Search for gravitational-wave bursts in LIGO data
from the fourth science run
B Abbott14, R Abbott14, R Adhikari14, J Agresti14,
P Ajith2, B Allen2,51, R Amin18, S B Anderson14,
W G Anderson51, M Arain39, M Araya14, H Armandula14,
M Ashley4, S Aston38, P Aufmuth36, C Aulbert1, S Babak1,
S Ballmer14, H Bantilan8, B C Barish14, C Barker15,
D Barker15, B Barr40, P Barriga50, M A Barton40,
K Bayer17, K Belczynski24, J Betzwieser17,
P T Beyersdorf27, B Bhawal14, I A Bilenko21,
G Billingsley14, R Biswas51, E Black14, K Blackburn14,
L Blackburn17, D Blair50, B Bland15, J Bogenstahl40,
L Bogue16, R Bork14, V Boschi14, S Bose52, P R Brady51,
V B Braginsky21, J E Brau43, M Brinkmann2, A Brooks37,
D A Brown14,6, A Bullington30, A Bunkowski2,
A Buonanno41, O Burmeister2, D Busby14, R L Byer30,
L Cadonati17, G Cagnoli40, J B Camp22, J Cannizzo22,
K Cannon51, C A Cantley40, J Cao17, L Cardenas14,
M M Casey40, G Castaldi46, C Cepeda14, E Chalkey40,
P Charlton9, S Chatterji14, S Chelkowski2, Y Chen1,
F Chiadini45, D Chin42, E Chin50, J Chow4,
N Christensen8, J Clark40, P Cochrane2, T Cokelaer7,
C N Colacino38, R Coldwell39, R Conte45, D Cook15,
T Corbitt17, D Coward50, D Coyne14, J D E Creighton51,
T D Creighton14, R P Croce46, D R M Crooks40,
A M Cruise38, A Cumming40, J Dalrymple31,
E D’Ambrosio14, K Danzmann36,2, G Davies7, D DeBra30,
J Degallaix50, M Degree30, T Demma46, V Dergachev42,
S Desai32, R DeSalvo14, S Dhurandhar13, M D́ıaz33,
J Dickson4, A Di Credico31, G Diederichs36, A Dietz7,
E E Doomes29, R W P Drever5, J.-C Dumas50,
R J Dupuis14, J G Dwyer10, P Ehrens14, E Espinoza14,
T Etzel14, M Evans14, T Evans16, S Fairhurst7,14, Y Fan50,
D Fazi14, M M Fejer30, L S Finn32, V Fiumara45,
N Fotopoulos51, A Franzen36, K Y Franzen39, A Freise38,
R Frey43, T Fricke44, P Fritschel17, V V Frolov16, M Fyffe16,
V Galdi46, J Garofoli15, I Gholami1, J A Giaime16,18,
S Giampanis44, K D Giardina16, K Goda17, E Goetz42,
L M Goggin14, G González18, S Gossler4, A Grant40,
S Gras50, C Gray15, M Gray4, J Greenhalgh26,
A M Gretarsson11, R Grosso33, H Grote2, S Grunewald1,
M Guenther15, R Gustafson42, B Hage36, D Hammer51,
http://arxiv.org/abs/0704.0943v3
Search for gravitational-wave bursts in LIGO data 2
C Hanna18, J Hanson16, J Harms2, G Harry17, E Harstad43,
T Hayler26, J Heefner14, I S Heng40, A Heptonstall40,
M Heurs2, M Hewitson2, S Hild36, E Hirose31, D Hoak16,
D Hosken37, J Hough40, E Howell50, D Hoyland38,
S H Huttner40, D Ingram15, E Innerhofer17, M Ito43,
Y Itoh51, A Ivanov14, D Jackrel30, B Johnson15,
W W Johnson18, D I Jones47, G Jones7, R Jones40, L Ju50,
P Kalmus10, V Kalogera24, D Kasprzyk38,
E Katsavounidis17, K Kawabe15, S Kawamura23,
F Kawazoe23, W Kells14, D G Keppel14, F Ya Khalili21,
C Kim24, P King14, J S Kissel18, S Klimenko39,
K Kokeyama23, V Kondrashov14, R K Kopparapu18,
D Kozak14, B Krishnan1, P Kwee36, P K Lam4,
M Landry15, B Lantz30, A Lazzarini14, B Lee50, M Lei14,
J Leiner52, V Leonhardt23, I Leonor43, K Libbrecht14,
P Lindquist14, N A Lockerbie48, M Longo45, M Lormand16,
M Lubinski15, H Lück36,2, B Machenschalk1, M MacInnis17,
M Mageswaran14, K Mailand14, M Malec36, V Mandic14,
S Marano45, S Márka10, J Markowitz17, E Maros14,
I Martin40, J N Marx14, K Mason17, L Matone10,
V Matta45, N Mavalvala17, R McCarthy15,
D E McClelland4, S C McGuire29, M McHugh20,
K McKenzie4, J W C McNabb32, S McWilliams22,
T Meier36, A Melissinos44, G Mendell15, R A Mercer39,
S Meshkov14, E Messaritaki14, C J Messenger40,
D Meyers14, E Mikhailov17, S Mitra13, V P Mitrofanov21,
G Mitselmakher39, R Mittleman17, O Miyakawa14,
S Mohanty33, G Moreno15, K Mossavi2, C MowLowry4,
A Moylan4, D Mudge37, G Mueller39, S Mukherjee33,
H Müller-Ebhardt2, J Munch37, P Murray40, E Myers15,
J Myers15, T Nash14, G Newton40, A Nishizawa23,
K Numata22, B O’Reilly16, R O’Shaughnessy24,
D J Ottaway17, H Overmier16, B J Owen32, Y Pan41,
M A Papa1,51, V Parameshwaraiah15, P Patel14,
M Pedraza14, J Pelc17, S Penn12, V Pierro46, I M Pinto46,
M Pitkin40, H Pletsch2, M V Plissi40, F Postiglione45,
R Prix1, V Quetschke39, F Raab15, D Rabeling4,
H Radkins15, R Rahkola43, N Rainer2, M Rakhmanov32,
M Ramsunder32, K Rawlins17, S Ray-Majumder51, V Re38,
H Rehbein2, S Reid40, D H Reitze39, L Ribichini2,
R Riesen16, K Riles42, B Rivera15, N A Robertson14,40,
C Robinson7, E L Robinson38, S Roddy16, A Rodriguez18,
A M Rogan52, J Rollins10, J D Romano7, J Romie16,
R Route30, S Rowan40, A Rüdiger2, L Ruet17, P Russell14,
K Ryan15, S Sakata23, M Samidi14,
L Sancho de la Jordana35, V Sandberg15, V Sannibale14,
S Saraf25, P Sarin17, B S Sathyaprakash7, S Sato23,
P R Saulson31, R Savage15, P Savov6, S Schediwy50,
R Schilling2, R Schnabel2, R Schofield43, B F Schutz1,7,
Search for gravitational-wave bursts in LIGO data 3
P Schwinberg15, S M Scott4, A C Searle4, B Sears14,
F Seifert2, D Sellers16, A S Sengupta7, P Shawhan41,
D H Shoemaker17, A Sibley16, J A Sidles49, X Siemens14,6,
D Sigg15, S Sinha30, A M Sintes35,1, B J J Slagmolen4,
J Slutsky18, J R Smith2, M R Smith14, K Somiya2,1,
K A Strain40, D M Strom43, A Stuver32,
T Z Summerscales3, K.-X Sun30, M Sung18, P J Sutton14,
H Takahashi1, D B Tanner39, M Tarallo14, R Taylor14,
R Taylor40, J Thacker16, K A Thorne32, K S Thorne6,
A Thüring36, M Tinto14, K V Tokmakov40, C Torres33,
C Torrie40, G Traylor16, M Trias35, W Tyler14, D Ugolini34,
C Ungarelli38, K Urbanek30, H Vahlbruch36, M Vallisneri6,
C Van Den Broeck7, M Varvella14, S Vass14, A Vecchio38,
J Veitch40, P Veitch37, A Villar14, C Vorvick15,
S P Vyachanin21, S J Waldman14, L Wallace14, H Ward40,
R Ward14, K Watts16, D Webber14, A Weidner2,
M Weinert2, A Weinstein14, R Weiss17, S Wen18, K Wette4,
J T Whelan1, D M Whitbeck32, S E Whitcomb14,
B F Whiting39, C Wilkinson15, P A Willems14,
L Williams39, B Willke36,2, I Wilmut26, W Winkler2,
C C Wipf17, S Wise39, A G Wiseman51, G Woan40,
D Woods51, R Wooley16, J Worden15, W Wu39,
I Yakushin16, H Yamamoto14, Z Yan50, S Yoshida28,
N Yunes32, M Zanolin17, J Zhang42, L Zhang14, C Zhao50,
N Zotov19, M Zucker17, H zur Mühlen36 and J Zweizig14
(LIGO Scientific Collaboration)
1 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-14476
Golm, Germany
2 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167
Hannover, Germany
3 Andrews University, Berrien Springs, MI 49104 USA
4 Australian National University, Canberra, 0200, Australia
5 California Institute of Technology, Pasadena, CA 91125, USA
6 Caltech-CaRT, Pasadena, CA 91125, USA
7 Cardiff University, Cardiff, CF24 3AA, United Kingdom
8 Carleton College, Northfield, MN 55057, USA
9 Charles Sturt University, Wagga Wagga, NSW 2678, Australia
10 Columbia University, New York, NY 10027, USA
11 Embry-Riddle Aeronautical University, Prescott, AZ 86301 USA
12 Hobart and William Smith Colleges, Geneva, NY 14456, USA
13 Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India
14 LIGO - California Institute of Technology, Pasadena, CA 91125, USA
15 LIGO Hanford Observatory, Richland, WA 99352, USA
16 LIGO Livingston Observatory, Livingston, LA 70754, USA
17 LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA
18 Louisiana State University, Baton Rouge, LA 70803, USA
19 Louisiana Tech University, Ruston, LA 71272, USA
20 Loyola University, New Orleans, LA 70118, USA
21 Moscow State University, Moscow, 119992, Russia
22 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
23 National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
24 Northwestern University, Evanston, IL 60208, USA
25 Rochester Institute of Technology, Rochester, NY 14623, USA
26 Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United
Kingdom
Search for gravitational-wave bursts in LIGO data 4
27 San Jose State University, San Jose, CA 95192, USA
28 Southeastern Louisiana University, Hammond, LA 70402, USA
29 Southern University and A&M College, Baton Rouge, LA 70813, USA
30 Stanford University, Stanford, CA 94305, USA
31 Syracuse University, Syracuse, NY 13244, USA
32 The Pennsylvania State University, University Park, PA 16802, USA
33 The University of Texas at Brownsville and Texas Southmost College,
Brownsville, TX 78520, USA
34 Trinity University, San Antonio, TX 78212, USA
35 Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
36 Universität Hannover, D-30167 Hannover, Germany
37 University of Adelaide, Adelaide, SA 5005, Australia
38 University of Birmingham, Birmingham, B15 2TT, United Kingdom
39 University of Florida, Gainesville, FL 32611, USA
40 University of Glasgow, Glasgow, G12 8QQ, United Kingdom
41 University of Maryland, College Park, MD 20742 USA
42 University of Michigan, Ann Arbor, MI 48109, USA
43 University of Oregon, Eugene, OR 97403, USA
44 University of Rochester, Rochester, NY 14627, USA
45 University of Salerno, 84084 Fisciano (Salerno), Italy
46 University of Sannio at Benevento, I-82100 Benevento, Italy
47 University of Southampton, Southampton, SO17 1BJ, United Kingdom
48 University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
49 University of Washington, Seattle, WA, 98195
50 University of Western Australia, Crawley, WA 6009, Australia
51 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
52 Washington State University, Pullman, WA 99164, USA
E-mail: pshawhan@umd.edu
Abstract. The fourth science run of the LIGO and GEO 600 gravitational-wave
detectors, carried out in early 2005, collected data with significantly lower noise
than previous science runs. We report on a search for short-duration gravitational-
wave bursts with arbitrary waveform in the 64–1600 Hz frequency range appearing
in all three LIGO interferometers. Signal consistency tests, data quality cuts, and
auxiliary-channel vetoes are applied to reduce the rate of spurious triggers. No
gravitational-wave signals are detected in 15.5 days of live observation time; we
set a frequentist upper limit of 0.15 per day (at 90% confidence level) on the rate
of bursts with large enough amplitudes to be detected reliably. The amplitude
sensitivity of the search, characterized using Monte Carlo simulations, is several
times better than that of previous searches. We also provide rough estimates
of the distances at which representative supernova and binary black hole merger
signals could be detected with 50% efficiency by this analysis.
PACS numbers: 04.80.Nn, 95.30.Sf, 95.85.Sz
Submitted to Classical and Quantum Gravity
1. Introduction
Large interferometers are now being used to search for gravitational waves with
sufficient sensitivity to be able to detect signals from distant astrophysical sources.
At present, the three detectors of the Laser Interferometer Gravitational-wave
Observatory (LIGO) project [1] have achieved strain sensitivities consistent with their
design goals, while the GEO 600 [2] and Virgo [3] detectors are in the process of being
commissioned and are expected to reach comparable sensitivities. Experience gained
with these detectors, TAMA300 [4], and several small prototype interferometers has
Search for gravitational-wave bursts in LIGO data 5
nurtured advanced designs for future detector upgrades and new facilities, including
Advanced LIGO [5], Advanced Virgo [6], and the Large-scale Cryogenic Gravitational-
wave Telescope (LCGT) proposed to be constructed in Japan [7]. The LIGO Scientific
Collaboration (LSC) carries out the analysis of data collected by the LIGO and
GEO 600 gravitational-wave detectors, and has begun to pursue joint searches with
other collaborations (see, for example, [8]) as the network of operating detectors
evolves.
As the exploration of the gravitational-wave sky can now be carried out with
greater sensitivity than ever before, it is important to search for all plausible signals
in the data. In addition to well-modeled signals such as those from binary inspirals [9]
and spinning neutron stars [10], some astrophysical systems may emit gravitational
waves which are modeled imperfectly (if at all) and therefore cannot reliably be
searched for using matched filtering. Examples of such imperfectly-modeled systems
include binary mergers (despite recent advances in the fidelity of numerical relativity
calculations for at least some cases; see, for example, [11]) and stellar core collapse
events. For the latter, several sets of simulations have been carried out in the past
(see, for example, [12] and [13]), but more recent simulations have suggested a new
resonant core oscillation mechanism, driven by in-falling material, which appears to
power the supernova explosion and also to emit strong gravitational waves [14, 15].
Given the current uncertainties regarding gravitational wave emission by systems such
as these, as well as the possibility of detectable signals from other astrophysical sources
which are unknown or for which no attempt has been made to model gravitational
wave emission, it is desirable to cast a wide net.
