Directed searches for continuous gravitational waves from spinning neutron stars in binary systems

by Grant David Meadors

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2013

Doctoral Committee: Professor John Keith Riles, Chair Professor Fred Adams Professor Timothy McKay Professor Stephen Rand Professor Nuria Calvet Research Scientist Harold Richard Gustafson c Grant David Meadors 2013 All Rights Reserved (space for a fancy dedication, such as the following) To the tree of Life, which took stardust and evolved into us. (Maybe in Latin, or Greek?) Pro arbore Vitae, ex nube stellarum ad nos evolvit.

ii ACKNOWLEDGEMENTS

This author should give thanks far beyond a simple page. It is too soon to write something so important.

iii TABLE OF CONTENTS

DEDICATION ...... ii

ACKNOWLEDGEMENTS ...... iii

LIST OF FIGURES ...... vii

CHAPTER

I. Introduction ...... 1

1.1 Gravitational waves in astrophysics ...... 1 1.1.1 Cosmic sources of gravitational waves ...... 2 1.1.2 History from general relativity ...... 6 1.1.3 Contrast with electromagnetic and particle astronomy ...... 6 1.2 Generalrelativity ...... 7 1.2.1 Symmetry and action principles ...... 8 1.2.2 Derivation of field equations ...... 8 1.2.3 Radiation from quadrupoles ...... 8 1.3 Astrophysicalestimates ...... 8 1.3.1 Sources: burst, continuous, inspiral and stochastic ...... 8 1.3.2 Continuous waves from neutron stars ...... 9 1.4 Laser Interferometer Gravitational-wave Observatories ...... 9 1.4.1 From Weber bars to interferometry ...... 9 1.4.2 Gravitational wave interferometry methods ...... 9 1.4.3 Advanced observatories and beyond ...... 11 1.4.4 Worldwide network ...... 11 1.5 Summary...... 11

II. Feedforward: Auxiliary MICH-PRC Subtraction ...... 12

2.1 Introduction ...... 12 2.2 Description of the feedforward method ...... 15 2.2.1 Auxiliary noise coherence at sensitive frequencies ...... 19 2.2.2 Estimatingfilters ...... 22 2.3 Feedforward in-loop and alternative program methods ...... 25 2.3.1 Manually designed rational filtering in-loop ...... 25 2.3.2 Vector-fittedfilterfunctions ...... 29 2.3.3 Wienerfilters ...... 29 2.3.4 Prospects for almost-real-time filtering ...... 30 2.4 Safeguardandvetomethods ...... 31 2.4.1 Runtime safeguards ...... 31 2.4.2 Post-processing safeguards ...... 32 2.5 Feedforward results and discussion: MICH and PRC channels ...... 34 2.5.1 Filter fitting across science segments ...... 35

iv 2.5.2 Post-processing diagnostics ...... 39 2.5.3 Feedforward benefits and potential ...... 44 2.6 Conclusion ...... 44

III. Squeezing: Quantum Vacuum Phase Noise ...... 49

3.1 Squeezingtheory...... 49 3.1.1 Quantum shot noise and radiation pressure ...... 49 3.1.2 Problems with lasers: thermal compensation ...... 49 3.1.3 Squeezing filter cavities against alternatives ...... 49 3.2 LIGO Hanford Observatory quantum vacuum squeezing ...... 49 3.2.1 Collaboration and contributions ...... 49 3.2.2 Success and Advanced LIGO prospects ...... 51

IV. TwoSpect: Search for Scorpius X-1 ...... 52

4.1 Neutron stars in binary systems ...... 52 4.1.1 Binary spin-up and detectable lifetime ...... 52 4.1.2 Detectionrateprojections ...... 52 4.2 TwoSpectall-skysearches ...... 52 4.2.1 Two spectra: a double Fourier transform ...... 52 4.2.2 Infering neutron stars with companions ...... 52 4.3 Scorpius X-1 and results from Directed TwoSpect ...... 53

V. Directed TwoSpect: Neutron Star Binaries ...... 54

5.1 DirectedTwoSpect ...... 54 5.1.1 Target, directed and all-sky search sensitivity ...... 54 5.1.2 Enhancements enabled by directed searching ...... 54 5.2 QuantifyingDirectedness ...... 54

VI. Exhibit: World Science Festival ...... 55

6.1 Prototypes: travelling kiosks and the Ann Arbor Hands-On Museum . . . . 55 6.2 World Science Festival interferometer manufacture ...... 55 6.2.1 Laser, optics and display ...... 55 6.2.2 Aluminum baseboard ...... 55 6.2.3 Plexiglassenclosure...... 55 6.3 Exhibitions: New York City, Portsmouth, Fort Wayne ...... 55 6.3.1 Exhibitoverview ...... 55 6.3.2 World Science Festival 2010 ...... 56 6.3.3 Portsmouth and Fort Wayne ...... 56 6.4 FutureLIGOoutreach ...... 56

VII. Conclusion ...... 57

7.1 Cyclesofscience...... 57 7.1.1 Improvements to observatories ...... 57 7.1.2 Understanding instruments ...... 57 7.1.3 Refiningdata ...... 57 7.1.4 Searching deep-space ...... 57 7.1.5 Reaching out, looking up ...... 57 7.2 Scientific merit: filtering and analysis ...... 57 7.2.1 Feedforward improvement to LIGO data ...... 58

v 7.2.2 TwoSpect directed search for neutron stars in binary systems . . . 58 7.3 Enteringtheadvanceddetectorera ...... 58 7.4 Visionofadarksky...... 58

BIBLIOGRAPHY ...... 59

vi LIST OF FIGURES

Figure

2.1 Gravitational wave strain h(t) is derived from differential arm motion (DARM), read-out from a photodio 2.2 Sample coherence measurements between Hoft and auxiliary control channels for LIGO Hanford Observatory 2.3 Sample transfer function measurements (amplitude and phase) from LIGO Hanford Observatory, H1: 2010 2.4 Sample Bode plots of fitted ZPK filter functions (amplitude and phase) for multiple 1024 s windows in a science 2.5 Sample subtracted spectra for one window, representing the applied feedforward corrections for each channel 2.6 Calibration line test: before-feedforward mean of the 393.1 Hz line and two neighboring FFT bins was 1.3082 2.7 Time-domain plot of diagnostic channels from a burst injection: the simultaneous envelope increase after 1.8 2.8 Cross-correlation pairwise between pre-, post-feedforward, and injection data: the extrema and zero-crossings 2.9 Feedforward subtraction pipeline to read in Hoft (calibrated DARM), MICH, PRC, and write out AMPS 2.10 A depiction of windowing for one cluster job, containing one LIGO science segment, illustrates windowing 2.11 Exemplar of a typical case, +1.1 Mpc (5.9% inspiral range) (GPS time 953164819 to 953165839, 2010 Mar 2.12 Best improvement seen in S6 for H1, +4.4 Mpc (29% inspiral range) (GPS 955187679 to 955188191, 2010 2.13 Harmonic mean, 200 jobs from GPS second 931.0 106 (2009 July 07) to 932.8 106 (2009 July 28): (befor 2.14 Inspiral range vs time for Science Run 6 (starting× 2009 July 07) before GPS time× 9.33e8 (2009 July 30): LIGO 2.15 Inspiral range fractional gain vs time for Science Run 6 (starting 2009 July 07) before GPS time 9.33e8 (2009 2.16 Screenshot of diagnostic web pages, indexed by window...... 47

vii CHAPTER I

Introduction

1.1 Gravitational waves in astrophysics

Space’s metric echoes with gravitational waves. Light told the tale of the cosmos for most of history; now, the earliest epochs and secret reaches of stars might be seen in light interfering after travels transformed by gravity. General Relativity and ensuing theories of gravitation posit that accelerating quadropolar masses will radiate, much as accelerating dipolar charges do electromagnetically. In those waves we might see black holes and neutron stars colliding, supernova, the dawn of the Big

Bang and rotating neutron stars – and the potential for unanticipated insights, into other objects or law of gravity, is too tantalazing to ignore. Hulse and Taylor observed a neutron star in a binary system, PSR 1913+16, with an orbit accelerating just as gravitational radiation would entail; kilometer-scale interferometers were build at the end of the last millenium to look. Laser light in these instruments travels orthogonal paths and is reflected back; shifts in the combined pattern are scrutinized for indications that gravitational waves stretched space itself. As yet, no direct detections are known. This thesis describes efforts to make that search more sensitive with quantum optics at the observatories, by filtering the data of noise and by refining a search for the promising candidate source of neutrons stars in binary systems.

1 2

Astronomy has grown from humanity’s first glimpses into the night sky with the

unaided eye. With every new instrument, from Galileo’s telescope through radio an-

tennae and neutrino detectors, our understanding of the cosmos has grown. Gravity

pervades the universe like no other force: we must hear its tale. We know some of

what to expect: astronomers have predictions for four categories of cosmic sources.

We know how it would be emitted: Einstein’s general relativity predicts the inten-

sity, speed and polarization of gravitational waves. Most recently, LIGO, VIRGO,

GEO600 and soon KAGRA have built gravitational wave antennae that stand on the

threshold of detection. This thesis will focus on those antennae. Their sensitivity

can be honed by tweaking Heisenberg’s uncertainty principle with quantum optical

squeezing, and even improved post-facto by feedforward filtering of recorded servo

data. Neutrons stars in low-mass X-ray binary systems would live astronomically

long lives, earning the attention of a dedicated Fourier-domain frequentist search.

Each project is an element of a field that promises to make audible the echoes of the

metric of space.

