (LAL) which contains GW data

analysis programs, and the open-source signal-processing software en-

vironment GStreamer. This provided an environment to concentrate

on writing GW data analysis specific software, whilst leaving the com-

mon digital signal processing details and data flow management to the

framework of GStreamer.

Several applications to facilitate the running of a new SPIIR search

pipeline were written. An application to create the IIR filter coefficients

from a given template bank was written (described in Section 5.3).

The SPIIR application itself, which searches for templates in multiple

detector data streams was realised as an offline version (described in

Section 5.1) and an online version (described in Section 6.1). The

offline SPIIR application searches for templates in multiple detector

data streams over a fixed period of time, with the end result a database

of triggers and related template information. Further processing of

7.2. THESIS AIM 153

the database, specifically ranking triggers by likelihood and calculating

their false alarm rates, was performed post-analysis. In the online

SPIIR application, ranking was performed immediately after a trigger

is found. If the false alarm rate of the trigger was found to be below

a given threshold, event information was sent to an external server for

follow-up (see Section 6.1.1).

Testing The SPIIR method was tested in a realistic search pipeline in two

parts, an offline analysis (Chapter 5), and an online analysis (Chapter

6). Although the online analysis was the ultimate goal of the SPIIR

method, an initial offline analysis was an important test to show how

the method would cope with realistic non-Gaussian detector data.

The SPIIR method was run in an offline configuration by executing a

search pipeline on both simulated Gaussian noise and real LIGO S5

data. Our χ2-consistency check (described in Section 5.1) managed to

successfully distinguish triggers due to glitches from those due to real

signals (results in Section 5.5). Running in an offline mode allowed the

injection of many simulated signals with randomly chosen parameters.

The measured detection efficiency was compared to that from the ex-

pected SNR obtained by the optimal matched filter assuming signals

were present in Gaussian noise. The SPIIR method does not have a

significantly different efficiency at the same level of confidence. The

parameter estimation also did not seem to be very different from what

was expected.

The final and most important testing phase of the SPIIR method was

described in Chapter 6, where the method was tested in an online

configuration. A real-time search for approximately 5000 inspiral tem-

plates was executed as part of LIGO’s second engineering run, which

occurred over roughly one month in July and August 2012. Problems

discovered during the engineering run related to the logging of the

search and length of time the search was performed. However, during

the run, many event triggers with latencies around 30 seconds were

submitted. This level of latency was about what would be expected


due to the implementation of the method through gstlal. A slightly

lower than expected event rate was observed, but given more time and

triggers collected, the background statistics could accurately estimate

the FAR.

7.3 Future work

Although the realisation of the SPIIR method as presented in Chapter 6

shows that the method can work as an inspiral search pipeline, longer tests

with larger template banks covering a greater parameter space would be ben-

eficial. A longer run time would allow a greater collection of non-coincident

events, ultimately giving a more accurate FAR estimation. Alternatively,

the pipeline could be run on previously collected ER2 or similar data for a

few weeks before searching live real detector data. A wider parameter space,

although more computationally expensive, would also result in more trials

to estimate the background statistics, leading to a better FAR estimation.

Also, the immediate need of better logging of the SPIIR application running

status is required to provide better information post-analysis.

Although the SPIIR method has in principle zero latency, the realisation

of the method through its implementation using the gstlal infrastructure

gives a latency of around 30 seconds. Improvements of maybe 5 or 10 seconds

might be possible whilst still using gstlal; however that may affect the

computational efficiency.

I believe it may be possible for the SPIIR method to search for a wider

parameter space without the need for greater computational resources. I can

see two approaches here; 1) reduce the amount of computation by reducing

the number of IIR filters per template, 2) use a dedicated hardware solu-

tion (such as graphical processing units — GPUs) to take advantage of the

embarrassingly parallel aspect of the SPIIR method.

The former seems like the immediately obvious choice at first, since most

of the IIR filter coefficient have similar values. As the goal of this thesis was

to have a working low-latency pipeline, I chose the first working coefficient

design, which is based on the Taylor expansion of the phase evolution, and

7.3. FUTURE WORK 155

then continued to developed the implementation. Because this method does

not take into account the parts of spectrum the detectors are most sensitive

to, it probably places IIR filters at the high frequency end that contribute

very little to the SNR. An improvement in this design may drop number

of filters by a small amount; however these are the most computationally

expensive.

Another way of reducing the number of IIR filters is to use an inter-

polation scheme as described in [4]. Although that description is for the

Newtonian order inspiral waveforms (and hence a one dimensional intrinsic

metric), it can be extended to higher order post-Newtonian expansions. The

essence of the scheme is to identify a subset of the full template bank, create

IIR filter coefficients for just these templates, and then calculate a series of

simply algebraic operations that can be applied to groups of IIR filters to

recover the rest of the templates. The planning, creation and execution of

the scheme would be a significant body of work in itself, but could result in

a significant reduction in computational cost (although is hard to estimate

here, since one would have to look at the computational cost gains across the

full parameter space).

The second strategy of using fewer computers is to use a dedicated hard-

ware solution to take advantage of the embarrassingly parallel aspect of the

SPIIR method. The SPIIR method is many simple single pole IIR filters (c.f.

Equation 3.30) operating independently of one another. Parallel processing

hardware such as GPUs can be employed to efficiently crunch the numbers.

An initial study has been performed to show the feasibility of such technol-

ogy on the SPIIR method [151]. Executing this design on a full scale inspiral

search on live detector data has yet to be performed.


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[3] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Physics

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