In this article, we report the results of a search for gravitational-wave “bursts”
that is designed to be able to detect short-duration (≪ 1 s) signals of arbitrary form
as long as they have significant signal power in the most sensitive frequency band
of LIGO, considered here to be 64–1600 Hz. This analysis uses LIGO data from
the fourth science run carried out by the LSC, called S4, and uses the same basic
methods as previous LSC burst searches [17, 18] that were performed using data from
the S2 and S3 science runs. (A burst search was performed using data from the S1
science run using different methods [16].) We briefly describe the instruments and
data collection in section 2. In sections 3 and 4 we review the two complementary
signal processing methods—one based on locating signal power in excess of the baseline
noise and the other based on cross-correlating data streams—that are used together
to identify gravitational-wave event candidates. We note where the implementations
have been improved relative to the earlier searches and describe the signal consistency
tests which are based on the outputs from these tools. Section 5 describes additional
selection criteria which are used to “clean up” the data sample, reducing the average
rate of spurious triggers in the data. The complete analysis “pipeline” finds no event
candidates that pass all of the selection criteria, so we present in section 6 an upper
limit on the rate of gravitational-wave events which would be detected reliably by our
pipeline.
The detectability of a given type of burst, and thus the effective rate limit for a
particular astrophysical source model, depends on the signal waveform and amplitude;
in general, the detection efficiency (averaged over sky positions and arrival times) is less
than unity. We do not attempt a comprehensive survey of possible astrophysical signals
in this paper, but use a Monte Carlo method with a limited number of ad-hoc simulated
signals to evaluate the amplitude sensitivity of our pipeline, as described in section 7.
Overall, this search has much better sensitivity than previous searches, mostly due to
Search for gravitational-wave bursts in LIGO data 6
Mode Cleaner
Smoothes out fluctuations
of the input beam,
passes only fundamental
Gaussian beam mode
Stabilized
Laser
Power Recycling Mirror
(2.7% transmission)
Increases the stored power
by a factor of ~45, reducing
the photostatistics noise
Fabry-Perot Arm Cavity
Increases the sensitivity
to small length changes by
a factor of ~140
Photodiode
Input Mirror End Mirror
Beam Splitter
(50% transmission)
2 km or 4 km
Figure 1. Simplified optical layout of a LIGO interferometer.
using lower-noise data and partly due to improvements in the analysis pipeline. In
section 8 we estimate the amplitude sensitivity for certain modeled signals of interest
and calculate approximate distances at which those signals could be detected with 50%
efficiency. This completed S4 search sets the stage for burst searches now underway
using data from the S5 science run of the LIGO and GEO 600 detectors, which benefit
from much longer observation time and will be able to detect even weaker signals.
2. Instruments and data collection
LIGO comprises two observatory sites in the United States with a total of three
interferometers. As shown schematically in figure 1, the optical design is a Michelson
interferometer augmented with additional partially-transmitting mirrors to form
Fabry-Perot cavities in the arms and to “recycle” the outgoing beam power by
interfering it with the incoming beam. Servo systems are used to “lock” the mirror
positions to maintain resonance in the optical cavities, as well as to control the mirror
orientations, laser frequency and intensity, and many other degrees of freedom of the
apparatus. Interference between the two beams recombining at the beam splitter is
detected by photodiodes, providing a measure of the difference in arm lengths that
would be changed by a passing gravitational wave. The large mirrors which direct
the laser beams are suspended from wires, with the support structures isolated from
ground vibrations using stacks of inertial masses linked by damped springs. Active
feed-forward and feedback systems provide additional suppression of ground vibrations
for many of the degrees of freedom. The beam path of the interferometer, excluding
the laser light source and the photodiodes, is entirely enclosed in a vacuum system.
The LIGO Hanford Observatory in Washington state has two interferometers within
the same vacuum system, one with arms 4 km long (called H1) and the other with
arms 2 km long (called H2). The LIGO Livingston Observatory in Louisiana has a
single interferometer with 4 km long arms, called L1.
The response of an interferometer to a gravitational wave arriving at local time
Search for gravitational-wave bursts in LIGO data 7
t depends on the dimensionless strain amplitude and polarization of the wave and its
arrival direction with respect to the arms of the interferometer. In the low-frequency
limit, the differential strain signal detected by the interferometer (effective arm length
difference divided by the length of an arm) can be expressed as a projection of the
two polarization components of the gravitational wave, h+(t) and h×(t), with antenna
response factors F+(α, δ, t) and F×(α, δ, t):
hdet(t) = F+(α, δ, t)h+(t) + F×(α, δ, t)h×(t) , (1)
where α and δ are the right ascension and declination of the source. F+ and F× are
distinct for each interferometer site and change slowly with t over the course of a
sidereal day as the Earth’s rotation changes the orientation of the interferometer with
respect to the source location.
The electrical signal from the photodiode is filtered and digitized continuously at a
rate of 16 384 Hz. The time series of digitized values, referred to as the “gravitational-
wave channel” (GW channel), is recorded in a computer file, along with a timestamp
derived from the Global Positioning System (GPS) and additional information. The
relationship between a given gravitational-wave signal and the digitized time series is
measured in situ by imposing continuous sinusoidal position displacements of known
amplitude on some of the mirrors. These are called “calibration lines” because they
appear as narrow line features in a spectrogram of the GW channel.
Commissioning the LIGO interferometers has required several years of effort and
was the primary activity through late 2005. Beginning in 2000, a series of short data
collection runs was begun to establish operating procedures, test the detector systems
with stable configurations, and provide data for the development of data analysis
techniques. The first data collection run judged to have some scientific interest,
science run S1, was conducted in August-September 2002 with detector noise more
than two orders of magnitude higher than the design goal. Science runs S2 and S3
followed in 2003 with steadily improving detector noise, but with a poor duty cycle
for L1 due primarily to low-frequency, large-amplitude ground motion from human
activities and weather. During 2004, a hydraulic pre-isolation system was installed
and commissioned at the Livingston site to measure the ground motion and counteract
it with a relative displacement between the external and internal support structures
for the optical components, keeping the internal components much closer to an inertial
frame at frequencies above 0.1 Hz. At the same time, several improvements were made
to the H1 interferometer at Hanford to allow the laser power to be increased to the
full design power of 10 W.
The S4 science run, which lasted from 22 February to 23 March 2005, featured
good overall “science mode” duty cycles of 80.5%, 81.4%, and 74.5% for H1, H2,
and L1, respectively, corresponding to observation times of 570, 576, and 528 hours.
Thanks to the improvements made after the S3 run, the detector noise during S4 was
within a factor of two of the design goal over most of the frequency band, as shown in
figure 2. The GEO 600 interferometer also collected data throughout the S4 run, but
was over a factor of 100 less sensitive than the LIGO interferometers at 200 Hz and
a factor of few at and above the 1 kHz frequency range. The analysis approach used
in this article effectively requires a gravitational-wave signal to be distinguishable
above the noise in each of a fixed set of detectors, so it uses only the three LIGO
interferometers and not GEO 600. There are a total of 402 hours of S4 during which
all three LIGO interferometers were simultaneously collecting science-mode data.
Search for gravitational-wave bursts in LIGO data 8
100 1000
LIGO Detector Sensitivities During S4 Science Run
Frequency (Hz)
LIGO SRD goal (4 km)
Figure 2. Best achieved detector noise for the three LIGO interferometers during
the S4 science run, in terms of equivalent gravitational wave strain amplitude
spectral density. “LIGO SRD goal” is the sensitivity goal for the 4-km LIGO
interferometers set forth in the 1995 LIGO Science Requirements Document [19].
3. Trigger generation
The first stage of the burst search pipeline is to identify times when the GW channels
of the three interferometers appear to contain signal power in excess of the baseline
noise; these times, along with parameters derived from the data, are called “triggers”
and are used as input to later processing stages. As in previous searches [17, 18],
the WaveBurst algorithm [20] is used for this purpose; it will only be summarized
here [21].
WaveBurst performs a linear wavelet packet decomposition, using the symlet
wavelet basis [22], on short intervals of gravitational-wave data from each
interferometer. This decomposition produces a time-frequency map of the data similar
to a windowed Fourier transformation. A time-frequency data sample is referred to as a
pixel. Pixels containing significant excess signal power are selected in a non-parametric
way by ranking them with other pixels at nearby times and frequencies. As in the S3
analysis, WaveBurst has been configured for S4 to use six different time resolutions
and corresponding frequency resolutions, ranging from 1/16 s by 8 Hz to 1/512 s by
256 Hz, to be able to closely match the natural time-frequency properties of a variety
of burst signals. The wavelet decomposition is restricted to 64–2048 Hz. At any
given resolution, significant pixels from the three detector data streams are compared
and coincident pixels are selected; these are used to construct “clusters”, potentially
spanning many pixels in time and/or frequency, within which there is evidence for
a common signal appearing in the different detector data streams. These coincident
clusters form the basis for triggers, each of which is characterized by a central time,
Search for gravitational-wave bursts in LIGO data 9
Entries  8325975
Mean    2.583
RMS    0.9666
0 5 10 15 20 25 30 35 40
Figure 3. Distribution of Zg values for all WaveBurst triggers. The arrow shows
the location of the initial significance cut, Zg > 6.7.
duration, central frequency, frequency range, and overall significance Zg as defined
in [23]. Zg is calculated from the pixels in the cluster and is roughly proportional to
the geometric average of the excess signal power measured in the three interferometers,
relative to the average noise in each interferometer at the relevant frequency. Thus,
a large value of Zg indicates that the signal power in those pixels is highly unlikely
to have resulted from usual instrumental noise fluctuations. In addition, the absolute
strength of the signal detected by each interferometer within the sensitive frequency
band of the search is estimated in terms of the root-sum-squared amplitude of the
detected strain,
hrssdet =
|hdet(t)|
dt . (2)
WaveBurst was run on time intervals during which all three LIGO interferometers
were in science mode, but omitting periods when simulated signals were injected into
the interferometer hardware, any photodiode readout experienced an overflow, or the
data acquisition system was not operating. In addition, the last 30 seconds of each
science-mode data segment were omitted because it was observed that loss of “lock”
is sometimes preceded by a period of instability. These selection criteria reduced the
amount of data processed by WaveBurst from 402 hours to 391 hours.
For this analysis, triggers found by WaveBurst are initially required to have a
frequency range which overlaps 64–1600 Hz. An initial significance cut, Zg ≥ 6.7, is
applied to reject the bulk of the triggers and limit the number passed along to later
stages of the analysis. Figure 3 shows the distribution of Zg prior to applying this
significance cut.
Besides identifying truly simultaneous signals in the three data streams,
WaveBurst applies the same pixel matching and cluster coincidence tests to the three
data streams with many discrete relative time shifts imposed between the Hanford
and Livingston data streams, each much larger than the maximum light travel time
between the sites and the duration of the signals targeted by this search. The time-
shifted triggers found in this way provide a large sample to allow the “background”
(spurious triggers produced in response to detector noise in the absence of gravitational
waves) to be studied, under the assumption that the detector noise properties do not
Search for gravitational-wave bursts in LIGO data 10
−150 −100 −50 0 50 100 150
Mean = 41.1
χ2 = 130.5
d.o.f. = 97
WaveBurst trigger rate versus time shift
Time shift (s)
Figure 4. WaveBurst trigger rate as a function of the relative time shift applied
between the Hanford and Livingston data streams. The horizontal line is a fit to
a constant value, yielding a χ2 of 130.5 for 97 degrees of freedom.
vary much over the span of a few minutes and are independent at the two sites.
The two Hanford data streams are not shifted relative to one another, so that any
local environmental effects which influence both detectors are preserved. In fact,
some correlation in time is observed between noise transients in the H1 and H2 data
streams.
Initially, WaveBurst found triggers for 98 time shifts in multiples of 3.125 s
between −156.25 and −6.25 s and between +6.25 and +156.25 s. These 5119 triggers,
called the “tuning set”, were used to choose the parameters of the signal consistency
tests and additional selection criteria described in the following two sections. As
shown in figure 4, the rate of triggers in the tuning set is roughly constant for all time
shifts, with a marginal χ2 value but without any gross dependence on time shift. The
unshifted triggers were kept hidden throughout the tuning process, in order to avoid
the possibility of human bias in the choice of analysis parameters.
4. Signal consistency tests
The WaveBurst algorithm requires only a rough consistency among the different
detector data streams—namely, some apparent excess power in the same pixels
in the wavelet decomposition—to generate a trigger. This section describes more
sophisticated consistency tests based on the detailed content of the GW channels.
These tests succeed in eliminating most WaveBurst triggers in the data, while keeping
essentially all triggers generated in response to simulated gravitational-wave signals
added to the data streams. (The simulation method is described in section 7.) Similar
tests were also used in the S3 search [18].
Search for gravitational-wave bursts in LIGO data 11
 ]Hz [ strain / rssdetH1 h
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 ]Hz [ strain / rssdetH1 h
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(b) (c)
Figure 5. (a) Two-dimensional histogram, with bin count indicated by greyscale,
of H2 vs. H1 amplitudes reconstructed by WaveBurst for the tuning set of
time-shifted triggers. (b) Two-dimensional histogram of H2 vs. H1 amplitudes
reconstructed for simulated sine-Gaussian signals with a wide range of frequencies
and amplitudes from sources uniformly distributed over the sky (see section 7). In
these plots, the diagonal lines show the limits of the H1/H2 amplitude consistency
cut: 0.5 < ratio < 2 . (c) Two-dimensional histogram of L1 vs. H1 amplitudes for
the same simulated sine-Gaussian signals. Diagonal lines are drawn at ratios of
0.5 and 2 only to guide the eye; no cut is applied using this pair of interferometers.
4.1. H1/H2 amplitude consistency test
Because the two Hanford interferometers are co-located and co-aligned, they will
respond identically (in terms of strain) to any given gravitational wave. Thus, the
overall root-sum-squared amplitudes of the detected signals, estimated by WaveBurst
according to equation (2), should agree well if the estimation method is reliable.
Figure 5a shows that the time-shifted triggers in the tuning set often have poor
agreement between the detected signal amplitudes in H1 and H2. In contrast,
simulated signals injected into the data are found with amplitudes which usually agree
within a factor of 2, as shown in figure 5b. Therefore, we keep a trigger only if the
ratio of estimated signal amplitudes is in the range 0.5 to 2.
The Livingston interferometer is roughly aligned with the Hanford interferome-
ters, but the curvature of the Earth makes exact alignment impossible. The antenna
responses to a given gravitational wave will tend to be similar, but not reliably enough
to allow a consistency test which is both effective at rejecting noise triggers and effi-
cient at retaining simulated signals, as shown in figure 5c.
Search for gravitational-wave bursts in LIGO data 12
4.2. Cross-correlation consistency tests
The amplitude consistency test described in the previous subsection simply compares
scalar quantities derived from the data, without testing whether the waveforms are
similar in detail. We use a program called CorrPower [24], also used in the S3 burst
search [18], to calculate statistics based on Pearson’s linear correlation statistic,
i=1(xi − x̄)(yi − ȳ)
i=1(xi − x̄)
i=1(yi − ȳ)
. (3)
In the above expression {xi} and {yi} are sequences selected from the two GW channel
time series, possibly with a relative time shift, and x̄ and ȳ are their respective mean
values. The length of each sequence, N samples, corresponds to a chosen time window
(see below) over which the correlation is to be evaluated. r assumes values between
−1 for fully anti-correlated sequences and +1 for fully correlated sequences.