1.1.1 Cosmic sources of gravitational waves

— Cosmic origins believed to generate GW. (note: should sprinkle citations as needed, not just where it says ”cite”) —

Gravity’s power to induce ripples in space is a matter of fact. Pulsar 1913+16, discovered by Hulse and Taylor, not only followed a pattern of orbital decay consistent with radiative loss of orbital decay to gravitational radiation – it continued to do so [88], even after Hulse and Taylor won the 1993 Nobel Prize in physics. We still may ask whether the waves are detectable on Earth. We may ask whether they appear in detectors in a way consistent with general relativity. The basic fact of their emission, however, appears settled. 3

Before delving into the specifics of general relativity, we might consider all the

astrophysical sources we expect to emit gravitational waves. Physics prompts our

search, but astronomy makes it exciting.

Gravitational waves (henceforth also abbreviated GW) searches presently focus on four distinct types of cosmic sources. This thesis concentrates on continuous waves, which are sine waves – perhaps modulated by orbital motion, spin-up or spin- down. Continuous waves are most likely to be detectable from neutron stars. Given a sufficiently large ellipticity, which might be on the order of ǫ 10−6 [CITE] or ≈ smaller for a neutron star rotating on the order of 1 kHz, a deformation of the crust would radiate sufficient gravitational radiation to be a plausibly-detectable source.

Indeed, the radiation would rapidly deplete the rotational energy of the neutron star [CITE], which is why binary systems, where the neutron star could be recycled and spun-up by a partner, prove a promising target [CITE]. Scorpius X-1 offers a canonical case, although our discussion of the TwoSpect analysis will elaborate the abundance of other low-mass X-ray binary (LMXB) systems of interest. Given the paucity of information on the interiors of collapsed stellar remnants, direct detection of gravitational waves from neutron stars could prove informative. We might infer details favoring one equation of state [CITE], might extract parameters suggesting the existence of quark stars or gravitars [CITE], and will have an unparalleled peek into the interior of the densest stable three-dimensional objects in the universe. Their simple waveforms might even facilitate the calibration of other types of gravitational wave data [CITE]. From any source, continuous waves are a conceptually-elegant and astronomically-enticing target.

Yet other sources of gravitational waves, as will be discussed in more detail later, have a comparable pull on our attention. Oft discussed, inspirals or compact binary 4

coalescences occur when two stellar remnants draw nearer in their orbits, radiating

gravitational radiation and finally merging in a titanic release of energy. While

sometimes invisible – the merging of black holes in short-hard gamma ray burts

(GRBs) proving an exception – these events compete eagerly with supernovae as

the most explosive in the modern universe. Were we to detect them directly with

gravitational waves, we would see their waveforms, which in turn can be predicted

through post-Newtonian approximation and numerical relativity. As the amplitude

would diminish linearly with distance, we would then have standard candles or ’sirens’

by which to calibrate and measure the universe. Advanced LIGO may prove sensitive

to neutron star-neutron star and stellar mass black hole-neutron star mergers, and,

if low-frequency sensitivity is sufficient and the sources exist, to intermediate-mass

black holes. Space-based observatories such as the longsuffering Laser Interferometer

Space Antenna (LISA), the DeciHertz Gravitational-wave Observatory (DECIGO),

Big Bang Observer (BBO) and proposed others could detect supermassive black hole

mergers. If fortunate, they would see a low-frequency noise floor due not to seismic

vibration, as in LIGO, but to white dwarf binaries throughout the galaxy. Since

the waveforms are well-predicted, we could even investigate deviations from general

relativity, perhaps seeing new physics in the ringdown of black holes.

Physical insight could also come from burst searches. Bursts share with inspiral searches the property of looking for a single event, as opposed to a source spread over duration. Burst is a somewhat general term, and analyses for them can sometimes be applied to inspiral or detector characterization tasks as well, yet the immediate focus lies with supernovae and perhaps gamma-ray bursts. Because the waveform is unknown, burst searches rely significantly more on the coincidence between multiple detectors to distinguish signal from noise. Just as with neutrino observations of su- 5

pernova 1987A, the burst program would hope for a fortuitously nearby cataclysm

to be seen simultaneously – or nearly so, the time of flight indicating a direction

– in a global gravitational-wave detector network. Due to the versatility of this

method, some researchers have proposed looking for non-general relativitic terms,

such as longitudinal polarization in addition to plus and cross orthogonal polariza-

tion. Any detection would be quiet exciting for probing systems still mysterious

with electromagnetic and neutrino measurements, and it would help, in conjunction

with multi-messenger coordinated searches with those observatories, to ascertain at

precisely what speed gravity travels through space-time and to what extent it is

attenuated or altered.

The background of space-time itself may itself how interesting physics and gravi- tational wave signatures. Searches for the stochastic gravitational wave background look not for events but for many months of correlated signals between networks of detectors. In doing so, they hope in particular to see the earliest turbulence of the universe – long before the cosmic electromagnetic background, now microwaves, was emitted 380000 years after the Big Bang, gravitational waves were travelling unim- peded. While the opacity of the infant cosmos conflates electromagnetic signals from different times and places, the transparency of the universe to gravity means that we might see the inflationary epoch or earlier, the Planck time, in gravitational waves.

Unfortunately, this signal is thought to be far below the sensitivity of existing detec- tors. While LIGO did set a new upper limit on the energy density of gravitational waves, measured as a fraction, Ωgw of the critical closure density of the universe [84], the inflationary background at LIGO frequencies is predicted to be about ten or- ders of magnitude lower. Alternative theories, such as ekpyrotic/cyclic universes, make other predictions, so an anomalously high stochastic background could prove 6

cosmologically significant.

All gravitational waves searches look for something. While the most exciting pos- sibility remains that we will see the unexpected, we think that our present divisions will permit serendipity while efficiently categorizing the computational challenges we do expect. Continuous wave and inspiral methods both search against waveform templates; burst and stochastic have no template and rely on correlation and coin- cidence. Continuous wave and stochastic analyze weeks, months, even years of data in search of persistant features; inspiral and burst look for transient events. In the abstract dimensions of search groups, we are complete. Our blind spots are in what data we provide to those groups – in the focus on audio frequencies of tens to a few thousand Hertz at present – blindness that will in time be rectified by CMB polarization, pulsar-timing and space-based interferometry for low frequencies and possibly by atom interferometry for high frequencies. To appreciate our choice of focus in these nascent days of the field, we must turn back a century to understand its origins in Einstein’s mathematics.

1.1.2 History from general relativity

Historical brief of Einstein.

It all began in 1915. In 1916, Einstein predicted gravitational waves, although he got some things wrong. Probably good to consult Gravitation [68] and Sean

Carroll [26] as well as proper history books.

1.1.3 Contrast with electromagnetic and particle astronomy

Contrast with electromagnetism, compare with radio/X-ray/et cetera astro. Can make analogy to radio waves and make note of the ease with which one can operate a small radio telescope, as I did in my thesis [65], compared the the difficulty of 7 gravitational wave detection. Muons from space were detected long ago, as in C.D.

Anderson’s 1949 paper on what was then called the mesotron, [56]. Compare with neutrinos as in the John Bahcall review paper [19], which were a well-established

field by the turn of the millenium, including the detection of supernova 1987A.

Might be worth comparing to the focus in new neutrino detectors that I had in my research and development work on them [22], [63]. Even with those detectors, how- ever, astrophysically-oriented detectors could be cross-referenced against terrestrial generators. Bahcall’s search for solar neutrinos, which were theoretical in 1964, [18], at least had the certainty that neutrinos had been detected from the Savannah River nuclear reactor. Yet when those solar neutrinos were detected, it a significant con-

firmation of nuclear theory. General relativity has been established by the 1913+16 pulsar [88] and would likely be much boosted by direct detection, and it might reach surprising new insights, analogous to neutrino oscillation found by looking at the solar neutrino spectrum.

1.2 General relativity

General relativity (the mathematics). The ideal reference here is Sean Carroll’s lecture notes on general relativity, [26], although I should also cite Will Farr’s the- sis [39] if it is elegant. Farr is good for citing things like the Palatini action. Of course, I should also ”dig down” and cite the original sources that they reference too, where applicable. Can also cite Misner, Thorne and Wheeler [68].

Intro to GR – right motivation is least action Ricci tensor, implying field equation and phase/time-of-flight implying interferometry. 8

1.2.1 Symmetry and action principles

Like electricity and magnetism, GR is the product of symmetry, action. This is the right place for [26].

1.2.2 Derivation of field equations

Derive field equations as extremized curvature.

It’s all about the Ricci tensor and the Einstein-Hilbert action. This might be the

right place for [?] and other sources.

1.2.3 Radiation from quadrupoles

Predict power from rotating quadrupole. Might also be a good place to invoke

Vladimir’s thesis [30] as well as original primary sources. Note that just as light

travels at the speed of light, which is measureable [69] and which can be derived

from electromagnetic theory [46], so we think that gravity should travel at the speed

of light. Note that this radiation should not be affected by the interstellar medium.

I think that Ostriker is the source to reference on the ISM [25], [61].

1.3 Astrophysical estimates

Astrophysical estimates and predictions.

1.3.1 Sources: burst, continuous, inspiral and stochastic

Describe the four types of sources: burst, continuous, inspiral and stochastic.

Mostly above, but clarify exact how much we should see. Cite the 2009 Nature

stochastic paper et al [84]. The importance to early universe cosmology was initially

handled by Maggiore [59]. Before that, of course, once can reference the Allen and

Romano methods paper [16], anything interesting from Nick Fotopoulos’s thesis [41],

and the various mid-2000s stochastic work that I was familiar with [6], [5]. Note the 9

interesting meaning of anisotropies [15] and point out how not only Stefan’s radiome-

ter search can find them but how it can be adapted to many other purposes, such as

spherical harmonics, which is useful for the cosmic microwave background [70] and

was briefly my work [64] and the Scorpius X-1 search; Stefan’s canonical radiometer

reference is in Classical Quantum Gravity [21].

1.3.2 Continuous waves from neutron stars

Continuous – specifically, binary neutron stars and rate predictions. Describe archetypical search methods as in the Abbott et al paper [10] and the most interesting to date, Vladimir’s PowerFlux [9]. Discuss earlier results from the earliest period [4] and an early search for Scorpius X-1 [8]. Reinhard Prix is probably another good source [73].

Rates go in here.

1.4 Laser Interferometer Gravitational-wave Observatories

LIGO observatories. The most fun part to write. Cite Nergis Mavalvala [60],

Stefan Ballmer [20], Rana Adhikari [12], Nicolas de Mateo Smith-Lefevbre [81] and,

of course, Peter Saulson [78].

1.4.1 From Weber bars to interferometry

History lesson: Weber bars progress to interferometers. Use Saulson, but Nic’s

thesis has links to some of the original sources [78], [81].

1.4.2 Gravitational wave interferometry methods

Michelson was one of the first to use interferometers [67]. He is famous for having

done so to try to measure the velocity of the Earth with respect to the luminerferous

ether and finding it to be unmeasurable. 10

Why GW interferometers work. Null measurements, a zeroed operating point.

The specifics are best handled by Saulson [78]. The idea of a Pound-Drever-Hall lock

is most elegantly explained by Black [23]. Rai Weiss may have some neat details,

possibly historical, albeit that it is in a presentation [89]. The details of Fabry-Perot

cavities in LIGO are handled by Rakhmanov, Savage et al [75], [74]. The motivation

for the evolution to Enhanced LIGO and its DC readout methods is covered well

in the corresponding CQG article [42] and specific details of its construction and

operation are in Tobin Fricke’s thesis [43].

Interferometer theory

GW interferometry theory: differential arm motion, key noise sources. Again, the main source is Saulson [78], although we also need another source in the Advanced

Detector Era.

Observatory operation

Operating LIGO: controls, Detector Characterization. One of the first sources from initial LIGO to read up on is a paper by Fritscel, Bork, Mavalvala et al [44].

Initially this system made a detection based on heterodyne readout using gravita- tional wave sidebands, which, among other troubles, could be unequal in the recycling cavity [62].

Detector characterization DetChar methods: omega scans, line hunting and glitches

Feedforward filtering Example of feedforward: 60 Hz magnetometer. The only source that mentions this is, I think, Nic’s thesis [81]. Yet the most pertinent example is one that has long been applied to LIGO: MICH and PRC feedforward. The specific needed are mention in the thesis of Adhikari [12] and Ballmer [20], but immediately 11 before Keita Kawabe and I began our project, these parameters had been tuned by

Jeff Kissel [52]

Phase camera Future devices: overview of phase camera with Vladimir. Vladimir’s thesis definitely talks about it [30]. We can discuss the fundamentals behind the need for angular stabilization and control from Nergis Mavalvala’s thesis [60], but we can refresh it with a modern reference from Kate Dooley’s thesis [33].

1.4.3 Advanced observatories and beyond

Advanced LIGO and beyond – squeezing and prospects?

1.4.4 Worldwide network

Allies: LIGO India, KAGRA, Advanced VIRGO, Einstein Telescope, LISA

1.5 Summary

Summary: strong motivation and instruments, need to find evidence of GW. CHAPTER II

Feedforward: Auxiliary MICH-PRC Subtraction

LIGO, the Laser Interferometer Gravitational-wave Observatory, has been de- signed and constructed to measure gravitational wave strain via differential arm length. The LIGO 4-km Michelson arms with Fabry-Perot cavities have auxiliary length control servos for suppressing Michelson motion of the beam-splitter and arm cavity input mirrors, which degrades interferometer sensitivity. We demonstrate how a post-facto pipeline called AMPS improves a data sample from LIGO Science Run 6 with feedforward subtraction. Dividing data into 1024-second windows, AMPS nu- merically fits filter functions representing the frequency-domain transfer functions from Michelson length channels into the gravitational-wave strain data channel for each window, then subtracts the filtered Michelson channel noise (witness) from the strain channel (target). In this paper we describe the algorithm, assess achievable improvements in sensitivity to astrophysical sources, and consider relevance to future interferometry.

2.1 Introduction

Antennae for gravitational wave observations [86] require precise understanding of noise sources to attain peak sensitivity. Some of these noises arise from auxil- iary degrees of freedom in interferometric antennae. Feedforward control can correct

12 13

these auxiliary control noises. Cluster computing on archived data makes previous

methods of feedforward correction scalable to year-long data-sets from science runs.

Computing can also adjust for the non-stationarity inherent in these noise couplings.

This paper describes such a computational method and the improvements it might

provide for searches with LIGO (Laser Interferometer Gravitational-wave Observa-

tory).

As a network with GEO600 [93, 50] and VIRGO [11], Enhanced LIGO [7, 42] produced data during LIGO Science Run 6 (S6) that was the most sensitive yet taken in the search for gravitational waves of astrophysical origin reaching the Earth: in this paper, we further enhance LIGO sensitivity via post-run software corrections. Radio astronomy of pulsar systems such as PSR 1913+16 [51] provides indirect evidence for gravitational radiation, and direct detection would inform the astrophysics of neutron stars [58,8], black holes [77], supernovae [27,71], cosmology [47], and related tests of the strong-field validity of general relativity [76]. This potential motivates new observatories, such as KAGRA [55], and improvements to existing observatories.

LIGO infers gravitational-wave strain h(t) at each of its two observatories [Han- ford, Washington and Livingston, Louisiana] from the length difference between 4-km

Michelson interferometer arms [78]. Each arm contains a Fabry-Perot resonant cav- ity locked using the Pound-Drever-Hall technique [24], comprised of an input test mass, near the Michelson beam-splitter, and an end test mass. A power-recycling mirror sits between the laser and the beam-splitter. These six core optics form cou- pled optical cavities with four length degrees of freedom, each of which is servoed to maintain optical resonance by minimizing motion. These are differential and com- mon motion of the arm length, differential Michelson length, and the power-recycling cavity length (see Figure 2.1). The effective change in the differential arm length 14

L− caused by gravitational waves is encoded in the intensity of the light reaching

the anti-symmetric port of the Michelson interferometer and is read out by DC ho-

modyne [42]. When the signal obtained, colloquially called DARM in LIGO, also

has a finite coupling to another degree of freedom, e.g., Michelson differential length,

any noise in the control of that degree of freedom is imprinted on DARM and com-

pounds a noise floor fundamentally limited by seismic, thermal suspension, and laser

shot noise. Auxiliary length control for the beam-splitter and input mirrors will

become more complex in future interferometers, such as Advanced LIGO, which

will add a signal recycling cavity. This paper describes post-facto software improve-

ments of detector noise using adaptive feedforward subtraction in a pipeline called

Auxiliary MICH-PRC Subtraction (AMPS) [57]: these improvements refine LIGO’s

gravitational-wave sensitivity to astrophysical sources.

AMPS improves LIGO S6 data (2009 July 07 to 2010 October 20), as this pa-

per will show. Feedforward subtraction corrects correlations between contaminated

strain signal (target channel) and noise (witness channels), yielding an estimate

of an uncontaminated signal [17]. In this paper, the S6 gravitational wave strain

channel is re-estimated based on noise witnessed in auxiliary length control servos.

Enhanced LIGO, used for S6, validated technologies, particularly high laser power

and DC readout, for Advanced LIGO [42]. The motion of the beam-splitter and

input mirrors of the Fabry-Perot cavities is known [12, 20] to cause cross-talk into

the gravitational wave strain channel, which is a calibrated readout primarily of dif-

ferential arm motion. Cross-talk was observed in S6 in channels for the differential

Michelson (MICH) as well as power-recycling cavity length (PRC)1. Methods [53] to

tune real-time feedforward filters for LIGO servo cross-talk are effective, but they

1The DARM readout, as explained in Section 2.2, is intrinsically sensitive to MICH divided by a factor of arm cavity gain, Equation 2.10. Theoretically, MICH responds to physical h(t), but the cavity gain and minuscule size of MICH make the effect about five orders of magnitude smaller than in DARM, so it is ignored. 15

require periodic manual retuning.

Post-facto, adaptive feedforward subtraction automates and simplifies cross-talk subtraction. The AMPS pipeline realizes this concept in Matlab 2012a [85]. The witness-to-target transfer function is estimated in discrete time windows of 1024 seconds and fit to a zero-pole-gain filter [32,49,48], with safeguards to ensure a statistically significant fit that does not further degrade the (target) signal. Noise from the witness channels passes through respective filters, then is subtracted from the strain target channel. AMPS increases gravitational-wave detector performance by lowering the noise floor. The lower noise floor could benefit any LIGO searches using this data.

2.2 Description of the feedforward method

Gravitational-wave antennae around the world share features and form a collabo- rative network. Amongst kilometer-scale Michelson interferometers, GEO600 [93] in Hannover, Germany uses folded arms with both power- and signal-recycling,

LIGO [7], and VIRGO [11] use Fabry-Perot cavities coupled with power- (and po- tentially signal-) recycling cavities. The Japanese interferometer KAGRA [55], un- der construction, will have a similar optical layout to LIGO and VIRGO but with cryogenically-cooled mirrors in an underground laboratory. Although nomenclature here pertains to LIGO, the core problem of this paper applies directly to all power- recycled Michelson interferometers with Fabry-Perot arms. It is theoretically exten- sible to other instruments with multiple degrees of freedom that obtain a signal from a particular target channel contaminated by control noise from auxiliary degrees of freedom, especially when those auxiliary witnesses are controlled using a lower signal-to-noise ratio error signal than the target and when the witnesses are highly 16

independent.