The r statistic measures the correlation between two data streams, such as
would be produced by a common gravitational-wave signal embedded in uncorrelated
detector noise [25]. It compares waveforms without being sensitive to the relative
amplitudes, and is thus complementary to the H1/H2 amplitude consistency test
described above. Furthermore, the r statistic may be used to test for a correlation
between H1 and L1 or between H2 and L1, even though these pairs consist of
interferometers with different antenna response factors, because each polarization
component will produce a measurable correlation for a suitable relative time delay
(unless the wave happens to arrive from one of the special directions for which one
of the detectors has a null response for that polarization component). In the special
case of a linearly polarized gravitational wave, the detected signals will simply differ
by a multiplicative factor, which can be either positive or negative depending on the
polarization angle and arrival direction.
Before calculating the r statistic for each detector pair, the data streams are
filtered to select the frequency band of interest (bandpass between 64 Hz and 1600 Hz)
and whitened to equalize the contribution of noise from all frequencies within this
band. The filtering is the same as was used in the S3 search [18] except for the
addition of a Q=10 notch filter, centered at 345 Hz, to avoid measuring correlations
from the prominent vibrational modes of the wires used to suspend the mirrors, which
are clustered around that frequency. The r statistic is then calculated over multiple
time windows with lengths of 20, 50, and 100 ms and a range of starting times,
densely placed (99% overlap) to cover the full duration of the trigger as reported by
WaveBurst; the maximum value from among these different time windows is used.
CorrPower [26] calculates two quantities, derived from the r statistic, which are
used to select triggers. The first of these, called R0, is simply the signed cross-
correlation between H1 and H2 with no relative time delay. Triggers with R0 < 0
are rejected. The second quantity, called Γ, combines the r-statistic values from the
three detector pairs, allowing relative time delays of up to 11 ms between H1 and L1
and between H2 and L1, and up to 1 ms between H1 and H2 (to allow for a possible
mismatch in time calibration). Specifically, Γ is the average of “confidence” values
calculated from the absolute value of each of the three individual r-statistic values.
A large value of Γ indicates that the data streams are correlated to an extent that
is highly unlikely to have resulted from normal instrumental noise fluctuations. This
quantity complements Zg, providing a different and largely independent means for
distinguishing real signals from background.
Search for gravitational-wave bursts in LIGO data 13
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(a) (b)
Figure 6. Plots of Γ versus Zg, after the H1/H2 amplitude consistency cut but
before any other cuts. (a) Scatter plot for all time-shifted triggers in the tuning
set. (b) Two-dimensional histogram, with bin count indicated by greyscale, for
simulated sine-Gaussian signals with a wide range of frequencies and amplitudes
from sources uniformly distributed over the sky (see section 7). In both plots, the
vertical dashed line indicates the initial WaveBurst significance cut at Zg=6.7.
Figure 6 shows plots of Γ vs. Zg for time-shifted triggers and for simulated
gravitational-wave signals after the H1/H2 amplitude consistency cut but before the
R0 cut. The time-shifted triggers with Γ < 12 and Zg < 20 are the tail of the bulk
distribution of triggers. The outliers with Γ > 12 all arise from a few distinct times
when large noise transients occurred in H1 and H2; these are found many times, paired
with different L1 time shifts, and have similar values of Γ because the calculation of
Γ is dominated by the H1-H2 pair in these cases. The outliers with Γ < 12 and
Zg > 20 are artefacts of sudden changes in the power line noise at 60 Hz and 180 Hz
which WaveBurst recorded as triggers. A cut on the value of Γ can eliminate many
of the time-shifted triggers in figure 6a, but at the cost of also rejecting weak genuine
gravitational-wave signals that may have the distribution in figure 6b. Therefore, the Γ
cut is chosen only after additional selection criteria have been applied; see section 5.3.
5. Additional selection criteria for event candidates
Environmental disturbances or instrumental misbehaviour occasionally produce non-
stationary noise in the GW channel of a detector which contributes to the recording of
a WaveBurst trigger. These triggers can sometimes pass the H1-H2 consistency and
cross-correlation consistency tests, particularly since an environmental disturbance
at the Hanford site affects both H1 and H2. As noted in the previous section, the
calculated value of Γ is susceptible to being dominated by the H1-H2 pair even if
there is minimal signal power in the L1 data stream. A significant background rate of
event candidates caused by environmental or instrumental effects could obscure the
rare gravitational-wave bursts that we seek, or else require us to apply more aggressive
cuts and thus lose sensitivity for weak signals.
This section describes the two general tactics we use to reject data with
identifiable problems and thereby reduce the rate of background triggers. First,
we make use of several “data quality flags” that have been introduced in order to
describe the status of the instruments and the quality of the recorded data over time
intervals ranging from seconds to hours. Second, we remove triggers attributed to
Search for gravitational-wave bursts in LIGO data 14
short-duration instrumental or environmental effects by applying “vetoes” based on
triggers generated from auxiliary channels which have been found to correlate with
transients in the GW channel. Applying data quality conditions and vetoes to the
data set reduces the amount of “live” observation time (or “livetime”) during which
an arriving gravitational-wave burst would be detected and kept as an event candidate
at the end of the analysis pipeline. Therefore, we must balance this loss (“deadtime”)
against the effectiveness for removing spurious triggers from the data sample.
Choosing data quality and veto conditions with reference to a sample of
gravitational-wave event candidates could introduce a selection bias and invalidate any
upper limit calculated from the sample. Therefore, we have evaluated the relevance
of potential data quality cuts and veto conditions using other trigger samples. In
addition to the tuning set of time-shifted WaveBurst triggers, we have applied the
KleineWelle [27] method to identify transients in each interferometer’s GW channel.
(We have also used KleineWelle to identify transients in numerous auxiliary channels
for veto studies, as described in 5.2.) Like WaveBurst, KleineWelle is a time-frequency
method utilizing multi-resolution wavelet decomposition, but it processes each data
channel independently [28]. In analyzing data, the time series is first whitened using
a linear predictor error filter [27]. Then the time-frequency decomposition is obtained
using the Haar wavelet transform. The squared wavelet coefficients normalized to the
scale’s (frequency’s) root-mean-square provide an estimate of the energy associated
with a certain time-frequency pixel. A clustering mechanism is invoked in order to
increase the sensitivity to signals with less than optimal shapes in the time-frequency
plane and a total normalized cluster energy is computed. The significance of a
cluster is then defined as the negative natural logarithm of the probability of the
computed total normalized cluster energy to have resulted from Gaussian white noise;
we apply a threshold on this significance to define KleineWelle triggers. The samples
of KleineWelle triggers from each detector, as well as the subsample of coincident
H1 and H2 triggers, are useful indicators of localized disturbances. They may in
principle contain one or more genuine gravitational-wave signals, but decisions about
data quality and veto conditions are based on the statistics of the entire sample which
is dominated by instrumental artefacts and noise fluctuations.
5.1. Data quality conditions
We wish to reject instances of clear hardware problems with the LIGO detectors
or conditions that could affect our ability to unequivocally register the passage of
gravitational-wave bursts. Various studies of the data, performed during and after
data collection, produced a catalog of conditions that might affect the quality of the
data. Each named condition, or “flag”, has an associated list of time intervals during
which the condition is present, derived either from one or more diagnostic channels
or from entries made in the electronic logbook by operators and scientific monitors.
We have looked for significant correlations between the flagged time intervals and
time-shifted WaveBurst triggers, and also between the flagged time intervals and
KleineWelle single-detector triggers (particularly the “outliers” with large significance
and the coincident H1 and H2 triggers). Based on these studies, we decided to impose
a number of data quality conditions.
We first require the calibration lines to be continuously present. On several
occasions when they dropped out briefly, due to a problem with the excitation engine,
the data is removed from the analysis. The livetime associated with these occurrences
Search for gravitational-wave bursts in LIGO data 15
is negligible while they are all correlated with transients appearing in the GW channel.
Local winds and sound from airplanes may couple to the instrument through
the ground and result in elevated noise and/or impulsive signals. A data quality
flag was established to identify intervals of local winds at the sites with speeds of
56 km/hour (35 miles per hour) and above. We studied the correlation of these times
with the single-detector triggers produced with KleineWelle. The correlation is more
apparent in the H2 detector, for which 7.4% of the most significant KleineWelle triggers
(threshold of 1600) coincide with the intervals of strong winds at the Hanford site. The
livetime that is rejected in this way is 0.66% of the H1-H2 coincident observation time
over which this study was performed. Thanks to improved acoustic isolation installed
after the S2 science run, acoustic noise from airplanes was not found to contribute
to triggers in the GW channel in general; however, a period of 300 seconds has been
rejected around a particularly loud time when a fighter jet passed over the Hanford
site.
Elevated low-frequency seismic activity has been observed to cause noise
fluctuations and transients in the GW channel. Data from several seismometers at the
Hanford observatory was band-pass filtered in various narrow bands between 0.4 Hz
and 2.4 Hz, and the root-mean-square signal in each band was tracked over time. A
set of particularly relevant seismometers and bands was selected, and time intervals
were flagged whenever a band in this set exceeded 7 times its median value. A follow
up analysis of the single instrument as well as coincident H1-H2 KleineWelle triggers
found significant correlation with the elevated seismic noise. The strongest correlation
is observed in the outlier triggers (KleineWelle significance of 1600 or greater) in H2,
of which 41.9% coincide with the seismic flags, compared to a deadtime of 0.6%.
In the two Hanford detectors, a diagnostic channel counting ADC overflows in the
length sensing and control subsystem was used to flag intervals for exclusion from the
analysis. One minute of livetime around these overflows is rejected. Such overflows
were indeed seen to correlate with single-detector outlier triggers in H1 (44.4% of
them, with 0.68% deadtime) and H2 (74.1% of them, with 0.41% deadtime).
Two data quality cuts are derived from “trend” data (summaries of minimum,
maximum, mean and root-mean-square values over each one-second period)
monitoring the interferometry used in the LIGO detectors. The first one is based
on occasional transient dips in the stored light in the arm cavities. These have been
identified by scanning the trend data for the relevant monitoring photodiodes, defining
the size of a dip as the fractional drop of the minimum in that second relative to the
average of the previous ten seconds, and applying various thresholds on the minimum
dip size. For the three LIGO detectors, thresholds of 5%, 4% and 5% respectively
for L1, H1 and H2 are used. High correlation of such light dips with single-detector
triggers is observed, while the deadtime resulting from them in each of the three LIGO
instruments is less than 0.6%. The second data quality cut of this type is based on the
DC level of light reaching the photodiode at the output of the interferometer, which
sees very little light when the interferometer is operating properly. By thresholding
on the trend data for this channel, intervals when its value was unusually high are
identified in H1 and L1. These intervals are seen to correlate with instrument outlier
triggers significantly. The deadtime resulting from them is 1.02% in H1 and 1.74% in
Altogether, these data quality cuts result in a net loss of observation time of 5.6%.
Search for gravitational-wave bursts in LIGO data 16
5.2. Auxiliary-channel vetoes
LIGO records thousands of auxiliary read-back channels of the servo control systems
employed in the instruments’ interferometric operation as well as auxiliary channels
monitoring the instruments’ physical environment. There are plausible couplings of
environmental disturbances or servo instabilities both to these monitoring channels
and to the GW channel; thus, transients appearing in these auxiliary channels may
be used to veto triggers seen simultaneously in the GW channel. This assumes that
a genuine gravitational-wave burst would not appear in these auxiliary channels, or
at least that any coupling is small enough to stay below the threshold for selecting
transients in these channels.
We have used KleineWelle to produce triggers from over 100 different auxiliary
channels that monitor the interferometry and the environment in the three LIGO
detectors. A first analysis of single-detector KleineWelle triggers from the L1 GW
channel and coincident KleineWelle triggers from the H1 and H2 GW channels
against respective auxiliary channels identified the ones that showed high GW channel
trigger rejection power with minimal livetime loss (in the vast majority of channels
much less that 1%). In addition to interferometric channels, environmental ones
(accelerometers and microphones) located on the optical tables holding the output
optics and photodiodes appeared to correlate with GW channel triggers recorded at
the same site.
Auxiliary interferometric channels (besides the GW channel) could in principle
be affected by a gravitational wave, and a veto condition derived from such a channel
could reject a genuine signal. Hardware signal injections imitating the passage of
gravitational waves through our detectors, performed at several pre-determined times
during the run, have been used to establish under what conditions each channel is
safe to use as a veto. Non-detection of a hardware injection by an auxiliary channel
suggests the unconditional safety of this channel as a veto in the search, assuming that
a reasonably broad selection of signal strengths and frequencies were injected. But
even if hardware injections are seen in the auxiliary channels, conditions can readily
be derived under which no triggers caused by the hardware injections are used as
vetoes. This involves imposing conditions on the significance of the trigger and/or
on the ratio of the signal strength seen in the auxiliary channel to that seen in the
GW channel. We have thus established the conditions under which several channels
involved in the length and angular sensing and control systems of the interferometers
can be used safely as vetoes. (The data quality conditions described in section 5.1
were also verified to be safe using hardware injections.)
The final choice of vetoes was made by examining the tuning set of time-
shifted triggers remaining in the WaveBurst search pipeline after applying the signal
consistency tests and data quality conditions. The ten triggers from the time-shifted
analysis with the largest values of Γ, plus the ten with the largest values of Zg, were
examined and six of them were found to coincide with transients in one or more of the
following channels: the in-phase and quadrature-phase demodulated signals from the
pick-off beam from the H1 beamsplitter, the in-phase demodulated pitch signal from
one of the wavefront sensors used in the H1 alignment sensing and control system, the
beam splitter pitch and yaw control signals, and accelerometer readings on the optical
tables holding the H1 and H2 output optics and photodiodes. KleineWelle triggers
produced from these seven auxiliary channels were clustered (with a 250 ms window)
and their union was taken. This defines the final list of veto triggers for this search,
Search for gravitational-wave bursts in LIGO data 17
each indicating a time interval (generally ≪ 1 s long) to be vetoed.
The total duration of the veto triggers considered in this analysis is at the level
of 0.15% of the total livetime. However, this does not reliably reflect the deadtime
of the search since a GW channel trigger is vetoed if it has any overlap with a veto
trigger. Thus, the actual deadtime of the search depends on the duration of the
signal being sought, as reconstructed by WaveBurst. We reproduce this effect in
the Monte Carlo simulation used to estimate the efficiency of the search (described
in section 7) by applying the same analysis pipeline and veto logic. The effective
deadtime depends on the morphology of the signal and on the signal amplitude, since
larger-amplitude signals tend to be assigned longer durations by WaveBurst. For the
majority of waveforms we considered in this search and for plausible signals strengths,
the resulting effective deadtime is of the order of 2%. Because this loss is signal-
dependent, in this analysis we consider it to be a loss of efficiency rather than a loss
of live observation time; in other words, the live observation time we state reflects the
data quality cuts applied but does not reflect the auxiliary-channel vetoes.
5.3. Gamma cut
The cuts described above cleaned up the outliers in the data considerably, as shown by
the sequence of scatter plots in figure 7. Following the data quality and veto criteria
we just described, the remaining time-shifted WaveBurst triggers (shown in figure 7d)
were used as the basis for choosing the cross correlation Γ threshold. As with previous
all-sky searches for gravitational-wave bursts with LIGO, we desire the number of
background triggers expected for the duration of the observation to be much less than
1 but not zero, typically of order ∼ 0.1. On that basis, we chose a threshold of Γ > 4
which results in 7 triggers in 98 time shifts, or 0.08 such triggers normalized to the
duration of the S4 observation time.
6. Search results
After all of the trigger selection criteria had been established using the tuning set of
time-shifted triggers, WaveBurst was re-run with a new, essentially independent set
of 100 time shifts, in increments of 5 s from −250 s to −5 s and from +5 s to +250 s,
in order to provide an estimate of the background which is minimally biased by the
choice of selection criteria. The total effective livetime for the time-shifted sample is
77.4 times the unshifted observation time, reflecting the reduced overlap of Hanford
and Livingston data segments when shifted relative to one another. The unshifted
triggers were looked at for the first time. Table 1 summarizes the trigger counts for
these time-shifted and unshifted triggers at each stage in the sequence of cuts. In
addition, the expected background at each stage (time-shifted triggers normalized to
the S4 observation time) is shown for direct comparison with the observed zero-lag
counts. Figure 8 shows a scatter plot of Γ vs. Zg and histograms of Γ for both time-
shifted and unshifted triggers after all other cuts. These new time-shifted triggers
are statistically consistent with the tuning set (figure 7d), although no triggers are
found with Zg > 15 in this case. Five unshifted triggers are found, distributed in a
manner reasonably consistent with the background. All five have Γ<4 and thus fail
the Γ cut. Three time-shifted triggers pass the Γ cut, corresponding to an estimated
average background of 0.04 triggers over the S4 observation time.
Search for gravitational-wave bursts in LIGO data 18
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(a) (b)
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(c) (d)
Figure 7. Scatter plots of Γ versus Zg for the tuning set of time-shifted triggers.
(a) All triggers; (b) after data quality cuts; (c) after data quality and H1-
H2 consistency cuts (amplitude ratio and R0); (d) after data quality, H1-H2
consistency, and auxiliary-channel vetoes.
Table 1. Counts of time-shifted and unshifted triggers as cuts are applied
sequentially. The column labeled “Normalized” is the time-shifted count divided
by 77.4, representing an estimate of the expected background for the S4
observation time.
Time-shifted
Unshifted
Cut Count Normalized Count
Data quality 3153 40.7 44
H1/H2 amplitude consistency 1504 19.4 14
R0 > 0 755 9.8 5
Auxiliary-channel vetoes 671 8.7 5
Γ > 4 3 0.04 0
With no unshifted triggers in the final sample, we place an upper limit on the
mean rate of gravitational-wave events that would be detected reliably (i.e., with
efficiency near unity) by this analysis pipeline. Since the background estimate is small
and is subject to some systematic uncertainties, we simply take it to be zero for
purposes of calculating the rate limit; this makes the rate limit conservative. With
15.5 days of observation time, the one-sided frequentist upper limit on the rate at 90%
confidence level is − ln (0.1)/T = 2.303/(15.5 days) = 0.15 per day. For comparison,
the S2 search [17] arrived at an upper limit of 0.26 per day. The S3 search [18] had
Search for gravitational-wave bursts in LIGO data 19
5 6 7 8 9 10 11 12 13 14 15
time-shifted
unshifted
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
unshifted
time-shifted
rms spread
(a) (b)
Figure 8. (a) Scatter plot of Γ vs. Zg for time-shifted triggers (grey circles)
and unshifted triggers (black circles) after all other analysis cuts. The vertical
dashed line indicates the initial WaveBurst significance cut at Zg=6.7. The
horizontal dashed line indicates the final Γ cut. (b) Overlaid histograms of Γ
for unshifted triggers (black circles) and mean background estimated from time-
shifted triggers (black stairstep with statistical error bars). The shaded bars
represent the expected root-mean-square statistical fluctuations on the number of
unshifted background triggers in each bin.
an observation time of only 8 days and did not state a rate limit.
7. Amplitude sensitivity of the search
The previous section presented a limit on the rate of a hypothetical population
of gravitational-wave signals for which the analysis pipeline has perfect detection
efficiency. However, the actual detection efficiency will depend on the signal waveform
and amplitude, being zero for very weak signals and generally approaching unity for
sufficiently strong signals. The signal processing methods used in this analysis are
expressly designed to be able to detect arbitrary waveforms as long as they have
short duration and frequency content in the 64–1600 Hz band which stands out above
the detector noise. Therefore, for any given signal of this general type, we wish to
determine a characteristic minimum signal amplitude for which the pipeline has good
detection efficiency. As in past analyses, we use a Monte Carlo technique with a
population of simulated gravitational wave sources. Simulated events are generated
at random sky positions and pseudo-random times (imposing a minimum separation
of 80 s) during the S4 run; the resulting signal waveforms in each interferometer are
calculated with the appropriate antenna factors and time delays. These simulated
signals are added to the actual detector data, and the summed data streams are
analyzed using the same pipeline with the same trigger selection criteria.
The intrinsic amplitude of a simulated gravitational wave may be characterized
by its root-sum-squared strain amplitude at the Earth, without folding in antenna
response factors:
hrss ≡
(|h+(t)|2 + |h×(t)|2) dt . (4)
This quantity has units of s1/2, or equivalently Hz−1/2. In general, the root-sum-
squared signal measured by a given detector, hrssdet, will be somewhat smaller. The
Search for gravitational-wave bursts in LIGO data 20
Monte Carlo approach taken for this analysis is to generate a set of signals all with
fixed hrss and then to add this set of signals to the data with several discrete scale
factors to evaluate different signal amplitudes. For a given signal morphology and hrss,
the efficiency of the pipeline is the fraction of simulated signals which are successfully
recovered.
For this analysis, we do not attempt to survey the complete spectrum of
astrophysically motivated signals, but rather we use a limited number of ad-hoc
waveforms to characterize the sensitivity of the search in terms of hrss. Similar
sensitivities may be expected for different waveforms with similar overall properties
(central frequency, bandwidth, duration); the degree to which this is true has been
investigated in [18] and [29]. The waveforms evaluated in the present analysis are:
• Sine-Gaussian: sinusoid with a given frequency f0 inside a Gaussian amplitude
envelope with dimensionless width Q and arrival time t0:
h(t0 + t) = h0 sin(2πf0t) exp
− (2πf0t)
. (5)
These are generated with linear polarization, with f0 ranging from 70 Hz to
1053 Hz and with Q equal to 3, 8.9, and 100. The signal consistency tests
described in section 4 were developed using an ensemble of sine-Gaussian signals
with all simulated frequencies and Q values.
• Gaussian: a simple unipolar waveform with a given width τ and linear
polarization:
h(t0 + t) = h0 exp(−t
2/τ2) . (6)
• Band-limited white noise burst: a random signal with two independent
polarization components that are white over a given frequency band, described
by a base frequency f0 and a bandwidth ∆f (i.e. containing frequencies from f0
to f0 + ∆f). The signal amplitude has a Gaussian time envelope with a width
τ . Because these waveforms have two uncorrelated polarizations (in a coordinate
system at some random angle), they provide a stringent check on the robustness
of our cross-correlation test.
In all cases, we generate each simulated signal with a random arrival direction and a
random angular relationship between the wave polarization basis and the Earth.
Figures 9 and 10 show the measured efficiency of the analysis pipeline as a function
of root-sum-squared strain amplitude, ǫ(hrss), for each simulated waveform. The
efficiency data points for each waveform are fit with a function of the form
ǫ(hrss) =
)α(1+β tanh(hrss/hmidrss ))
, (7)
where ǫmax corresponds to the efficiency for strong signals (normally very close to
unity), hmidrss is the hrss value corresponding to an efficiency of ǫmax/2, β is the
parameter that describes the asymmetry of the sigmoid (with range −1 to +1), and
α describes the slope. Data points with efficiency below 0.05 are excluded from the
fit because they do not necessarily follow the functional form, while data points with
efficiency equal to 1.0 are excluded because their asymmetric statistical uncertainties
are not handled properly in the chi-squared fit. The empirical functional form in
equation 7 has been found to fit the remaining efficiency data points well.
Note that the Gaussian waveform with τ = 6.0 ms has efficiency less than 0.8
even for the largest simulated amplitude. This broad waveform, with little signal
Search for gravitational-wave bursts in LIGO data 21
]Hz [strain/rssh
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70 Hz
100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=3
]Hz [strain/rssh
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70 Hz
100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=8.9
]Hz [strain/rssh
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1 100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=100
Figure 9. Efficiency curves for simulated gravitational-wave signals: linearly-
polarized sine-Gaussian waves with (a) Q=3; (b) Q=8.9; (c) Q=100. Statistical
errors are comparable to the size of the plot symbols.
Search for gravitational-wave bursts in LIGO data 22
]Hz [strain/rssh
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0.1 ms
0.25 ms
0.5 ms
1.0 ms
2.5 ms
4.0 ms
6.0 ms
Gaussians
]Hz [strain/rssh
-2210 -2110 -2010 -1910
100-110 Hz, 0.1 s
100-200 Hz, 0.1 s
100-200 Hz, 0.01 s
250-260 Hz, 0.1 s
250-350 Hz, 0.1 s
250-350 Hz, 0.01 s
1000-1010 Hz, 0.1 s
1000-1100 Hz, 0.1 s
1000-1100 Hz, 0.01 s
1000-2000 Hz, 0.1 s
1000-2000 Hz, 0.01 s
1000-2000 Hz, 0.001 s
Band-limited white noise bursts
Figure 10. Efficiency curves for simulated gravitational-wave signals: (a)
linearly-polarized Gaussian waves; (b) band-limited white-noise bursts with two
independent polarization components. Note that four curves in the latter plot
are nearly identical: 100–110 Hz, 0.1 s; 100–200 Hz, 0.1 s; 250–260 Hz, 0.1 s;
and 250–350 Hz, 0.01 s. Statistical errors are comparable to the size of the plot
symbols.
power at frequencies above 64 Hz (the lower end of the nominal search range), is at
the limit of what the search method can detect. For some of the other waveforms,
the efficiency levels off at a value slightly less than 1.0 due to the application of the
auxiliary-channel vetoes, which randomly coincide in time with some of the simulated
signals. This effect is most pronounced for the longest-duration simulated signals due
to the veto logic used in this analysis, which rejects a trigger if there is any overlap
between the reconstructed trigger duration and a vetoed time interval. The 70-Hz
sine-Gaussian with Q=100 has a duration longer than 1 s and is reconstructed quite
poorly; it is omitted from figure 9c and from the following results.
The analytic expressions of the fits are used to determine the signal strength hrss
for which efficiencies of 50% and 90% are reached. These fits are subject to statistical
Search for gravitational-wave bursts in LIGO data 23
Table 2. hrss values corresponding to 50% and 90% detection efficiencies for
simulated sine-Gaussian signals with various central frequencies and Q values.
The 70 Hz sine-Gaussian with Q=100 is not detected reliably.
hrss (10
−21 Hz−1/2)
50% efficiency 90% efficiency
Central
frequency (Hz) Q=3 Q=8.9 Q=100 Q=3 Q=8.9 Q=100
70 3.4 5.8 — 19.2 52.0 —
100 1.8 1.7 2.6 10.4 9.4 17.7
153 1.5 1.4 1.7 8.2 8.3 8.7
235 1.6 1.7 1.9 11.0 9.8 12.6
361 2.4 2.7 3.2 11.5 16.7 20.9
554 3.3 3.2 3.2 16.1 17.9 20.4
849 5.9 4.9 4.5 28.4 28.9 24.9
1053 8.3 7.2 6.6 39.3 37.5 37.5
Table 3. hrss values corresponding to 50% and 90% detection efficiencies for
simulated Gaussian signals with various widths. The waveform with τ=6.0 ms
does not reach an efficiency of 90% within the range of signal amplitudes
simulated.
hrss (10
−21 Hz−1/2)
τ (ms) 50% efficiency 90% efficiency
0.05 6.6 33.9
0.1 4.4 25.3
0.25 3.0 14.3
0.5 2.2 13.5
1.0 2.2 10.6
2.5 3.4 20.5
4.0 8.3 43.3
6.0 39.0 —
errors from the limited number of simulations performed to produce the efficiency data
points. Also, the overall amplitude scale is subject to the uncertainty in the calibration
of the interferometer response, conservatively estimated to be 10% [30]. We increase
the nominal fitted hrss values by the amount of these systematic uncertainties to arrive
at conservative hrss values at efficiencies of 50% and 90%, summarized in tables 2,
3, and 4. The sine-Gaussian hrss values are also displayed graphically in figure 11,
showing how the frequency dependence generally follows that of the instrumental
noise.
Event rate limits as a function of waveform type and signal amplitude can be
represented by an “exclusion diagram”. Each curve in an exclusion diagram indicates
what the rate limit would be for a population of signals with a fixed hrss, as a
function of hrss. The curves in figure 12 illustrate, using selected sine-Gaussian and
Gaussian waveforms that were also considered in the S1 and S2 analyses, that the
amplitude sensitivities achieved by this S4 analysis are at least an order of magnitude
better than the sensitivities achieved by the S2 analysis. For instance, the 50%
efficiency hrss value for 235 Hz sine-Gaussians with Q=8.9 is 1.5 × 10
−20 Hz−1/2
for S2 and 1.7× 10−21 Hz−1/2 for S4. (Exclusion curves were not generated for the S3
Search for gravitational-wave bursts in LIGO data 24
Table 4. hrss values corresponding to 50% and 90% detection efficiencies for
simulated “white noise burst” signals with various base frequencies, bandwidths,
and durations.
hrss (10
−21 Hz−1/2)
Base frequency Bandwidth Duration
(Hz) (Hz) (s) 50% eff. 90% eff.
100 10 0.1 1.8 4.7
100 100 0.1 1.9 4.1
100 100 0.01 1.3 2.9
250 10 0.1 1.8 4.5
250 100 0.1 2.4 5.4
250 100 0.01 1.8 4.3
1000 10 0.1 6.5 15.8
1000 100 0.1 7.9 16.7
1000 100 0.01 5.5 12.7
1000 1000 0.1 19.2 42.6
1000 1000 0.01 9.7 22.3
1000 1000 0.001 9.5 23.7
100 1000
Sensitivities for Sine−Gaussian Waveforms
Frequency (Hz)
90% for Q=3
90% for Q=8.9
90% for Q=100
50% for Q=3
50% for Q=8.9
50% for Q=100
H2 noise
L1 noise
H1 noise
Figure 11. Sensitivity of the analysis pipeline for sine-Gaussian waveforms as a
function of frequency and Q. Symbols indicate the hrss values corresponding to
50% and 90% efficiency, taken from table 2. The instrumental sensitivity curves
from figure 2 are shown for comparison.
analysis, but the S3 sensitivity was 9 × 10−21 Hz−1/2 for this particular waveform.)
The improvement is greatest for lower-frequency sine-Gaussians and for the widest
Gaussians, due to the reduced low-frequency detector noise and the explicit extension
of the search band down to 64 Hz.