LIGO core optics include the beam-splitter (BS) and power-recycling mirror (PRM), which is situated between the laser and the beam-splitter. The four LIGO mirror test masses (TM) are named by arm (X or Y) and input (I) vs end (E) of the Fabry-Perot cavities. LIGO controls four optical pathlength degrees of freedom [7,44]. DARM is a signal of differential arm length, which is calibrated into the primary part of the gravitational strain measurement, h(t). CARM yields common arm length, and is controlled with a common mode servo using laser frequency. MICH Michelson and

PRC power-recycling cavity length refer only to input test masses. MICH and PRC must be servoed for LIGO to work, but sensing noise in the servo leads to elevated control noise, which induces noise in the physical Michelson and power-recycling cavity lengths. This self-inflicted noise motivates methods [17,12] for real-time feed- forward to cancel cross-talk into DARM. Post-facto feedforward can improve this cancellation: in this paper we regress to a transfer function of the cross-talk, convert it into a linear filter, safeguard the data against misfit filters, and quantify LIGO sensitivity improvements.

δ (L−(t)) (2.1) Strain: h(t)= , L h +i

δ(L + L ) (2.2) Common arm length: CARM δ(L )= y x , ∝ + 2

(2.3) Differential arm length: DARM δ(L−)= δ(L L ), ∝ y − x δ(l + l ) (2.4) Power-recycling cavity length: PRC δ(l )= y x , ∝ + 2

(2.5) (Inner) Michelson length: MICH δ(l−)= δ(l l ), ∝ y − x 17

(2.6) L z(ETMY) z(ITMY), y ≡ − (2.7) L z(ETMX) z(ITMX), x ≡ − (2.8) l z(ITMY) z(RM), y ≡ − (2.9) l z(ITMX) z(RM). x ≡ −

The value L is the nominal average arm length, about 4 km in LIGO. The h +i distance function z( ) indicates the distance2, along the optical path, from the laser X to an optic ; the variation δ denotes a change with respect to nominal value. X DARM length is thus defined as δ(L L ) and MICH length as δ(l l ). In y − x y − x practice, DARM and MICH are the names given to the channels that predominantly measure those quantities. These channels are not so neatly delineated as the lengths.

Unless stated otherwise, the terms DARM, MICH, PRC and CARM will refer to the measured channels, which are related to the lengths through calibration and are cross-contaminated (e.g., DARM = L− + π/(2 )l−, where is cavity fineesse). The F F terms will not refer to the physical lengths in Equations 2.2 through 2.5.

As Equation 2.3 and 2.5 imply and Figure 2.1 illustrates, MICH noise makes the

physical interpretation of DARM ambiguous. An arm cavity gain r′ /r 139/0.990, c c ≃ ′ where rc is the arm cavity reflectivity for the LIGO laser carrier frequency and rc is the derivative of rc with respect to round trip phase [44, 20], amplifies DARM motion for Initial and Enhanced LIGO. Where 219 is the cavity finesse, the F ≃ gain is given by Equation 2.10:

r′ 2 (2.10) c = F (139/0.990). rc π ≃ 2Note that z(ITMY) is a function of both the ITMY and BS position 18

Figure 2.1: Gravitational wave strain h(t) is derived from differential arm motion (DARM), read- out from a photodiode downstream of the antisymmetric port. An internal reflection off an anti-reflective coating, on either the beam-splitter (BS) or an input test mass (ITM), provides the Michelson (MICH) channel. The DARM readout channel predominantly measures the small change in different arm length, δ(L−) δ(L L ), while MICH ≡ y − x measures that in the Michelson length δ(l−) δ(l l ). There is also a small coupling ≡ y − x from δ(l−) to the DARM channel. To a lesser extent, changes in the length of PRC, which is defined as δ(l+) δ(ly + lx)/2 and is measured in quadrature demodulation with respect to the MICH≡ pick-off, also add noise to DARM. 19

A priori MICH noise will leak into measurements of DARM with a transfer func- tion equal to the inverse of Equation 2.10 [79]. Empirically, coherence measurements confirm this coupling is the dominant part of the transfer function, but residuals suggest other effects exist. PRC is also indirectly correlated with DARM. These correlations are physical consequences of the interferometer design.

In Enhanced LIGO, DARM is measured with a photodiode at the interferometer

‘dark’ antisymmetric port of the beams-splitter. Independent photodiodes for MICH and PRC, used for feedback on their respective auxiliary length control servos, pro- vide the witness channels for canceling cross-talk into DARM. The MICH and PRC photodiodes receive a beam from an internal reflection in the beam-splitter. This beam carries a radio-frequency modulation; one demodulation quadrature provides

MICH, the other PRC.

2.2.1 Auxiliary noise coherence at sensitive frequencies

Cross-talk can be quantified with coherence, the Fourier frequency-dependent ana- log of statistical correlation. On a scale of 0 (none) to 1 (full), coherence represents the normalized fraction of power of a frequency bin in the spectrum of one channel that can be found in the same frequency bin in the spectrum of another channel.

First, we must define the cross-power of a two time-series. Where Pxy is cross-power spectral density as in Equation 2.11.

Continuous and discrete definitions of cross-power P follow; the operator E de- notes an expectation value, and p and q are discrete indices: 20

+∞ +∞ ∗ i2πft (2.11) Pyx(f)= y (τ)x(t + τ)dτ e dt; −∞ −∞ Z Z  t −2πifq (2.12) Pyx(f, t)=Σq=−tRyx(q)e ,

∗ (2.13) Ryx(q)= E yp+qxp .   Now we can describe how the coherence at a given frequency f and time t [91] is

given by Equation 2.14. False positive rate for a coherence measure is governed by

the probability that an observed coherence Cobs is greater than the actual coherence

C. For a window factor w (0.95 for a Hann window) and N sample averages with

Gaussian noise, the false positive rate is described in [66] by Equation 2.15

2 Pxy(f, t) (2.14) Cxy(f, t)= | | , sPxx(f, t)Pyy(f, t) − (2.15) Prob C2 >C2 = 1 C2 w(N 1) obs −

Uncorrelated channels have low coherence. The calibrated strain channel, Hoft3, is measurably coherent with MICH and PRC, as seen in Figure 2.2. MICH-DARM coherence, hence noise coupling, is sometimes as large as 0.1 in the 100 to 300 Hz band for the LIGO Hanford Observatory detector H1. Problematically, this is the most sensitive band for Initial and Enhanced LIGO.

Allen, Hua, and Ottewill [17] first posited the filtering scheme that this paper employs. Where there is a strong correlation between a signal (target) channel and a noise (witness) channel, the noise can be partially cancelled if a witness-to- target transfer function, convolved with the witness, is applied to the measured target. Auxiliary MICH-PRC Subtraction (AMPS), a Matlab [85] pipeline, realizes this procedure.

3For clarity, we use Hoft to refer to the discrete-time calibrated strain data channel and h(t) to refer to the corresponding physical strain. 21

AMPS fits an infinite impulse response (IIR) filter onto the most coherent fre- quency band of the witness-to-target transfer function. It processes LIGO Science

Run 6 segment-by-segment, subdiving each segment into windows up to 1024 seconds in duration. S6 LIGO science segments exhibit MICH and PRC cross-talk primarily from 50 to 400 Hz. DARM is calibrated through a linear, frequency-dependent fil- ter [2] into the scientific channel Hoft. Since coherence and the technique of Allen,

Hua, and Ottewill are both linear and transitive, AMPS has been designed to analyze

Hoft data rather than analyzing DARM and duplicating the calibration filter.

Transfer function fitting is weighted toward 50-400 Hz by design in AMPS. Each transfer function for a given 1024 s of data is the average of 1024 independent ratios- of-Fourier-transforms (2047 Hann-windowed, 50%-overlapping FFTs of 1 s samples of the 1024 s). Since the error in an FFT scales with the inverse square root of the number of averages, the relative accuracy for the transfer function is 1024−1/2 . O AMPS can proceed with less data, as little as 32 s, which yield 32−1/2 relative
O accuracy. Outside the 50-400 Hz band, the program deweights the fit to the
transfer

α function and pre-processes it, suppressing it by factors of (f/fknee) , where α = 8 at low frequencies and 8 at high, where f was respectively 50 and 400 Hz. − knee AMPS smooths and deweights (Figure 2.3) known spectral peaks, including 60 Hz harmonics, the LIGO suspension violin modes, and calibration lines. Violin modes are internal resonances of the LIGO mirror suspensions caused by thermal noise; calibration lines are intentionally introduced narrow peaks in the spectrum used to help physically calibrate the interferometer in unit counts-to-length. De-weighting and pre-processsing, when carefully tuned, prevent AMPS from confusing the filter design with statistically insignificant transfer function data, which could introduce noise, as well as supporting faster and more accurate fit convergence with fewer free 22

parameters.

2.2.2 Estimating filters

Parameters for the transfer function are determined by Vectfit, developed by Gus- tavsen et al. [32,48,49]. Vectfit iteratively fits a state-space model to the numerically- estimated transfer function. AMPS calls Vectfit on one channel at a time, as will be discussed in Section 2.5. Each iteration of Vectfit uses pole-shifting to optimize a state-space model of the transfer function according to a least-squares fit. Empir- ically, convergence on a model was achieved after about five iterations on S6 data; we chose fifteen iterations for a safety margin, enforcing stable poles on the complex left half-plane. After sufficient iterations, the transfer function model (for AMPS,

32nd order) is extracted.

Before applying the filter, different representations of the transfer function model are used to validate key parameters. Zero-pole-gain (ZPK) format is used to trim out-of-band zeroes and poles and multiply by a 2nd order Butterworth low-pass

filter just below the Nyquist frequency of 8192 Hz, placing poles at 7 kHz to ensure causality. An overall scale factor is applied to the low-pass to keep the filter gain at

150 Hz the same value as the regression in Vectfit. AMPS converts the ZPK model to second-order-section (SOS) digital filter format for numerical stability. Instead of the inverse Fourier transform of Equation 2.17, this SOS model is converted from continuous time (or s-domain) to discrete time (or z-domain). Finally, using the discrete-time SOS representation, the IIR filter is applied to its respective witness channel. The estimated true Hoft target equals the original Hoft measurement minus the filtered witness channels.

This procedure makes two assumptions. It assumes that the dominant coupling from each witness channel into Hoft is linear. Further, it assumes that 2nd order 23

coupling, from one witness channel into the other and subsequently into Hoft, is neg-

ligible. Expectations and spot-check measurements confirm that these assumptions

are justified approximations.

Equations 2.16 through 2.18 capture this method. It is analogous to frequency- domain principal component analysis (PCA) using Gram-Schmidt orthogonalization.

Allen, Hua, and Ottewill established the mathematical theory, using superscript (b) to indicate a frequency band that we denote as domain (f). Their Equations 7, 8, and 11 correspond, respectively, to Equations 2.11, 2.18 and 2.16 here.

A transfer function T can be expressed as the cross-power ratio of arbitrary chan- nels x and y:

Pyx(f) (2.16) Txy(f)= . Pxx(f)

The estimated feedforward filter g can be viewed as an inverse Fourier transform

−1, decoupling signal (target, subscript s) from noise (witness, subscript n): F

(2.17) g(t)= −1 (fit [T (f)]) . F sn

Finally, the post-filtering signal (target)s ˆ can be written in terms of convolution (the

sign) with γ, the transfer function coupling noise (witness) into signal (target), s × pre-filter signal, n noise, and with channels indexed by j and curly brackets indicating

an observable quantity:

(2.18) sˆ(t)= s +Σ (γ n ) (t) Σ (g (t) n (t)) . { j j × j } − j j ×{ j}

Blind application of this method could theoretically produce unwarranted noise

reduction. Application of this paper’s method to uncorrelated channels would lead

to unwarranted noise reduction by an analytic factor of (1 1/F ) [17], where F − 24

is the number of bins in a fitted frequency span, or, in time-domain, the number

of averages. Given 1-s windowing with 50%-overlap on 1024 s, F = 2047, for an

unwarranted noise reduction of about 0.05%. The ideal of 1024-s windows is not

always achievable with LIGO duty cycles. In these cases, AMPS incorporates some

filters estimated on as little as 32 s of data, for which the unwarranted reduction

would be 3%, but only when these filters are averaged together with longer-duration

(512 s or greater) filters. No isolated filter is estimated on less than 60 s of data, which

could yield an uncorrelated noise reduction of 1.5%. Allen, Hua, and Ottewill clarify

that subtraction is tenable so long as covariance (in frequency-domain, coherence)

is present at a statistically significant level. They set a benchmark of an order-of-

magnitude above the magnitude-square covariance expectation value of 1/F . Since

the AMPS pipeline emphasize fits in regions where the coherence is greater than 3%,

and often 10% or more, it usually satisfies their criterion.

As described in Section 2.5.1, the filters calculated for each 1024-s window are

blended to generate the estimated target for a whole science segment. Figure 2.4

illustrates the short-term consistency of transfer functions during a few-hour science

segment, presenting a Bode plot of Hoft-MICH transfer function fits for consecutive

windows from 2010 March 21 at LIGO Hanford Observatory (magnitude, top and

phase, bottom vs frequency [Hz]). These transfer functions varied more markedly

over longer timescales of months, reflected in the fluctuating amount of improvement

seen in Figures 2.11 and 2.12 (covering 4 106 seconds of data). Figure 2.5 presents × a sample subtraction for one 1024-s window. These figures show that AMPS success- fully, efficiently realizes the intent of Allen, Hua, and Ottewill on operational data for a kilometer-scale gravitational wave interferometer.

Section 2.3 sets this method in context, Section 2.4 discusses safeguards and vetos, 25

Figure 2.2: Sample coherence measurements between Hoft and auxiliary control channels for LIGO Hanford Observatory, H1: 2010 March 21. Hoft-to-MICH coherence on left, Hoft-to- PRC coherence on right. Statistically significant coherence justifies fitting; in frequency bands, about 80 to 400 Hz, where coherence rose above background levels, the transfer function fit was weighted more heavily. Units of coherence spectral density (Hz−1/2) vs frequency (Hz).

and Section 2.5 discusses the details of implementation and verification.

2.3 Feedforward in-loop and alternative program methods

During LIGO running, real-time feedforward subtraction for MICH and PRC noise must meet operational constraints. Existing manually designed filters have certain tradeoffs compared to Vectfit-designed filtering; new methods, such as the Wiener

filtering now being considered for seismic and gravity-gradient cancellation, are also possible [34].

2.3.1 Manually designed rational filtering in-loop

Manual designs of feedforward functions prove time-consuming. Transfer func- tions can be manually fit using the LIGO Foton filtering system [80] or a comparable

filter design tool. Several additional transfer function measurements are needed for 26

Figure 2.3: Sample transfer function measurements (amplitude and phase) from LIGO Hanford Observatory, H1: 2010 March 21; MICH on left, PRC on right. Transfer function fit in coherent band – note the difference between raw data residual and the ‘pre-processed residual’, which has been smoothed and weighted to emphasize known-coherent bands. Units of amplitude spectral density (Hz−1/2) and phase (degrees) vs frequency (Hz). 27

Figure 2.4: Sample Bode plots of fitted ZPK filter functions (amplitude and phase) for multiple 1024 s windows in a science segment, at LIGO Hanford Observatory, H1: 2010 March 21; MICH on left, PRC on right. The similarity in the high-coherence, 80 to 400 Hz band leads us to conclude that the filter design is fairly stable throughout a science segment. Units of amplitude spectral density (Hz−1/2) and phase (degrees) vs frequency (Hz). 28

Figure 2.5: Sample subtracted spectra for one window, representing the applied feedforward cor- rections for each channel during that window, at LIGO Hanford Observatory, H1: 2010 March 21; MICH on left, PRC on right. Units of amplitude spectral density (Hz−1/2) vs frequency (Hz).

a filter deployed in-loop, in order to compensate for servo loop gain and actuation

functions. Manual design is not an efficient alternative to algorithmic transfer func-

tion design: it is labor-intensive to repeat regularly, and the results of our run over

S6 suggest that filter redesign should be performed often.

While labor-intensive, manual-designed rational filtering of MICH and PRC in- loop provides a necessary part of LIGO servo controls. The auxiliary control channels intrinsically introduce ‘self-inflicted’ noise into the DARM channel, and without real- time correction, the performance of its science runs would be much worse than design sensitivity. The majority of MICH/PRC subtraction comes from this real-time cor- rection; AMPS as used on S6 data is a comparatively small perturbation, one to two orders of magnitude smaller. 29

2.3.2 Vector-fitted filter functions

Algorithmic fitters such as Vectfit allow automated design of transfer functions.

Because it is automated, the filter can be made adaptive by periodic re-design. Since the dynamic range in magnitude for LIGO length control channels, and consequently transfer functions, can vary over tens of orders of magnitude, one must take care to weight and pre-process data, especially when operating in the most sensitive band from 100 to 300 Hz. Although not the only possibility, Vectfit, applied to a single witness-target pair, is straightforward. Vectfit iterates a rough prior set of pole locations, translating it across the complex plane according to a least-squares fit of the rational function onto transfer function data. About five iterations achieve good convergence. AMPS employs Vectfit’s report on cumulative goodness-of-fit, root- mean-square (RMS) error, as a statistic to decide whether to reject Vectfit’s filter regression. Using this statistic allows AMPS to isolate a failure in regression to one channel. From empirical studies, RMS error above 1 10−18 indicates a poor fit, × where the fit residual of frequency bin i is ∆ (i [1, ..., N]), and RMS error σ is i ∈ defined by Equation 2.19:

N 2 Σi=1 ∆i (2.19) σ = | | . q √N

2.3.3 Wiener filters

Wiener filtering [92] searches for an optimal filter of a particular kind: it mini- mizes the squared error of the residual. However, the error is over the entire spectrum.

Coherence at low-frequencies in MICH-PRC applications is low, but a Wiener filter by default will fit their high-power at the expense of lower-power, higher frequency bands. That fit would be statistical unsubstantiated and counterproductive, if cal- 30

culated for the entire spectrum at once. Filters that intentionally focus on bands

with high RMS power, such as seismic and gravity gradient noise, can use Wiener

filtering without difficulties. Several methods can be used to treat this challenge

for more general cases. Noise whitening [35,31] can use frequency-weighted cost

functions to limit noise injection from out-of-band. Data over the dynamic range of

LIGO’s bucket may also be addressed by band-passing the spectrum into sub-spectra.

Evaluating the error over sub-spectra could circumvent contamination from the high

spectral power present at low-frequencies. Transforms that break the spectrum up

into many bands, such as wavelet transforms [54], seem promising.