Search for gravitational-wave bursts in LIGO data 25
]Hz [strain/rssh
-2210 -2110 -2010 -1910 -1810 -1710 -1610
70 Hz
153 Hz
235 Hz
554 Hz
849 Hz
1053 Hz
S4 LIGO-TAMA
(a) Sine-Gaussians with Q=8.9
]Hz [strain/rssh
-2210 -2110 -2010 -1910 -1810 -1710 -1610
0.05 ms
0.1 ms
0.25 ms
1.0 ms
2.5 ms
4.0 ms
(b) Gaussians
Figure 12. Exclusion diagrams (rate limit at 90% confidence level, as a function
of signal amplitude) for (a) sine-Gaussian and (b) Gaussian simulated waveforms
for this S4 analysis compared to the S1 and S2 analyses (the S3 analysis did not
state a rate limit). These curves incorporate conservative systematic uncertainties
from the fits to the efficiency curves and from the interferometer response
calibration. The 849 Hz curve labeled “LIGO-TAMA” is from the joint burst
search using LIGO S2 with TAMA DT8 data [8], which included data subsets
with different combinations of operating detectors with a total observation time
of 19.7 days and thereby achieved a lower rate limit. The hrss sensitivity of the
LIGO-TAMA search was nearly constant for sine-Gaussians over the frequency
range 700–1600 Hz.
Search for gravitational-wave bursts in LIGO data 26
8. Astrophysical reach estimates
In order to set an astrophysical scale to the sensitivity achieved by this search, we
can ask what amount of mass converted into gravitational-wave burst energy at a
given distance would be strong enough to be detected by the search pipeline with 50%
efficiency. We start with the expression for the instantaneous energy flux emitted by a
gravitational wave source in the two independent polarizations h+(t) and h×(t) [31],
d2EGW
(ḣ+)
2 + (ḣ×)
, (8)
and follow the derivations in [32]. Plausible astrophysical sources will, in general, emit
gravitational waves anisotropically, but here we will assume isotropic emission in order
to get simple order-of-magnitude estimates. The above formula, when integrated over
the signal duration and over the area of a sphere at radius r (assumed not to be at
a cosmological distance), yields the total energy emitted in gravitational waves for a
given signal waveform. For the case of a sine-Gaussian with frequency f0 and Q ≫ 1,
we find
EGW =
(2πf0)
2h2rss . (9)
Taking the waveform for which we have the best hrss sensitivity, a 153 Hz sine-
Gaussian with Q=8.9, and assuming a typical Galactic source distance of 10 kpc, the
above formula relates the 50%-efficiency hrss = 1.4× 10
−21 Hz−1/2 to 10−7 solar mass
equivalent emission into a gravitational-wave burst from this hypothetical source and
under the given assumptions. For a source in the Virgo galaxy cluster, approximately
16 Mpc away, the same hrss would be produced by an energy emission of roughly
0.25M⊙c
2 in a burst with this highly favourable waveform.
We can draw more specific conclusions about detectability for models of
astrophysical sources which predict the absolute energy and waveform emitted. Here
we consider the core-collapse supernova simulations of Ott et al. [15] and a binary black
hole merger waveform calculated by the Goddard numerical relativity group [11] (as a
representative example of the similar merger waveforms obtained by several groups).
While the Monte Carlo sensitivity studies in section 7 did not include these particular
waveforms, we can relate the modeled waveforms to qualitatively similar waveforms
that were included in the Monte Carlo study and thus infer the approximate sensitivity
of the search pipeline for these astrophysical models.
Ott et al. simulated core collapse for three progenitor models and calculated
the resulting gravitational wave emission, which was dominated by oscillations of
the protoneutron star core driven by accretion [15]. Their s11WW model, based
on a non-spinning 11-M⊙ progenitor, produced a total gravitational-wave energy
emission of 1.6× 10−8M⊙c
2 with a characteristic frequency of ∼654 Hz and duration
of several hundred milliseconds. If this were a sine-Gaussian, it would have a Q
of several hundred; table 2 shows that our sensitivity does not depend strongly on
Q, so we might expect 50% efficiency for a signal at this frequency with hrss of
∼3.7 × 10−21 Hz−1/2. However, the signal is not monochromatic, and its increased
time-frequency volume may degrade the sensitivity by up to a factor of ∼2. Using
this EGW and hrss ≈ 7 × 10
−21 Hz−1/2 in equation 9, we find that our search
has an approximate “reach” (distance for which the signal would be detected with
50% efficiency by the analysis pipeline) of ∼0.2 kpc for this model. The m15b6
model, based on a spinning 15-M⊙ progenitor, yields a very similar waveform and
Search for gravitational-wave bursts in LIGO data 27
essentially the same reach. The s25WW model, based on a 25-M⊙ progenitor, was
found to emit vastly more energy in gravitational waves, 8.2 × 10−5M⊙c
2, but with
a higher characteristic frequency of ∼937 Hz. With respect to the Monte Carlo
results in section 7, we may consider this similar to a high-Q sine-Gaussian, yielding
hrss ≈ 5.5×10
−21 Hz−1/2, or to a white noise burst with a bandwidth of ∼100 Hz and
a duration of > 0.1 s, yielding hrss ≈ 8 × 10
−21 Hz−1/2. Using the latter, we deduce
an approximate reach of 8 kpc for this model.
A pair of merging black holes emits gravitational waves with very high efficiency;
for instance, numerical evolutions of equal-mass systems without spin have found the
radiated energy from the merger and subsequent ringdown to be 3.5% or more of the
total mass of the system [11]. From figure 8 of that paper, the frequency of the signal
at the moment of peak amplitude is seen to be
fpeak ≈
15 kHz
(Mf/M⊙)
, (10)
where Mf is the final mass of the system. Very roughly, we can consider the
merger+ringdown waveform to be similar to a sine-Gaussian with central frequency
fpeak and Q ≈ 2 for purposes of estimating the reach of this search pipeline for binary
black hole mergers. (Future analyses will include Monte Carlo efficiency studies using
complete inspiral-merger-ringdown waveforms.) Thus, a binary system of two 10-M⊙
black holes (i.e. Mf ≈ 20M⊙) has fpeak ≈ 750 Hz, and from table 2 we can estimate
the hrss sensitivity to be ∼5.5×10
−21 Hz−1/2. Using EGW = 0.035Mfc
2, we conclude
that the reach for such a system is roughly 1.4 Mpc. Similarly, a binary system with
Mf = 100M⊙ has fpeak ≈ 150 Hz, a sensitivity of ∼1.5×10
−21 Hz−1/2, and a resulting
reach of roughly 60 Mpc.
9. Discussion
The search reported in this paper represents the most sensitive search to date for
gravitational-wave bursts in terms of strain amplitude, reaching hrss values below
10−20 Hz−1/2, and covers a broad frequency range, 64–1600 Hz, with a live observation
time of 15.5 days.
Comparisons with previous LIGO [16, 17] and LIGO-TAMA [8] searches have
already been shown graphically in figure 12. The LIGO-TAMA search targeted
millisecond-duration signals with frequency content in the 700–2000 Hz frequency
regime (i.e., partially overlapping the present search) and had a detection efficiency
of at least 50% (90%) for signals with hrss greater than ∼ 2 × 10
−19 Hz−1/2
(10−18 Hz−1/2). Among other searches with broad-band interferometric detectors [33,
34, 35], the most recent one by the TAMA collaboration reported an upper limit of
0.49 events per day at the 90% confidence level based on an analysis of 8.1 days of the
TAMA300 instrument’s ninth data taking run (DT9) in 2003–04. The best sensitivity
of this TAMA search was achieved when looking for narrow-band signals at TAMA’s
best operating frequency, around 1300 Hz, and it was at hrss ≈ 10
−18 Hz−1/2 for 50%
detection efficiency [35]. Although we did not measure the sensitivity of the S4 LIGO
search with narrow-band signals at 1300 Hz, LIGO’s noise at that frequency range
varies slowly enough so that we do not expect it to be significantly worse than the
sensitivity for 1053 Hz sine-Gaussian signals described in section 7, which stands at
about 7× 10−21 Hz−1/2.
Search for gravitational-wave bursts in LIGO data 28
Comparisons with results from resonant mass detectors were detailed in our
previous publications [16, 17]. The upper limit of ∼ 4×10−3 events per day at the 95%
confidence level on the rate of gravitational wave bursts set by the IGEC consortium
of five resonant mass detectors still represents the most stringent rate limit for hrss
signal strengths of order 10−18 Hz−1/2 and above [36]. This upper limit quickly falls off
and becomes inapplicable to signals weaker than 10−19 Hz−1/2 (see figure 14 in [17].)
Furthermore, with the improvement in our search sensitivity, the signal strength of the
events corresponding to the slight excess seen by the EXPLORER and NAUTILUS
resonant mass detectors in their 2001 data [37] falls well above the 90% sensitivity of
our current S4 search: as described in [17], the optimal orientation signal strength of
these events assuming a Gaussian morphology with τ=0.1 ms corresponds to a hrss
of 1.9 × 10−19 Hz−1/2. For such Gaussians our S4 search all-sky 90% sensitivity is
2.5 × 10−20 Hz−1/2 (see Table 3) and when accounting for optimal orientation, this
improves by roughly a factor of 3, to 9.3×10−21 Hz−1/2. The rate of the EXPLORER
and NAUTILUS events was of order 200 events/year (or 0.55 events per day) [37, 38].
A steady flux of gravitational-wave bursts at this rate is excluded by our present
measurement at the 99.9% confidence level. Finally, in more recent running of the
EXPLORER and NAUTILUS detectors, an analysis of 149 days of data collected in
2003 set an upper limit of 0.02 events per day at the 95% confidence level and with a
hrss sensitivity of ∼ 2× 10
−19 Hz−1/2 [39].
The S5 science run, which began in November 2005 and is expected to continue
until late 2007, has a goal of collecting a full year of coincident LIGO science-mode
data. Searches for gravitational-wave bursts using S5 data are already underway and
will be capable of detecting any sufficiently strong signals which arrive during that
time, or else placing an upper limit on the rate of such signals on the order of a few
per year. Furthermore, the detector noise during the S5 run has reached the design
goals for the current LIGO interferometers, and so the amplitude sensitivity of S5
burst searches is expected to be roughly a factor of two better than the sensitivity of
this S4 search.
Another direction being pursued with the S5 data is to make appropriate use
of different detector network configurations. In addition to the approach used
in the S4 analysis reported here, which requires a signal to appear with excess
power in a time-frequency map in all three LIGO interferometers, data from two-
detector combinations is also being analyzed to maximize the total observation
time. Furthermore, using LIGO data together with simultaneous data from other
interferometers can significantly improve confidence in a signal candidate and allow
more properties of the signal to be deduced. The GEO 600 interferometer has joined
the S5 run for full-time observing in May 2006, and we look forward to the time
when VIRGO begins operating with sensitivity comparable to the similarly-sized
LIGO interferometers. Members of the LSC are currently implementing coherent
network analysis methods using maximum likelihood approaches for optimal detection
of arbitrary burst signal (see, for example, [40]) and for robust signal consistency
tests [41, 42]. Such methods will make the best use of the data collected from the
global network of detectors to search for gravitational-wave bursts.
Search for gravitational-wave bursts in LIGO data 29
Acknowledgments
The authors gratefully acknowledge the support of the United States National Science
Foundation for the construction and operation of the LIGO Laboratory and the
Science and Technology Facilities Council of the United Kingdom, the Max-Planck-
Society, and the State of Niedersachsen/Germany for support of the construction
and operation of the GEO600 detector. The authors also gratefully acknowledge the
support of the research by these agencies and by the Australian Research Council, the
Council of Scientific and Industrial Research of India, the Istituto Nazionale di Fisica
Nucleare of Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria
d’Economia, Hisenda i Innovació of the Govern de les Illes Balears, the Scottish
Funding Council, the Scottish Universities Physics Alliance, The National Aeronautics
and Space Administration, the Carnegie Trust, the Leverhulme Trust, the David
and Lucile Packard Foundation, the Research Corporation, and the Alfred P. Sloan
Foundation. This document has been assigned LIGO Laboratory document number
LIGO-P060016-C-Z.
References
[1] Sigg D (for the LSC) 2006 Class. Quantum Grav. 23 S51–6
[2] Lück H et al 2006 Class. Quantum Grav. 23 S71–8
[3] Acernese F et al 2006 Class. Quantum Grav. 23 S63–9
[4] Ando M and the TAMA Collaboration 2005 Class. Quantum Grav. 22 S881–9
[5] Fritschel P 2003 Gravitational-Wave Detection: Proc. SPIE vol 4856 ed Cruise M and Saulson
P (SPIE) p 282–91
[6] Acernese F et al 2006 Class. Quantum Grav. 23 S635–42
[7] Kuroda K et al 2003 Proc. 28th Int. Cosmic Ray Conf. ed Kajita T et al (Universal Academy
Press) p 3103
[8] Abbott B et al (LSC) and Akutsu T et al (TAMA Collaboration) 2006 Phys. Rev. D 73 102002
[9] Blanchet L, Damour T, Esposito-Farèse G and Iyer BR 2004 Phys. Rev. Lett. 93 091101
[10] Jaranowski P, Królak A and Schutz BF 1998 Phys. Rev. D 58 063001
[11] Baker J G, Centrella J, Choi D, Koppitz M and van Meter J 2006 Phys. Rev. D 73 104002
[12] Dimmelmeier H, Font J A and Müller E 2001 Astrophys. J. Lett. 560 L163–6
[13] Ott C D, Burrows A, Livne E and Walder R 2004 Astrophys. J. 600 834–64
[14] Burrows A, Livne E, Dessart L, Ott C D and Murphy J 2006 Astrophys. J. 640 878–90
[15] Ott C D, Burrows A, Dessart L and Livne E 2006 Phys. Rev. Lett. 96 201102
[16] Abbott B et al (LSC) 2004 Phys. Rev. D 69 102001
[17] Abbott B et al (LSC) 2005 Phys. Rev. D 72 062001
[18] Abbott B et al (LSC) 2006 Class. Quantum Grav. 23 S29–39
[19] Lazzarini A and Weiss R 1995 LIGO Science Requirements Document (SRD) LIGO technical
document LIGO-E950018-02-E
[20] Klimenko S and Mitselmakher G 2004 Class. Quantum Grav. 21 S1819–30
[21] The version of WaveBurst used for this analysis may be found at
http://ldas-sw.ligo.caltech.edu/cgi-bin/cvsweb.cgi/Analysis/WaveBurst/S4/?cvsroot=GDS
with the CVS tag “S4”
[22] Daubechies I 1992 Ten Lectures on Wavelets (Philadelphia: SIAM)
[23] Klimenko S, Yakushin I, Rakhmanov M and Mitselmakher G 2004 Class. Quantum Grav. 21
S1685–94
[24] Cadonati L and Márka S 2005 Class. Quantum Grav. 22 S1159–67
[25] Cadonati L 2004 Class. Quantum Grav. 21 S1695–703
[26] The version of CorrPower used for this analysis may be found at
http://www.lsc-group.phys.uwm.edu/cgi-bin/cvs/viewcvs.cgi/matapps/src/searches/burst/CorrPower/?cvsroot=lscsoft
with the CVS tag “CorrPower-080605”
[27] Chatterji S, Blackburn L, Martin G and Katsavounidis E 2004 Class. Quantum Grav. 21 S1809–
[28] The version of KleineWelle used for this analysis may be found at
http://ldas-sw.ligo.caltech.edu/-cgi-bin/-cvsweb.cgi/-Analysis/-WaveBurst/-S4/-?cvsroot=GDS
http://www.lsc-group.phys.uwm.edu/-cgi-bin/-cvs/-viewcvs.cgi/-matapps/-src/-searches/-burst/-CorrPower/-?cvsroot=lscsoft
Search for gravitational-wave bursts in LIGO data 30
http://ldas-sw.ligo.caltech.edu/cgi-bin/cvsweb.cgi/gds/Monitors/kleineWelle/?cvsroot=GDS
dated January 20, 2005
[29] Beauville F et al 2006 “A comparison of methods for gravitational wave burst searches from
LIGO and Virgo”, submitted to Phys. Rev. D, preprint gr-qc/0701026
[30] Dietz A, Garofoli J, González G, Landry M, O’Reilly B and Sung M 2006 LIGO technical
document LIGO-T050262-01-D
[31] Shapiro S L and Teukolsky S A 1983 Black Holes, White Dwarfs, and Neutron Stars (New York:
John Wiley & Sons)
[32] Riles K 2004 LIGO technical document LIGO-T040055-00-Z
[33] Nicholson D et al 1996 Phys. Lett. A 218 175
[34] Forward R L 1978 Phys. Rev. D 17 379
[35] Ando M et al (TAMA Collaboration) 2005 Phys. Rev. D 71 082002
[36] Astone P et al (International Gravitational Event Collaboration) 2003 Phys. Rev. D 68 022001
[37] Astone P et al 2002 Class. Quantum Grav. 19 5449
[38] Coccia E, Dubath F and Maggiore M 2004 Phys. Rev. D 70 084010
[39] Astone P et al 2006 Class. Quantum Grav. 23 S169–78
[40] Klimenko S, Mohanty S, Rakhmanov M and Mitselmakher G 2005 Phys. Rev. D 72 122002
[41] Wen L and Schutz B 2005 Class. Quantum Grav. 22 S1321–35
[42] Chatterji S, Lazzarini A, Stein L, Sutton P J, Searle A and Tinto M 2006 Phys. Rev. D 74
082005
http://ldas-sw.ligo.caltech.edu/-cgi-bin/-cvsweb.cgi/-gds/-Monitors/-kleineWelle/-?cvsroot=GDS
http://arxiv.org/abs/gr-qc/0701026
	Introduction
	Instruments and data collection
	Trigger generation
	Signal consistency tests
	H1/H2 amplitude consistency test
	Cross-correlation consistency tests
	Additional selection criteria for event candidates
	Data quality conditions
	Auxiliary-channel vetoes
	Gamma cut
	Search results
	Amplitude sensitivity of the search
	Astrophysical reach estimates
	Discussion

  The fourth science run of the LIGO and GEO 600 gravitational-wave detectors,
carried out in early 2005, collected data with significantly lower noise than
previous science runs. We report on a search for short-duration
gravitational-wave bursts with arbitrary waveform in the 64-1600 Hz frequency
range appearing in all three LIGO interferometers. Signal consistency tests,
data quality cuts, and auxiliary-channel vetoes are applied to reduce the rate
of spurious triggers. No gravitational-wave signals are detected in 15.5 days
of live observation time; we set a frequentist upper limit of 0.15 per day (at
90% confidence level) on the rate of bursts with large enough amplitudes to be
detected reliably. The amplitude sensitivity of the search, characterized using
Monte Carlo simulations, is several times better than that of previous
searches. We also provide rough estimates of the distances at which
representative supernova and binary black hole merger signals could be detected
with 50% efficiency by this analysis.