At the Caltech 40m prototype gravitational wave antenna, Wiener filtering has

been used to prototype measures for dealing with Newtonian and seismic noise in

advanced interferometers [34]. Wiener filtering is theoretically the optimal kind of

filtering for balancing the gain from feedforward correction against the harm done

when witness channels are themselves noisy. In the case of MICH and PRC, the

channels are relatively good witnesses, but in other situations, Wiener filtering might

provide superior performance. There is no reason to think that Wiener filtering would

not provide MICH and PRC feedforward subtraction equal to what is obtained using

a numerically determined transfer function (fit to IIR form using Vectifit). The latter

was computationally-compelling for the AMPS pipeline.

2.3.4 Prospects for almost-real-time filtering

Auxiliary MICH-PRC Subtraction (AMPS) code, as presented by this paper, is designed to provide out-of-loop, post-facto Vectfit feedforward subtraction that meets criteria for statistically soundness and accuracy. The software must also be efficient.

Auxiliary MICH-PRC Subtraction code runs a few times faster than real-time on

2013 hardware whilst conducting extensive tests and safeguards, documented in Sec- 31

tion 2.4. Unfortunately, failing the veto triggers extends the time-to-completion.

Long program time is acceptable for a side-channel generating in data when a refer-

ence Hoft, measuring h(t), already exists. This arrangement is not ideal for in-loop,

real-time production of h(t). Even if the vetos are not triggered, the maximum time

delay between a sample and its final, corrected form is the time at which the fil-

ter is generated, up to a window (1024-s) in the future. This lag is undesirable for

electromagnetic follow-up, part of the multi-messenger astronomy [38,13] desired for

Advanced LIGO.

A modified system that could operate in almost-real-time batches, refreshing peri- odically with prior data from the same science segment, might be useful for countering upconversion and other non-linear features of cross-coupling. One method for true real-time use of an AMPS-derived pipeline would be to output filters periodically for use in upcoming real-time data, that is, to use the windowing system to designate training sets. These features, however, are not yet implemented.

2.4 Safeguard and veto methods

2.4.1 Runtime safeguards

Gravitational wave strain, h(t), is the most important data channel for LIGO; the

AMPS pipeline takes several measures to ensure its integrity. Post-facto adjustments are unusual; when individual LIGO searches process h(t), they typically do not apply any correction based on auxiliary channel data. Such corrections have been the province of real-time servos. Real-time servo techniques work well, and if a problem is detected, it can often be fixed within a short commissioning break. Sometimes, the problem is an ephemeral ‘glitch’ with no straightforward solution. Should time and resources for an immediate fix not be available, it is advantageous to have methods for revisiting the data and attempting a correction after the fact. AMPS is useful 32

for providing this kind of correction, and by improving h(t) itself – rather than pre- processing data for a particular search – it potentially benefits all LIGO analyses.

Given the new approach here, AMPS verifies data integrity through several safe- guards and vetoes against known and possible issues. These issues include making data worse, offsetting and incorrectly time-stamping the data, incorrectly subtracting h(t) from itself, and introducing Fourier transform windowing artifacts.

After filtering each window, AMPS calculates its amplitude spectral density in

Matlab using Welch’s method (pwelch in Matlab) [85,90], which enables calculating inspiral range , an astrophysical measure of performance [40] and quantifying the R change in spectrum. The window’s filtered data is used, if post-filter is at least R 99.9% of unfiltered , and if none of the 40 points in a ‘comb’ (each point being 5/16 R Hz wide) of quiet bands is noisier in AMPS h(t) value than 1.2 times uncorrected,

Hoft h(t). If the window fails this test, it is passed back through the windowing method and combined with baseline, unfiltered LIGO data from the same window until it passed the veto. This process allows for progressive diminuation of the influence of any abrupt non-stationary features in the transfer function that may be causing the test to fail. Eight such ‘re-normalizing’ cycles are tried – if those fail, it then writes what it has and proceeds to the next segment. Empirically, it is found that these criteria ensure that almost all AMPS data is better, on average, than Hoft raw data, because it selects the better of either baseline or filtered data.

2.4.2 Post-processing safeguards

Besides imposing these runtime safeguards, the AMPS project computes diagnos- tics to check whether calibration lines (Figure 2.6) are diminished – which would indicate that AMPS was incorrectly subtracting h(t) from the Hoft channel. It also checks whether injections (Figures 2.7 and 2.8) are recovered at the correct times. 33

These injection studies examine compact binary coalescence and sine-Gaussian hard- ware injections [3] for GPS seconds 931.0 106 (2009 July 07) to 932.8 106 (2009 × × July 28) and produced basic time-domain plots and cross-correlations. In addition, to verify that the ability to observe compact binary coalescences is not decreased by AMPS, we calculate the matched-filter signal-to-noise ratio of each of the com- pact binary coalescence hardware injections in the S6 data. This is done by filtering each hardware injection against the compact binary coalescence waveform that was used when the hardware injection was performed. The matched filter signal-to- noise ratio is directly proportional to the distance at which such a signal can be observed, and therefore also to the overall sensitivity of the instrument to signals of that type [14,72]. These tests affirmed the result that feedforward data contains recoverable (and slightly higher signal-to-noise ratio) injections.

Calibration line studies seek to answer two questions: whether feedforward dis- torts the signal and whether it adds noise. The latter question is answered in the affir- mative if we find windowing artifacts, such as spectral combs with spacing of 1/1024 or 1/512 Hz, that might be generated around a prominent line. In post-processing, so-called ‘Short Fourier Transforms’ (SFTs) were made with a finer frequency resolu- tion (1/1800 Hz, by taking 1800 s stretches of data) from the generated AMPS time series. These SFTs are short in the sense of being shorter than the duration of a (po- tentially year-long) science run, and the 1800 s duration was chosen because it is the standard for LIGO continuous wave searches. The fine frequency resolution of SFTs should have made any artifacts visible. Post-processing tools saw no new combs in our 1800-s, 50%-overlapping Hann-windowed SFTs before/after comparison of ap- proximately 106 s of H1 science time between GPS times 931.0 106 (2009 July 07) × 34

and 932.8 106 (2009 July 28). The 393.1 Hz calibration line4 exhibited no noticeable × comb artifacts. The signal-distortion question is answered by taking the mean of the three 1/1800 bins in the range of [393.1 - 1/1800, 393.1 + 1/1800] Hz: for the 106 s of

H1 science time analyzed, the before-feedforward mean was 1.3082 10−21(Hz)−1/2, × whereas afterward it was 1.3128 10−21(Hz)−1/2. Feedforward made the calibration × line region noisier by 4.6 10−24(Hz)−1/2 or 0.35%. Evaluating the three bins around × the calibration line helps to account for possible spectral leakage, since the central line is much larger than its neighbors.

We conclude that we are not subtracting Hoft/DARM from itself. The additional noise is perhaps due to the low weighting of the AMPS transfer function fit at the

393.1 Hz line: that part of the spectrum is not tightly fit, because it could bias more sensitive parts of the spectrum.

From the nearly equal height of the calibration line before and after and the

identical lag of the before and after feedforward strains with respect to the injections,

we have confidence that AMPS is not subtracting h(t) from itself, and that it is not

shifting the samples in time.

2.5 Feedforward results and discussion: MICH and PRC channels

Realizing a post-facto, linear filtering engine, AMPS includes aletheia.m, a Mat- lab function in the Auxiliary MICH-PRC Subtraction package that fits and applies a filter to either MICH or PRC (specified by an argument). The fit occurs in the frequency-domain. The filter is applied in the time domain. From a high-level per- spective, it is a system for reading in Hoft frames, correcting them with MICH and

PRC frames, and writing AMPS data frames, accompanied by data quality and

4Strictly speaking, the line visible in Hoft is a residual artifact from the imperfect cancellation of control and error signals used in the construction of Hoft ∝ h(t) from the DARM error and DARM control signals. The accidental nature of the line does not affect our analysis, because neither MICH nor PRC affect the line. 35

Figure 2.6: Calibration line test: before-feedforward mean of the 393.1 Hz line and two neighboring FFT bins was 1.3082 10−21, after was 1.3128 10−21. Feedforward made the calibration line region noisier by× 4.6 10−24 or 0.35%,× suggesting that we correctly apply Hann- windowed feedforward without× subtracting true h(t). Moreover, no spectral line combs are observed to either side of the calibration line peak at 393.1 Hz, indicating that the method does not introduce windowing artifacts. state vector channels, and producing diagnostic graphs. The pipeline is shown in

Figure 2.9.

2.5.1 Filter fitting across science segments

So as to run efficiently on a LIGO computing cluster, the AMPS pipeline processes one filtering job per science segment. LIGO science segments range in duration from seconds to days, with the median segment typically lasting hours. During each segment, the interferometer is continuously locked, meaning it is held fixed on one interferometric fringe, by servo controlling all degrees of freedom so that they are stationary. If lock is lost, the segment ends. Serious noise degradations can also define the end of a segment. Each segment is managed by the main AMPS program, eleutheria.m. This program windows the segment, with windows up to 1024 s long, passes them to the filtering function and merges them smoothly together using 36

Figure 2.7: Time-domain plot of diagnostic channels from a burst injection: the simultaneous enve- lope increase after 1.8 s indicates that the burst injection is correctly time-stamped in the new data. ‘Before feedforward’ and ‘after feedforward’ traces occult each other in the graph, because they are almost identical. ‘Before feedforward’ is Hoft data; ‘after feedforward‘ is AMPS data with feedforward subtraction. ‘Injection estimated strain’ is the digital injection as intended to be introduced into strain, but the actual injection is made on the end test mass X (ETMX), so the calibrated ‘Injection estimated ETMX’ is also displayed, along with the raw ‘DARM’ differential arm channel and raw ‘ETMX’ channel.