Introduction
Large interferometers are now being used to search for gravitational waves with
sufficient sensitivity to be able to detect signals from distant astrophysical sources.
At present, the three detectors of the Laser Interferometer Gravitational-wave
Observatory (LIGO) project [1] have achieved strain sensitivities consistent with their
design goals, while the GEO 600 [2] and Virgo [3] detectors are in the process of being
commissioned and are expected to reach comparable sensitivities. Experience gained
with these detectors, TAMA300 [4], and several small prototype interferometers has
Search for gravitational-wave bursts in LIGO data 5
nurtured advanced designs for future detector upgrades and new facilities, including
Advanced LIGO [5], Advanced Virgo [6], and the Large-scale Cryogenic Gravitational-
wave Telescope (LCGT) proposed to be constructed in Japan [7]. The LIGO Scientific
Collaboration (LSC) carries out the analysis of data collected by the LIGO and
GEO 600 gravitational-wave detectors, and has begun to pursue joint searches with
other collaborations (see, for example, [8]) as the network of operating detectors
evolves.
As the exploration of the gravitational-wave sky can now be carried out with
greater sensitivity than ever before, it is important to search for all plausible signals
in the data. In addition to well-modeled signals such as those from binary inspirals [9]
and spinning neutron stars [10], some astrophysical systems may emit gravitational
waves which are modeled imperfectly (if at all) and therefore cannot reliably be
searched for using matched filtering. Examples of such imperfectly-modeled systems
include binary mergers (despite recent advances in the fidelity of numerical relativity
calculations for at least some cases; see, for example, [11]) and stellar core collapse
events. For the latter, several sets of simulations have been carried out in the past
(see, for example, [12] and [13]), but more recent simulations have suggested a new
resonant core oscillation mechanism, driven by in-falling material, which appears to
power the supernova explosion and also to emit strong gravitational waves [14, 15].
Given the current uncertainties regarding gravitational wave emission by systems such
as these, as well as the possibility of detectable signals from other astrophysical sources
which are unknown or for which no attempt has been made to model gravitational
wave emission, it is desirable to cast a wide net.
In this article, we report the results of a search for gravitational-wave “bursts”
that is designed to be able to detect short-duration (≪ 1 s) signals of arbitrary form
as long as they have significant signal power in the most sensitive frequency band
of LIGO, considered here to be 64–1600 Hz. This analysis uses LIGO data from
the fourth science run carried out by the LSC, called S4, and uses the same basic
methods as previous LSC burst searches [17, 18] that were performed using data from
the S2 and S3 science runs. (A burst search was performed using data from the S1
science run using different methods [16].) We briefly describe the instruments and
data collection in section 2. In sections 3 and 4 we review the two complementary
signal processing methods—one based on locating signal power in excess of the baseline
noise and the other based on cross-correlating data streams—that are used together
to identify gravitational-wave event candidates. We note where the implementations
have been improved relative to the earlier searches and describe the signal consistency
tests which are based on the outputs from these tools. Section 5 describes additional
selection criteria which are used to “clean up” the data sample, reducing the average
rate of spurious triggers in the data. The complete analysis “pipeline” finds no event
candidates that pass all of the selection criteria, so we present in section 6 an upper
limit on the rate of gravitational-wave events which would be detected reliably by our
pipeline.
The detectability of a given type of burst, and thus the effective rate limit for a
particular astrophysical source model, depends on the signal waveform and amplitude;
in general, the detection efficiency (averaged over sky positions and arrival times) is less
than unity. We do not attempt a comprehensive survey of possible astrophysical signals
in this paper, but use a Monte Carlo method with a limited number of ad-hoc simulated
signals to evaluate the amplitude sensitivity of our pipeline, as described in section 7.
Overall, this search has much better sensitivity than previous searches, mostly due to
Search for gravitational-wave bursts in LIGO data 6
Mode Cleaner
Smoothes out fluctuations
of the input beam,
passes only fundamental
Gaussian beam mode
Stabilized
Laser
Power Recycling Mirror
(2.7% transmission)
Increases the stored power
by a factor of ~45, reducing
the photostatistics noise
Fabry-Perot Arm Cavity
Increases the sensitivity
to small length changes by
a factor of ~140
Photodiode
Input Mirror End Mirror
Beam Splitter
(50% transmission)
2 km or 4 km
Figure 1. Simplified optical layout of a LIGO interferometer.
using lower-noise data and partly due to improvements in the analysis pipeline. In
section 8 we estimate the amplitude sensitivity for certain modeled signals of interest
and calculate approximate distances at which those signals could be detected with 50%
efficiency. This completed S4 search sets the stage for burst searches now underway
using data from the S5 science run of the LIGO and GEO 600 detectors, which benefit
from much longer observation time and will be able to detect even weaker signals.
2. Instruments and data collection
LIGO comprises two observatory sites in the United States with a total of three
interferometers. As shown schematically in figure 1, the optical design is a Michelson
interferometer augmented with additional partially-transmitting mirrors to form
Fabry-Perot cavities in the arms and to “recycle” the outgoing beam power by
interfering it with the incoming beam. Servo systems are used to “lock” the mirror
positions to maintain resonance in the optical cavities, as well as to control the mirror
orientations, laser frequency and intensity, and many other degrees of freedom of the
apparatus. Interference between the two beams recombining at the beam splitter is
detected by photodiodes, providing a measure of the difference in arm lengths that
would be changed by a passing gravitational wave. The large mirrors which direct
the laser beams are suspended from wires, with the support structures isolated from
ground vibrations using stacks of inertial masses linked by damped springs. Active
feed-forward and feedback systems provide additional suppression of ground vibrations
for many of the degrees of freedom. The beam path of the interferometer, excluding
the laser light source and the photodiodes, is entirely enclosed in a vacuum system.
The LIGO Hanford Observatory in Washington state has two interferometers within
the same vacuum system, one with arms 4 km long (called H1) and the other with
arms 2 km long (called H2). The LIGO Livingston Observatory in Louisiana has a
single interferometer with 4 km long arms, called L1.
The response of an interferometer to a gravitational wave arriving at local time
Search for gravitational-wave bursts in LIGO data 7
t depends on the dimensionless strain amplitude and polarization of the wave and its
arrival direction with respect to the arms of the interferometer. In the low-frequency
limit, the differential strain signal detected by the interferometer (effective arm length
difference divided by the length of an arm) can be expressed as a projection of the
two polarization components of the gravitational wave, h+(t) and h×(t), with antenna
response factors F+(α, δ, t) and F×(α, δ, t):
hdet(t) = F+(α, δ, t)h+(t) + F×(α, δ, t)h×(t) , (1)
where α and δ are the right ascension and declination of the source. F+ and F× are
distinct for each interferometer site and change slowly with t over the course of a
sidereal day as the Earth’s rotation changes the orientation of the interferometer with
respect to the source location.
The electrical signal from the photodiode is filtered and digitized continuously at a
rate of 16 384 Hz. The time series of digitized values, referred to as the “gravitational-
wave channel” (GW channel), is recorded in a computer file, along with a timestamp
derived from the Global Positioning System (GPS) and additional information. The
relationship between a given gravitational-wave signal and the digitized time series is
measured in situ by imposing continuous sinusoidal position displacements of known
amplitude on some of the mirrors. These are called “calibration lines” because they
appear as narrow line features in a spectrogram of the GW channel.
Commissioning the LIGO interferometers has required several years of effort and
was the primary activity through late 2005. Beginning in 2000, a series of short data
collection runs was begun to establish operating procedures, test the detector systems
with stable configurations, and provide data for the development of data analysis
techniques. The first data collection run judged to have some scientific interest,
science run S1, was conducted in August-September 2002 with detector noise more
than two orders of magnitude higher than the design goal. Science runs S2 and S3
followed in 2003 with steadily improving detector noise, but with a poor duty cycle
for L1 due primarily to low-frequency, large-amplitude ground motion from human
activities and weather. During 2004, a hydraulic pre-isolation system was installed
and commissioned at the Livingston site to measure the ground motion and counteract
it with a relative displacement between the external and internal support structures
for the optical components, keeping the internal components much closer to an inertial
frame at frequencies above 0.1 Hz. At the same time, several improvements were made
to the H1 interferometer at Hanford to allow the laser power to be increased to the
full design power of 10 W.
The S4 science run, which lasted from 22 February to 23 March 2005, featured
good overall “science mode” duty cycles of 80.5%, 81.4%, and 74.5% for H1, H2,
and L1, respectively, corresponding to observation times of 570, 576, and 528 hours.
Thanks to the improvements made after the S3 run, the detector noise during S4 was
within a factor of two of the design goal over most of the frequency band, as shown in
figure 2. The GEO 600 interferometer also collected data throughout the S4 run, but
was over a factor of 100 less sensitive than the LIGO interferometers at 200 Hz and
a factor of few at and above the 1 kHz frequency range. The analysis approach used
in this article effectively requires a gravitational-wave signal to be distinguishable
above the noise in each of a fixed set of detectors, so it uses only the three LIGO
interferometers and not GEO 600. There are a total of 402 hours of S4 during which
all three LIGO interferometers were simultaneously collecting science-mode data.
Search for gravitational-wave bursts in LIGO data 8
100 1000
LIGO Detector Sensitivities During S4 Science Run
Frequency (Hz)
LIGO SRD goal (4 km)
Figure 2. Best achieved detector noise for the three LIGO interferometers during
the S4 science run, in terms of equivalent gravitational wave strain amplitude
spectral density. “LIGO SRD goal” is the sensitivity goal for the 4-km LIGO
interferometers set forth in the 1995 LIGO Science Requirements Document [19].
3. Trigger generation
The first stage of the burst search pipeline is to identify times when the GW channels
of the three interferometers appear to contain signal power in excess of the baseline
noise; these times, along with parameters derived from the data, are called “triggers”
and are used as input to later processing stages. As in previous searches [17, 18],
the WaveBurst algorithm [20] is used for this purpose; it will only be summarized
here [21].
WaveBurst performs a linear wavelet packet decomposition, using the symlet
wavelet basis [22], on short intervals of gravitational-wave data from each
interferometer. This decomposition produces a time-frequency map of the data similar
to a windowed Fourier transformation. A time-frequency data sample is referred to as a
pixel. Pixels containing significant excess signal power are selected in a non-parametric
way by ranking them with other pixels at nearby times and frequencies. As in the S3
analysis, WaveBurst has been configured for S4 to use six different time resolutions
and corresponding frequency resolutions, ranging from 1/16 s by 8 Hz to 1/512 s by
256 Hz, to be able to closely match the natural time-frequency properties of a variety
of burst signals. The wavelet decomposition is restricted to 64–2048 Hz. At any
given resolution, significant pixels from the three detector data streams are compared
and coincident pixels are selected; these are used to construct “clusters”, potentially
spanning many pixels in time and/or frequency, within which there is evidence for
a common signal appearing in the different detector data streams. These coincident
clusters form the basis for triggers, each of which is characterized by a central time,
Search for gravitational-wave bursts in LIGO data 9
Entries  8325975
Mean    2.583
RMS    0.9666
0 5 10 15 20 25 30 35 40
Figure 3. Distribution of Zg values for all WaveBurst triggers. The arrow shows
the location of the initial significance cut, Zg > 6.7.
duration, central frequency, frequency range, and overall significance Zg as defined
in [23]. Zg is calculated from the pixels in the cluster and is roughly proportional to
the geometric average of the excess signal power measured in the three interferometers,
relative to the average noise in each interferometer at the relevant frequency. Thus,
a large value of Zg indicates that the signal power in those pixels is highly unlikely
to have resulted from usual instrumental noise fluctuations. In addition, the absolute
strength of the signal detected by each interferometer within the sensitive frequency
band of the search is estimated in terms of the root-sum-squared amplitude of the
detected strain,
hrssdet =
|hdet(t)|
dt . (2)
WaveBurst was run on time intervals during which all three LIGO interferometers
were in science mode, but omitting periods when simulated signals were injected into
the interferometer hardware, any photodiode readout experienced an overflow, or the
data acquisition system was not operating. In addition, the last 30 seconds of each
science-mode data segment were omitted because it was observed that loss of “lock”
is sometimes preceded by a period of instability. These selection criteria reduced the
amount of data processed by WaveBurst from 402 hours to 391 hours.