Figure 2.8: Cross-correlation pairwise between pre-, post-feedforward, and injection data: the extrema and zero-crossings match. Note both before-feedforward (blue) and after- feedforward (green) strain traces are almost identical and therefore overlap. The strains appear inverted, but in the same way, due to a sign error in the hardware injections at this time. 37

Figure 2.9: Feedforward subtraction pipeline to read in Hoft (calibrated DARM), MICH, PRC, and write out AMPS (clean calibrated DARM). Data flows schematically from left to right; the MICH-Hoft stage output is used as input for the PRC-Hoft stage, then AMPS data frames are finally written. Code can be found in the LIGO Matapps SVN: matapps/packages/detchar/AMPS/trunk/aletheia.m 38

50%-overlapping Hann windowing. Windowing is idempotent for raw Hoft; the only

difference from window to window is the correction added to Hoft. The first 512

s are entirely driven by the first window; every 512 s thereafter, a new window is

introduced, as in Figure 2.10.

Using Equation 2.18 with filters g, target S = s +Σ (γ n ) and witness N = { j j × j } j n , we can evaluates ˆ(t). Since the filters for different channels are calculated in { j} series, with transfer functions T , Equation 2.20 has g T but g T − × . 1 ∼ S,N1 2 ∼ (S g1 N1),N2

In AMPS, S, N1 and N2 are respectively the Hoft, MICH and PRC channels.

(2.20) sˆ(t) = S(t) Σ (g (t) N(t)) , − j j × (2.21) = S(t) g (t) N (t) g (t) N (t), − 1 × 1 − 2 × 2 −1 −1 (2.22) S(t) fit [T ] N (t) fit T − × N (t). ∼ − F S,N1 × 1 − F (S g1 N1),N2 × 2  

Since N1(t) and N2(t) are added linearly to S(t), we can analyze them inde-

pendently. For clarity, analyze only the first two terms of Equation 2.21 and take

N(t) = N1(t). Let gA and gB be the earlier and later filters for N(t) being time-

domain merged in a Hann-window; they are respectively calculated from overlapping

data sets [SA, NA] and [SB, NB]. The sets are sample-for-sample identical at a given

time t, so S(t)= SA(t)= SB(t), N(t)= NA(t)= NB(t). AMPS merges data streams

sˆA ands ˆB over the window τ = 1024 s to effect the windowing, per Equation 2.23: 39

sˆ (t) 2π(t + τ ) sˆ (t) 2π(t + τ) (2.23) sˆ(t) = A 1 cos 2 + B 1 cos , 2 − τ 2 − τ     1 2πt (2.24) = sˆ (t)+ˆs (t)+cos [ˆs (t) sˆ (t)] , 2 A B τ A − B   2S(t) (g + g ) N(t) g g 2πt (2.25) = − A B × A − B N(t)cos , 2 − 2 × τ 1 2πt 2πt (2.26) = S(t) g 1+cos + g 1 cos N(t). − 2 A τ B − τ ×      Equation 2.26 shows that the windowing process is analogous to evolving filter coefficients with a 512 s cadence. Iterating, we substitutes ˆ(t) into S(t) with N(t)=

N2(t) for the next noise channel to obtain the same result.

Because of computing cluster constraints on job time, AMPS breaks up long science segments so that no job processes more than 16384 s of data; these subdivided science segments overlap slightly so that each side would calculate identical filters for the overlap(s), but only the latter half writes the frames for the overlap, avoiding race conditions5. Moreover, AMPS does not process science segments shorter than

60 s in the segment list, and science segments with dangling windows shorter than 32 s (after segment subdivision) have those windows rolled into their predecessors so as not to generate a filter based on insufficient data. Finally, AMPS implements range and comb tests to veto filtered windows if they look worse than unfiltered data, as discussed in Section 2.4.

2.5.2 Post-processing diagnostics

Spectra showing the lowering of the noise floor reveal the most visible sign of improvement. Figures 2.11 and 2.12 exemplify cases where the LIGO spectra look quiet after filtering, especially in the sensitive frequencies. Studies of many similar 5 Explicitly, label gW , gX , gY , gZ the final filters calculated in job 1; gA, gB , gC , gD are the first in job 2. Where each parenthesis contains 512 s and the addition sign denotes Hann-windowing of the filters, the end of job 1 is . . . (gW + gX )(gX + gY )(gY + gZ ) and the start of job 2 is (gA + gB )(gB + gC )(gC + gD). By overlap, we mean that filter gA is derived from the exact same data as filter gY , and likewise gB ≃ gZ . Thus (gA + gB )=(gY + gZ ). 40

Figure 2.10: A depiction of windowing for one cluster job, containing one LIGO science segment, illustrates windowing after an initial half-window offset. Filters are calculated for 1024-s windows, then the 50%-overlapping Hann windows merged, whereupon AMPS frames are written with a corrected measurement of h(t). Code can be found in the LIGO Matapps SVN: matapps/packages/detchar/AMPS/trunk/eleutheria.m 41

Figure 2.11: Exemplar of a typical case, +1.1 Mpc (5.9% inspiral range) (GPS time 953164819 to 953165839, 2010 March 21) spectra and tests against known injections suggest that auxiliary length feedforward correction improves spectra with elevated witness noise levels without degradation. It extends the sensitivity less noticeably when the interferometer is already performing optimally. This behavior accords with the understanding of LIGO thermal and shot noise, which AMPS cannot alter. Auxiliary length servos produce many glitches in

LIGO data, but these glitches contaminate the strain channel less when the auxiliary servo-to-strain coupling is minimized. As long as any non-stationarity in the coupling evolves more slowly than the 512-s timescale of our windowing, adaptive filtering appears to reduce the impact of such glitches. 42

Figure 2.12: Best improvement seen in S6 for H1, +4.4 Mpc (29% inspiral range) (GPS 955187679 to 955188191, 2010 April 13). Such a loud cross-coupling would be noticed in real-time by the on-site staff. 43

Figure 2.13: Harmonic mean, 200 jobs from GPS second 931.0 106 (2009 July 07) to 932.8 106 (2009 July 28): (before-after) (L), (before-after)/before× (R); greater than zero× is improvement.

To be precise, post-processing tests also compute average spectra over many win- dows, relying on the aforementioned Short Fourier Transforms (SFTs) of LHO data.

GPS seconds are measured from 1980 January 01: from GPS second 931.0 106 × (2009 July 07) to 932.8 106 (2009 July 28, roughly the first 200 science segments × of S6), we computed a harmonic mean of the spectra and compared differences and ratios, shown in Figure 2.13.

Note that the SFTs are high-pass filtered with a knee at 38 Hz. The harmonic mean spectrum is less susceptible to outliers than an arithmetic mean would be, and it shows a clear, several percent improvement in the band from about 80 Hz up to the violin mode frequencies. Above 400 Hz, there is slight degradation, but it is much smaller proportionally than the benefit. 44

2.5.3 Feedforward benefits and potential

Inspiral range , the maximum distance at which coalescing neutron stars are R likely to be detected, averaged over the sky, is one of the main LIGO figures of merit [40]. Range increases noticeably for both LIGO observatories when data from

Science Run 6 is feedforward filtered; post facto feedforward noise subtraction does improve performance. In future evolutions of this project, an explicit second-order correction for MICH-PRC coupling could yield marginally better performance. Real- time implementation and/or non-linear (frequency against frequency) methods might achieve more dramatic gains. Regardless, feedforward helps S6 sensitivity, and it should likewise help any observatory or science run with noise over a broad band due to residual contamination from auxiliary length control channels.

From Figures 2.14 and 2.15 we can infer the variation in the MICH and PRC couplings over a 12-day sample of S6 data. Figure 2.16 shows a screenshot of a webpage where LIGO data analysts can access summary graphs of the feedforward performance.

2.6 Conclusion

Auxiliary MICH-PRC Subtraction (AMPS) has been used to regenerate LIGO

S6 data, yielding better strain sensitivity and inspiral range. Cleaned of MICH and PRC noise, data frames currently reside on the LIGO Data Grid. Frequency- domain-derived, time-domain-applied feedforward correction removes these auxiliary length control noises by fitting a rational transfer function between witness & target.

Second order sections filter the noise (witness channels), which then subtract from the measured (target) signal to yield an improved estimate of gravitational wave strain. Matlab source code can be found at the Matapps repository [57]. Diagnostics 45

Figure 2.14: Inspiral range vs time for Science Run 6 (starting 2009 July 07) before GPS time 9.33e8 (2009 July 30): LIGO Hanford Observatory, H1 (top) gains 0.23 Mpc; LIGO Livingston Observatory, L1 (bottom) gains 0.84 Mpc. This graph shows the first month of S6; detector performance improved throughout the run. 46

Figure 2.15: Inspiral range fractional gain vs time for Science Run 6 (starting 2009 July 07) before GPS time 9.33e8 (2009 July 30): LIGO Hanford Observatory, H1 (top) 1.68% better; LIGO Livingston Observatory, L1 (bottom) 7.00% better 47

Figure 2.16: Screenshot of diagnostic web pages, indexed by window.

confirm that the corrected measurement of h(t) benefits from dynamic, adaptive,

algorithmic post facto feedforward subtraction, gaining several percent in detectable

inspiral range. Such an improvement potentially enhances the performance of any

search using LIGO data.