For this analysis, triggers found by WaveBurst are initially required to have a
frequency range which overlaps 64–1600 Hz. An initial significance cut, Zg ≥ 6.7, is
applied to reject the bulk of the triggers and limit the number passed along to later
stages of the analysis. Figure 3 shows the distribution of Zg prior to applying this
significance cut.
Besides identifying truly simultaneous signals in the three data streams,
WaveBurst applies the same pixel matching and cluster coincidence tests to the three
data streams with many discrete relative time shifts imposed between the Hanford
and Livingston data streams, each much larger than the maximum light travel time
between the sites and the duration of the signals targeted by this search. The time-
shifted triggers found in this way provide a large sample to allow the “background”
(spurious triggers produced in response to detector noise in the absence of gravitational
waves) to be studied, under the assumption that the detector noise properties do not
Search for gravitational-wave bursts in LIGO data 10
−150 −100 −50 0 50 100 150
Mean = 41.1
χ2 = 130.5
d.o.f. = 97
WaveBurst trigger rate versus time shift
Time shift (s)
Figure 4. WaveBurst trigger rate as a function of the relative time shift applied
between the Hanford and Livingston data streams. The horizontal line is a fit to
a constant value, yielding a χ2 of 130.5 for 97 degrees of freedom.
vary much over the span of a few minutes and are independent at the two sites.
The two Hanford data streams are not shifted relative to one another, so that any
local environmental effects which influence both detectors are preserved. In fact,
some correlation in time is observed between noise transients in the H1 and H2 data
streams.
Initially, WaveBurst found triggers for 98 time shifts in multiples of 3.125 s
between −156.25 and −6.25 s and between +6.25 and +156.25 s. These 5119 triggers,
called the “tuning set”, were used to choose the parameters of the signal consistency
tests and additional selection criteria described in the following two sections. As
shown in figure 4, the rate of triggers in the tuning set is roughly constant for all time
shifts, with a marginal χ2 value but without any gross dependence on time shift. The
unshifted triggers were kept hidden throughout the tuning process, in order to avoid
the possibility of human bias in the choice of analysis parameters.
4. Signal consistency tests
The WaveBurst algorithm requires only a rough consistency among the different
detector data streams—namely, some apparent excess power in the same pixels
in the wavelet decomposition—to generate a trigger. This section describes more
sophisticated consistency tests based on the detailed content of the GW channels.
These tests succeed in eliminating most WaveBurst triggers in the data, while keeping
essentially all triggers generated in response to simulated gravitational-wave signals
added to the data streams. (The simulation method is described in section 7.) Similar
tests were also used in the S3 search [18].
Search for gravitational-wave bursts in LIGO data 11
 ]Hz [ strain / rssdetH1 h
-2210 -2110 -2010 -1910
-2210
-2110
-2010
-1910
 ]Hz [ strain / rssdetH1 h
-2210 -2110 -2010 -1910
-2210
-2110
-2010
-1910
 ]Hz [ strain / rssdetH1 h
-2210 -2110 -2010 -1910
-2210
-2110
-2010
-1910
(b) (c)
Figure 5. (a) Two-dimensional histogram, with bin count indicated by greyscale,
of H2 vs. H1 amplitudes reconstructed by WaveBurst for the tuning set of
time-shifted triggers. (b) Two-dimensional histogram of H2 vs. H1 amplitudes
reconstructed for simulated sine-Gaussian signals with a wide range of frequencies
and amplitudes from sources uniformly distributed over the sky (see section 7). In
these plots, the diagonal lines show the limits of the H1/H2 amplitude consistency
cut: 0.5 < ratio < 2 . (c) Two-dimensional histogram of L1 vs. H1 amplitudes for
the same simulated sine-Gaussian signals. Diagonal lines are drawn at ratios of
0.5 and 2 only to guide the eye; no cut is applied using this pair of interferometers.
4.1. H1/H2 amplitude consistency test
Because the two Hanford interferometers are co-located and co-aligned, they will
respond identically (in terms of strain) to any given gravitational wave. Thus, the
overall root-sum-squared amplitudes of the detected signals, estimated by WaveBurst
according to equation (2), should agree well if the estimation method is reliable.
Figure 5a shows that the time-shifted triggers in the tuning set often have poor
agreement between the detected signal amplitudes in H1 and H2. In contrast,
simulated signals injected into the data are found with amplitudes which usually agree
within a factor of 2, as shown in figure 5b. Therefore, we keep a trigger only if the
ratio of estimated signal amplitudes is in the range 0.5 to 2.
The Livingston interferometer is roughly aligned with the Hanford interferome-
ters, but the curvature of the Earth makes exact alignment impossible. The antenna
responses to a given gravitational wave will tend to be similar, but not reliably enough
to allow a consistency test which is both effective at rejecting noise triggers and effi-
cient at retaining simulated signals, as shown in figure 5c.
Search for gravitational-wave bursts in LIGO data 12
4.2. Cross-correlation consistency tests
The amplitude consistency test described in the previous subsection simply compares
scalar quantities derived from the data, without testing whether the waveforms are
similar in detail. We use a program called CorrPower [24], also used in the S3 burst
search [18], to calculate statistics based on Pearson’s linear correlation statistic,
i=1(xi − x̄)(yi − ȳ)
i=1(xi − x̄)
i=1(yi − ȳ)
. (3)
In the above expression {xi} and {yi} are sequences selected from the two GW channel
time series, possibly with a relative time shift, and x̄ and ȳ are their respective mean
values. The length of each sequence, N samples, corresponds to a chosen time window
(see below) over which the correlation is to be evaluated. r assumes values between
−1 for fully anti-correlated sequences and +1 for fully correlated sequences.
The r statistic measures the correlation between two data streams, such as
would be produced by a common gravitational-wave signal embedded in uncorrelated
detector noise [25]. It compares waveforms without being sensitive to the relative
amplitudes, and is thus complementary to the H1/H2 amplitude consistency test
described above. Furthermore, the r statistic may be used to test for a correlation
between H1 and L1 or between H2 and L1, even though these pairs consist of
interferometers with different antenna response factors, because each polarization
component will produce a measurable correlation for a suitable relative time delay
(unless the wave happens to arrive from one of the special directions for which one
of the detectors has a null response for that polarization component). In the special
case of a linearly polarized gravitational wave, the detected signals will simply differ
by a multiplicative factor, which can be either positive or negative depending on the
polarization angle and arrival direction.
Before calculating the r statistic for each detector pair, the data streams are
filtered to select the frequency band of interest (bandpass between 64 Hz and 1600 Hz)
and whitened to equalize the contribution of noise from all frequencies within this
band. The filtering is the same as was used in the S3 search [18] except for the
addition of a Q=10 notch filter, centered at 345 Hz, to avoid measuring correlations
from the prominent vibrational modes of the wires used to suspend the mirrors, which
are clustered around that frequency. The r statistic is then calculated over multiple
time windows with lengths of 20, 50, and 100 ms and a range of starting times,
densely placed (99% overlap) to cover the full duration of the trigger as reported by
WaveBurst; the maximum value from among these different time windows is used.
CorrPower [26] calculates two quantities, derived from the r statistic, which are
used to select triggers. The first of these, called R0, is simply the signed cross-
correlation between H1 and H2 with no relative time delay. Triggers with R0 < 0
are rejected. The second quantity, called Γ, combines the r-statistic values from the
three detector pairs, allowing relative time delays of up to 11 ms between H1 and L1
and between H2 and L1, and up to 1 ms between H1 and H2 (to allow for a possible
mismatch in time calibration). Specifically, Γ is the average of “confidence” values
calculated from the absolute value of each of the three individual r-statistic values.
A large value of Γ indicates that the data streams are correlated to an extent that
is highly unlikely to have resulted from normal instrumental noise fluctuations. This
quantity complements Zg, providing a different and largely independent means for
distinguishing real signals from background.
Search for gravitational-wave bursts in LIGO data 13
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(a) (b)
Figure 6. Plots of Γ versus Zg, after the H1/H2 amplitude consistency cut but
before any other cuts. (a) Scatter plot for all time-shifted triggers in the tuning
set. (b) Two-dimensional histogram, with bin count indicated by greyscale, for
simulated sine-Gaussian signals with a wide range of frequencies and amplitudes
from sources uniformly distributed over the sky (see section 7). In both plots, the
vertical dashed line indicates the initial WaveBurst significance cut at Zg=6.7.
Figure 6 shows plots of Γ vs. Zg for time-shifted triggers and for simulated
gravitational-wave signals after the H1/H2 amplitude consistency cut but before the
R0 cut. The time-shifted triggers with Γ < 12 and Zg < 20 are the tail of the bulk
distribution of triggers. The outliers with Γ > 12 all arise from a few distinct times
when large noise transients occurred in H1 and H2; these are found many times, paired
with different L1 time shifts, and have similar values of Γ because the calculation of
Γ is dominated by the H1-H2 pair in these cases. The outliers with Γ < 12 and
Zg > 20 are artefacts of sudden changes in the power line noise at 60 Hz and 180 Hz
which WaveBurst recorded as triggers. A cut on the value of Γ can eliminate many
of the time-shifted triggers in figure 6a, but at the cost of also rejecting weak genuine
gravitational-wave signals that may have the distribution in figure 6b. Therefore, the Γ
cut is chosen only after additional selection criteria have been applied; see section 5.3.
5. Additional selection criteria for event candidates
Environmental disturbances or instrumental misbehaviour occasionally produce non-
stationary noise in the GW channel of a detector which contributes to the recording of
a WaveBurst trigger. These triggers can sometimes pass the H1-H2 consistency and
cross-correlation consistency tests, particularly since an environmental disturbance
at the Hanford site affects both H1 and H2. As noted in the previous section, the
calculated value of Γ is susceptible to being dominated by the H1-H2 pair even if
there is minimal signal power in the L1 data stream. A significant background rate of
event candidates caused by environmental or instrumental effects could obscure the
rare gravitational-wave bursts that we seek, or else require us to apply more aggressive
cuts and thus lose sensitivity for weak signals.
This section describes the two general tactics we use to reject data with
identifiable problems and thereby reduce the rate of background triggers. First,
we make use of several “data quality flags” that have been introduced in order to
describe the status of the instruments and the quality of the recorded data over time
intervals ranging from seconds to hours. Second, we remove triggers attributed to
Search for gravitational-wave bursts in LIGO data 14
short-duration instrumental or environmental effects by applying “vetoes” based on
triggers generated from auxiliary channels which have been found to correlate with
transients in the GW channel. Applying data quality conditions and vetoes to the
data set reduces the amount of “live” observation time (or “livetime”) during which
an arriving gravitational-wave burst would be detected and kept as an event candidate
at the end of the analysis pipeline. Therefore, we must balance this loss (“deadtime”)
against the effectiveness for removing spurious triggers from the data sample.
Choosing data quality and veto conditions with reference to a sample of
gravitational-wave event candidates could introduce a selection bias and invalidate any
upper limit calculated from the sample. Therefore, we have evaluated the relevance
of potential data quality cuts and veto conditions using other trigger samples. In
addition to the tuning set of time-shifted WaveBurst triggers, we have applied the
KleineWelle [27] method to identify transients in each interferometer’s GW channel.
(We have also used KleineWelle to identify transients in numerous auxiliary channels
for veto studies, as described in 5.2.) Like WaveBurst, KleineWelle is a time-frequency
method utilizing multi-resolution wavelet decomposition, but it processes each data
channel independently [28]. In analyzing data, the time series is first whitened using
a linear predictor error filter [27]. Then the time-frequency decomposition is obtained
using the Haar wavelet transform. The squared wavelet coefficients normalized to the
scale’s (frequency’s) root-mean-square provide an estimate of the energy associated
with a certain time-frequency pixel. A clustering mechanism is invoked in order to
increase the sensitivity to signals with less than optimal shapes in the time-frequency
plane and a total normalized cluster energy is computed. The significance of a
cluster is then defined as the negative natural logarithm of the probability of the
computed total normalized cluster energy to have resulted from Gaussian white noise;
we apply a threshold on this significance to define KleineWelle triggers. The samples
of KleineWelle triggers from each detector, as well as the subsample of coincident
H1 and H2 triggers, are useful indicators of localized disturbances. They may in
principle contain one or more genuine gravitational-wave signals, but decisions about
data quality and veto conditions are based on the statistics of the entire sample which
is dominated by instrumental artefacts and noise fluctuations.
5.1. Data quality conditions
We wish to reject instances of clear hardware problems with the LIGO detectors
or conditions that could affect our ability to unequivocally register the passage of
gravitational-wave bursts. Various studies of the data, performed during and after
data collection, produced a catalog of conditions that might affect the quality of the
data. Each named condition, or “flag”, has an associated list of time intervals during
which the condition is present, derived either from one or more diagnostic channels
or from entries made in the electronic logbook by operators and scientific monitors.
We have looked for significant correlations between the flagged time intervals and
time-shifted WaveBurst triggers, and also between the flagged time intervals and
KleineWelle single-detector triggers (particularly the “outliers” with large significance
and the coincident H1 and H2 triggers). Based on these studies, we decided to impose
a number of data quality conditions.
We first require the calibration lines to be continuously present. On several
occasions when they dropped out briefly, due to a problem with the excitation engine,
the data is removed from the analysis. The livetime associated with these occurrences
Search for gravitational-wave bursts in LIGO data 15
is negligible while they are all correlated with transients appearing in the GW channel.
Local winds and sound from airplanes may couple to the instrument through
the ground and result in elevated noise and/or impulsive signals. A data quality
flag was established to identify intervals of local winds at the sites with speeds of
56 km/hour (35 miles per hour) and above. We studied the correlation of these times
with the single-detector triggers produced with KleineWelle. The correlation is more
apparent in the H2 detector, for which 7.4% of the most significant KleineWelle triggers
(threshold of 1600) coincide with the intervals of strong winds at the Hanford site. The
livetime that is rejected in this way is 0.66% of the H1-H2 coincident observation time
over which this study was performed. Thanks to improved acoustic isolation installed
after the S2 science run, acoustic noise from airplanes was not found to contribute
to triggers in the GW channel in general; however, a period of 300 seconds has been
rejected around a particularly loud time when a fighter jet passed over the Hanford
site.
Elevated low-frequency seismic activity has been observed to cause noise
fluctuations and transients in the GW channel. Data from several seismometers at the
Hanford observatory was band-pass filtered in various narrow bands between 0.4 Hz
and 2.4 Hz, and the root-mean-square signal in each band was tracked over time. A
set of particularly relevant seismometers and bands was selected, and time intervals
were flagged whenever a band in this set exceeded 7 times its median value. A follow
up analysis of the single instrument as well as coincident H1-H2 KleineWelle triggers
found significant correlation with the elevated seismic noise. The strongest correlation
is observed in the outlier triggers (KleineWelle significance of 1600 or greater) in H2,
of which 41.9% coincide with the seismic flags, compared to a deadtime of 0.6%.
In the two Hanford detectors, a diagnostic channel counting ADC overflows in the
length sensing and control subsystem was used to flag intervals for exclusion from the
analysis. One minute of livetime around these overflows is rejected. Such overflows
were indeed seen to correlate with single-detector outlier triggers in H1 (44.4% of
them, with 0.68% deadtime) and H2 (74.1% of them, with 0.41% deadtime).