The subtraction leads to the lowest noise floor in strain sensitivity around 150

Hz of any time or interferometer so far6. This record may remain until Advanced

LIGO begins operation. When that time comes, adaptive feedforward filters, either real-time or post facto, can be applied to mitigate contamination from the noisy-but- inescapable parts of the tightly-coupled interferometer servo system. Signal recycling and filter cavities will only add to the complex challenge commissioning scientists face. Angular and length sensing degrees of freedom will both need finer control servos. Advanced LIGO will also contain many more physical and environmental monitors, from seismic and accelerometric to magnetic, which could provide witness channels for non-control-related noise. Altogether, many more auxiliary channels

6The highest performance to date at shot-noise limited frequencies has been obtained very differently, with quantum optical squeezing [1, 36] 48 and control loops will exist in addition to MICH and PRC, and while they may require more sophisticated, non-linear methods, the subtraction technique presented here is a point of reference. Sensitive interferometry in years to come will benefit from simple yet effective methods of suppressing auxiliary instrumental influences.

LIGO was constructed by the California Institute of Technology and Massachusetts

Institute of Technology with funding from the National Science Foundation and op- erates under cooperative agreement PHY-0757058. This paper carries LIGO Doc- ument Number LIGO-P1300193. This research was also made possible by the gen- erous support of the National Science Foundation, awards 0855422 and 1205173,

LIGO Hanford Observatory, the LIGO Scientific Collaboration, and the University of Michigan. The authors wish to thank Gregory Mendell as well as Stuart Anderson,

Juan Barayoga and Dan Kozak for grid computing expertise, Ian Harry for inves- tigating signal recovery before and after injections and providing conclusions about signal-to-noise ratio for matched filtering, and Jeff Kissel for refining MICH and PRC subtraction by hand. Tobin Fricke wrote the converter function from second-order- system to ZPK filtering and also reviewed this manuscript, as did Rana Adhikari and

Jenne Driggers, who developed many of these methods at the Caltech 40 m inter- ferometer. Michael Coughlin, Jan Harms, and Nicol´as Smith-Lefebvre all generously provided comments. CHAPTER III

Squeezing: Quantum Vacuum Phase Noise

(Fill in bits about the squeezer)

3.1 Squeezing theory

Squeezing theory.

3.1.1 Quantum shot noise and radiation pressure

Carlton Caves, quantum shot noise and radiation pressure.

3.1.2 Problems with lasers: thermal compensation

Experience (some firsthand) with thermal compensation.

3.1.3 Squeezing filter cavities against alternatives 3.2 LIGO Hanford Observatory quantum vacuum squeezing

Quantum vacuum squeezing at LIGO Hanford Observatory. Naturally, a great deal of description and background will come from Sheila Dwyer’s thesis [37] and

Sheon Chua’s thesis [28].

3.2.1 Collaboration and contributions

Contributions: table, in-vacuum installation, electronics, range est.

49 50

Optical table support assembly

Table legs (me) and results of Sheon and Robert’s shakers.

Here might be good place to put old AutoCAD drawings to use.

Faraday isolator measurement

Faraday isolator measurement (me with Keita, Matt, Lisa).

What were the results of the measurement? Show e-log entries, comment on in- and-out-of-vacuum performance and what it says about the need for low loss to be a top priority in future squeezing efforts.

In-vacuum installation

In-vacuum Faraday and baffle installation with ””.

Show pictures of the installation, connect to the issues with stray and perhaps backscattered light.

Discuss the repair of the output mode cleaner, which is mentioned (citation 50) in Nic’s thesis [81]. The technical report corresponding to it is by Waldman and

Chua [87].

Data digitization

Electronic cabling and analog-to-digital converter installation.

May want helpful diagram.

Figures of merit: inspiral range

Range estimation after squeezing

From the improved shot noise, we can see that squeezing at high frequencies

bought enhanced LIGO a megaparsec of inspiral range. This number is impressive

in several respects: our goal was to acheive a squeezing factor of perhaps as much 51

as 3 dB, but to do it in the shot noise-limited region, at high frequencies, where

the inspiral range equations (MATH: add the inspiral range equation if not already

shown for feedforward!) count for much less. Morever, that range figure reflects the

acheivement of squeezing down to 150 Hz (CITE: can we use this number?), which

is the lowest yet achieved for a gravitational wave interferometer.

3.2.2 Success and Advanced LIGO prospects

Results and hopes for aLIGO+ squeezing.

The squeezer group has a paper pending in review for Nature, written by Lisa

Barsotti, in which we discuss our acheivement of perhaps 2 dB worth of squeezing

(need to cite and check whether it is OK to use this number) [1]. It builds on the previous success of GEO600 in squeezing [83].

Discuss Sheila Dwyer’s [36] and Sheon Chua’s [29] papers, since I am an author on both of them. We have a preliminary understanding now of at least two ma- jor problems: the quadrature phase noise fluctatuations and backscattered light.

Backscattered light can be resolved in several ways. Phase noise must be progres- sively improved, as Sheila discusses, because we can hope to acheive the mature filter cavity design proposed below.

Discuss Lisa Barsotti’s talk about the future prospect for LIGO using filter cavi- ties, work that Tomoki Isogai is doing. With filter cavities, we can acheive frequency- dependent squeezing, having the best of both works by reducing quantum radiation pressure noise at low frequencies and shot noise at high, by using the filter cavity to produce a squeezed vacuum with a squeeze angle that varies as a function of fre- quency. Though as yet this filter cavity has yet to be constructed, it is in the works at MIT. CHAPTER IV

TwoSpect: Search for Scorpius X-1

4.1 Neutron stars in binary systems

Astrophysical prospects for binary pulsar detection. Binary pulsars are perhaps our best hope for detecting continuous gravitational waves.

4.1.1 Binary spin-up and detectable lifetime

GW pulsar lifetime alone vs companion.

4.1.2 Detection rate projections

aLIGO rate projections.

4.2 TwoSpect all-sky searches

TwoSpect methods as-is. These are described in detail in Evan Goetz’s thesis [45].

Note that the code is located on the web freely accessible in the LALApps reposi- tory [82].

4.2.1 Two spectra: a double Fourier transform

’Two spectra’ – FFT of periodograms reveals modulation of sine waves.

4.2.2 Infering neutron stars with companions

Infer whether modulation is due to a companion star.

52 53

4.3 Scorpius X-1 and results from Directed TwoSpect

Preliminary results of a directed search (possibly simulation-only).

We have a great deal of material here already, just need to pull from figures and commentary from the Sco X-1 wiki. Keith recommends paralleling the Sco X-1 paper. CHAPTER V

Directed TwoSpect: Neutron Star Binaries

5.1 Directed TwoSpect

TwoSpect improvements myself (to do).

5.1.1 Target, directed and all-sky search sensitivity

Targeted (known object) vs directed (region)vs all-sky (everything).

5.1.2 Enhancements enabled by directed searching

Modifications for directed search.

5.2 Quantifying Directedness

Quantify how good the improvements are in directed TwoSpect.

Cite Ethan’s thesis [?] and our paper, maybe by using generation functions to make better TwoSpect statistics.

54 CHAPTER VI

Exhibit: World Science Festival

6.1 Prototypes: travelling kiosks and the Ann Arbor Hands-On Museum

Prototypes: Ann Arbor Hands-On Museum and travelling kiosk.

6.2 World Science Festival interferometer manufacture

World Science Festival interferometer in isolation.

6.2.1 Laser, optics and display

Laser and optics (and display).

6.2.2 Aluminum baseboard

Aluminum base plate.

6.2.3 Plexiglass enclosure

Plexiglass, many lessons learned.

6.3 Exhibitions: New York City, Portsmouth, Fort Wayne

World Science Festival interferometer installed.

6.3.1 Exhibit overview

NYC exhibit overview: design, walls, kiosks, displays, interactivities.

55 56

6.3.2 World Science Festival 2010

Success in WSF 2010.

6.3.3 Portsmouth and Fort Wayne

Secondary installations and future outreach potential.

6.4 Future LIGO outreach

Future LIGO outreach? How to explain a new astronomy.

————————————– CHAPTER VII

Conclusion

7.1 Cycles of science

How it all fits together.

7.1.1 Improvements to observatories

Enhancements like enhanced/advanced LIGO and squeezing.

7.1.2 Understanding instruments

....necessitate detector characterization, like scans and filters

7.1.3 Refining data

....automated feedforward filters yield own enhancements

7.1.4 Searching deep-space

...TwoSpect and other searches benefit

7.1.5 Reaching out, looking up

...Outreach makes research accessible to public.

7.2 Scientific merit: filtering and analysis

Core projects.

57 58

7.2.1 Feedforward improvement to LIGO data

Evaluate success of feedforward.

7.2.2 TwoSpect directed search for neutron stars in binary systems

...and TwoSpect-directed.

One thought that might develop into something more fruitful is as follows.

Someday deconvolve, maybe Bayesian, the skymaps and parameter space spread of TwoSpect with simulation to understand what we are really seeing. Cannot do all templates in paramter space, but only need a few to compare – in a way, it already does.

7.3 Entering the advanced detector era

Advanced LIGO: how much better can we do?

7.4 Vision of a dark sky

Why GW astronomy at all? What could be out there? BIBLIOGRAPHY

59 60

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Directed searches for continuous gravitational waves from spinning neutron stars in

binary systems

by

Grant David Meadors

Chair: John Keith Riles

Gravitational wave (GW) detectors would reveal the universe in a way unlike any existing kind of telescope. These waves, predicted by general relativity, radiate from accelerating gravitational quadrupoles, such as black holes, neutron stars and the Big

Bang. Indirect evidence for GW comes from the work of Hulse and Taylor; the Laser

Interferometer Gravitational Wave Observatory (LIGO) and allies seek to observe gravitational waves directly. In this thesis, I discuss the goals and history of the

LIGO project and the theory and practice of its operation, including my contributions to the scientific collaboration in feedforward signal filtering, directed binary pulsar star searches, and scientific outreach and education to the next generation of young scientists. (Probably conclude with a sentence describing numerical results of my projects, when known.)