Two data quality cuts are derived from “trend” data (summaries of minimum,
maximum, mean and root-mean-square values over each one-second period)
monitoring the interferometry used in the LIGO detectors. The first one is based
on occasional transient dips in the stored light in the arm cavities. These have been
identified by scanning the trend data for the relevant monitoring photodiodes, defining
the size of a dip as the fractional drop of the minimum in that second relative to the
average of the previous ten seconds, and applying various thresholds on the minimum
dip size. For the three LIGO detectors, thresholds of 5%, 4% and 5% respectively
for L1, H1 and H2 are used. High correlation of such light dips with single-detector
triggers is observed, while the deadtime resulting from them in each of the three LIGO
instruments is less than 0.6%. The second data quality cut of this type is based on the
DC level of light reaching the photodiode at the output of the interferometer, which
sees very little light when the interferometer is operating properly. By thresholding
on the trend data for this channel, intervals when its value was unusually high are
identified in H1 and L1. These intervals are seen to correlate with instrument outlier
triggers significantly. The deadtime resulting from them is 1.02% in H1 and 1.74% in
Altogether, these data quality cuts result in a net loss of observation time of 5.6%.
Search for gravitational-wave bursts in LIGO data 16
5.2. Auxiliary-channel vetoes
LIGO records thousands of auxiliary read-back channels of the servo control systems
employed in the instruments’ interferometric operation as well as auxiliary channels
monitoring the instruments’ physical environment. There are plausible couplings of
environmental disturbances or servo instabilities both to these monitoring channels
and to the GW channel; thus, transients appearing in these auxiliary channels may
be used to veto triggers seen simultaneously in the GW channel. This assumes that
a genuine gravitational-wave burst would not appear in these auxiliary channels, or
at least that any coupling is small enough to stay below the threshold for selecting
transients in these channels.
We have used KleineWelle to produce triggers from over 100 different auxiliary
channels that monitor the interferometry and the environment in the three LIGO
detectors. A first analysis of single-detector KleineWelle triggers from the L1 GW
channel and coincident KleineWelle triggers from the H1 and H2 GW channels
against respective auxiliary channels identified the ones that showed high GW channel
trigger rejection power with minimal livetime loss (in the vast majority of channels
much less that 1%). In addition to interferometric channels, environmental ones
(accelerometers and microphones) located on the optical tables holding the output
optics and photodiodes appeared to correlate with GW channel triggers recorded at
the same site.
Auxiliary interferometric channels (besides the GW channel) could in principle
be affected by a gravitational wave, and a veto condition derived from such a channel
could reject a genuine signal. Hardware signal injections imitating the passage of
gravitational waves through our detectors, performed at several pre-determined times
during the run, have been used to establish under what conditions each channel is
safe to use as a veto. Non-detection of a hardware injection by an auxiliary channel
suggests the unconditional safety of this channel as a veto in the search, assuming that
a reasonably broad selection of signal strengths and frequencies were injected. But
even if hardware injections are seen in the auxiliary channels, conditions can readily
be derived under which no triggers caused by the hardware injections are used as
vetoes. This involves imposing conditions on the significance of the trigger and/or
on the ratio of the signal strength seen in the auxiliary channel to that seen in the
GW channel. We have thus established the conditions under which several channels
involved in the length and angular sensing and control systems of the interferometers
can be used safely as vetoes. (The data quality conditions described in section 5.1
were also verified to be safe using hardware injections.)
The final choice of vetoes was made by examining the tuning set of time-
shifted triggers remaining in the WaveBurst search pipeline after applying the signal
consistency tests and data quality conditions. The ten triggers from the time-shifted
analysis with the largest values of Γ, plus the ten with the largest values of Zg, were
examined and six of them were found to coincide with transients in one or more of the
following channels: the in-phase and quadrature-phase demodulated signals from the
pick-off beam from the H1 beamsplitter, the in-phase demodulated pitch signal from
one of the wavefront sensors used in the H1 alignment sensing and control system, the
beam splitter pitch and yaw control signals, and accelerometer readings on the optical
tables holding the H1 and H2 output optics and photodiodes. KleineWelle triggers
produced from these seven auxiliary channels were clustered (with a 250 ms window)
and their union was taken. This defines the final list of veto triggers for this search,
Search for gravitational-wave bursts in LIGO data 17
each indicating a time interval (generally ≪ 1 s long) to be vetoed.
The total duration of the veto triggers considered in this analysis is at the level
of 0.15% of the total livetime. However, this does not reliably reflect the deadtime
of the search since a GW channel trigger is vetoed if it has any overlap with a veto
trigger. Thus, the actual deadtime of the search depends on the duration of the
signal being sought, as reconstructed by WaveBurst. We reproduce this effect in
the Monte Carlo simulation used to estimate the efficiency of the search (described
in section 7) by applying the same analysis pipeline and veto logic. The effective
deadtime depends on the morphology of the signal and on the signal amplitude, since
larger-amplitude signals tend to be assigned longer durations by WaveBurst. For the
majority of waveforms we considered in this search and for plausible signals strengths,
the resulting effective deadtime is of the order of 2%. Because this loss is signal-
dependent, in this analysis we consider it to be a loss of efficiency rather than a loss
of live observation time; in other words, the live observation time we state reflects the
data quality cuts applied but does not reflect the auxiliary-channel vetoes.
5.3. Gamma cut
The cuts described above cleaned up the outliers in the data considerably, as shown by
the sequence of scatter plots in figure 7. Following the data quality and veto criteria
we just described, the remaining time-shifted WaveBurst triggers (shown in figure 7d)
were used as the basis for choosing the cross correlation Γ threshold. As with previous
all-sky searches for gravitational-wave bursts with LIGO, we desire the number of
background triggers expected for the duration of the observation to be much less than
1 but not zero, typically of order ∼ 0.1. On that basis, we chose a threshold of Γ > 4
which results in 7 triggers in 98 time shifts, or 0.08 such triggers normalized to the
duration of the S4 observation time.
6. Search results
After all of the trigger selection criteria had been established using the tuning set of
time-shifted triggers, WaveBurst was re-run with a new, essentially independent set
of 100 time shifts, in increments of 5 s from −250 s to −5 s and from +5 s to +250 s,
in order to provide an estimate of the background which is minimally biased by the
choice of selection criteria. The total effective livetime for the time-shifted sample is
77.4 times the unshifted observation time, reflecting the reduced overlap of Hanford
and Livingston data segments when shifted relative to one another. The unshifted
triggers were looked at for the first time. Table 1 summarizes the trigger counts for
these time-shifted and unshifted triggers at each stage in the sequence of cuts. In
addition, the expected background at each stage (time-shifted triggers normalized to
the S4 observation time) is shown for direct comparison with the observed zero-lag
counts. Figure 8 shows a scatter plot of Γ vs. Zg and histograms of Γ for both time-
shifted and unshifted triggers after all other cuts. These new time-shifted triggers
are statistically consistent with the tuning set (figure 7d), although no triggers are
found with Zg > 15 in this case. Five unshifted triggers are found, distributed in a
manner reasonably consistent with the background. All five have Γ<4 and thus fail
the Γ cut. Three time-shifted triggers pass the Γ cut, corresponding to an estimated
average background of 0.04 triggers over the S4 observation time.
Search for gravitational-wave bursts in LIGO data 18
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(a) (b)
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 50
(c) (d)
Figure 7. Scatter plots of Γ versus Zg for the tuning set of time-shifted triggers.
(a) All triggers; (b) after data quality cuts; (c) after data quality and H1-
H2 consistency cuts (amplitude ratio and R0); (d) after data quality, H1-H2
consistency, and auxiliary-channel vetoes.
Table 1. Counts of time-shifted and unshifted triggers as cuts are applied
sequentially. The column labeled “Normalized” is the time-shifted count divided
by 77.4, representing an estimate of the expected background for the S4
observation time.
Time-shifted
Unshifted
Cut Count Normalized Count
Data quality 3153 40.7 44
H1/H2 amplitude consistency 1504 19.4 14
R0 > 0 755 9.8 5
Auxiliary-channel vetoes 671 8.7 5
Γ > 4 3 0.04 0
With no unshifted triggers in the final sample, we place an upper limit on the
mean rate of gravitational-wave events that would be detected reliably (i.e., with
efficiency near unity) by this analysis pipeline. Since the background estimate is small
and is subject to some systematic uncertainties, we simply take it to be zero for
purposes of calculating the rate limit; this makes the rate limit conservative. With
15.5 days of observation time, the one-sided frequentist upper limit on the rate at 90%
confidence level is − ln (0.1)/T = 2.303/(15.5 days) = 0.15 per day. For comparison,
the S2 search [17] arrived at an upper limit of 0.26 per day. The S3 search [18] had
Search for gravitational-wave bursts in LIGO data 19
5 6 7 8 9 10 11 12 13 14 15
time-shifted
unshifted
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
unshifted
time-shifted
rms spread
(a) (b)
Figure 8. (a) Scatter plot of Γ vs. Zg for time-shifted triggers (grey circles)
and unshifted triggers (black circles) after all other analysis cuts. The vertical
dashed line indicates the initial WaveBurst significance cut at Zg=6.7. The
horizontal dashed line indicates the final Γ cut. (b) Overlaid histograms of Γ
for unshifted triggers (black circles) and mean background estimated from time-
shifted triggers (black stairstep with statistical error bars). The shaded bars
represent the expected root-mean-square statistical fluctuations on the number of
unshifted background triggers in each bin.
an observation time of only 8 days and did not state a rate limit.
7. Amplitude sensitivity of the search
The previous section presented a limit on the rate of a hypothetical population
of gravitational-wave signals for which the analysis pipeline has perfect detection
efficiency. However, the actual detection efficiency will depend on the signal waveform
and amplitude, being zero for very weak signals and generally approaching unity for
sufficiently strong signals. The signal processing methods used in this analysis are
expressly designed to be able to detect arbitrary waveforms as long as they have
short duration and frequency content in the 64–1600 Hz band which stands out above
the detector noise. Therefore, for any given signal of this general type, we wish to
determine a characteristic minimum signal amplitude for which the pipeline has good
detection efficiency. As in past analyses, we use a Monte Carlo technique with a
population of simulated gravitational wave sources. Simulated events are generated
at random sky positions and pseudo-random times (imposing a minimum separation
of 80 s) during the S4 run; the resulting signal waveforms in each interferometer are
calculated with the appropriate antenna factors and time delays. These simulated
signals are added to the actual detector data, and the summed data streams are
analyzed using the same pipeline with the same trigger selection criteria.
The intrinsic amplitude of a simulated gravitational wave may be characterized
by its root-sum-squared strain amplitude at the Earth, without folding in antenna
response factors:
hrss ≡
(|h+(t)|2 + |h×(t)|2) dt . (4)
This quantity has units of s1/2, or equivalently Hz−1/2. In general, the root-sum-
squared signal measured by a given detector, hrssdet, will be somewhat smaller. The
Search for gravitational-wave bursts in LIGO data 20
Monte Carlo approach taken for this analysis is to generate a set of signals all with
fixed hrss and then to add this set of signals to the data with several discrete scale
factors to evaluate different signal amplitudes. For a given signal morphology and hrss,
the efficiency of the pipeline is the fraction of simulated signals which are successfully
recovered.
For this analysis, we do not attempt to survey the complete spectrum of
astrophysically motivated signals, but rather we use a limited number of ad-hoc
waveforms to characterize the sensitivity of the search in terms of hrss. Similar
sensitivities may be expected for different waveforms with similar overall properties
(central frequency, bandwidth, duration); the degree to which this is true has been
investigated in [18] and [29]. The waveforms evaluated in the present analysis are:
• Sine-Gaussian: sinusoid with a given frequency f0 inside a Gaussian amplitude
envelope with dimensionless width Q and arrival time t0:
h(t0 + t) = h0 sin(2πf0t) exp
− (2πf0t)
. (5)
These are generated with linear polarization, with f0 ranging from 70 Hz to
1053 Hz and with Q equal to 3, 8.9, and 100. The signal consistency tests
described in section 4 were developed using an ensemble of sine-Gaussian signals
with all simulated frequencies and Q values.
• Gaussian: a simple unipolar waveform with a given width τ and linear
polarization:
h(t0 + t) = h0 exp(−t
2/τ2) . (6)
• Band-limited white noise burst: a random signal with two independent
polarization components that are white over a given frequency band, described
by a base frequency f0 and a bandwidth ∆f (i.e. containing frequencies from f0
to f0 + ∆f). The signal amplitude has a Gaussian time envelope with a width
τ . Because these waveforms have two uncorrelated polarizations (in a coordinate
system at some random angle), they provide a stringent check on the robustness
of our cross-correlation test.
In all cases, we generate each simulated signal with a random arrival direction and a
random angular relationship between the wave polarization basis and the Earth.
Figures 9 and 10 show the measured efficiency of the analysis pipeline as a function
of root-sum-squared strain amplitude, ǫ(hrss), for each simulated waveform. The
efficiency data points for each waveform are fit with a function of the form
ǫ(hrss) =
)α(1+β tanh(hrss/hmidrss ))
, (7)
where ǫmax corresponds to the efficiency for strong signals (normally very close to
unity), hmidrss is the hrss value corresponding to an efficiency of ǫmax/2, β is the
parameter that describes the asymmetry of the sigmoid (with range −1 to +1), and
α describes the slope. Data points with efficiency below 0.05 are excluded from the
fit because they do not necessarily follow the functional form, while data points with
efficiency equal to 1.0 are excluded because their asymmetric statistical uncertainties
are not handled properly in the chi-squared fit. The empirical functional form in
equation 7 has been found to fit the remaining efficiency data points well.
Note that the Gaussian waveform with τ = 6.0 ms has efficiency less than 0.8
even for the largest simulated amplitude. This broad waveform, with little signal
Search for gravitational-wave bursts in LIGO data 21
]Hz [strain/rssh
-2210 -2110 -2010 -1910
70 Hz
100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=3
]Hz [strain/rssh
-2210 -2110 -2010 -1910
70 Hz
100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=8.9
]Hz [strain/rssh
-2210 -2110 -2010 -1910
1 100 Hz
153 Hz
235 Hz
361 Hz
554 Hz
849 Hz
1053 Hz
Sine-Gaussians, Q=100
Figure 9. Efficiency curves for simulated gravitational-wave signals: linearly-
polarized sine-Gaussian waves with (a) Q=3; (b) Q=8.9; (c) Q=100. Statistical
errors are comparable to the size of the plot symbols.
Search for gravitational-wave bursts in LIGO data 22
]Hz [strain/rssh
-2210 -2110 -2010 -1910
0.05 ms
0.1 ms
0.25 ms
0.5 ms
1.0 ms
2.5 ms
4.0 ms
6.0 ms
Gaussians
]Hz [strain/rssh
-2210 -2110 -2010 -1910
100-110 Hz, 0.1 s
100-200 Hz, 0.1 s
100-200 Hz, 0.01 s
250-260 Hz, 0.1 s
250-350 Hz, 0.1 s
250-350 Hz, 0.01 s
1000-1010 Hz, 0.1 s
1000-1100 Hz, 0.1 s
1000-1100 Hz, 0.01 s
1000-2000 Hz, 0.1 s
1000-2000 Hz, 0.01 s
1000-2000 Hz, 0.001 s
Band-limited white noise bursts
Figure 10. Efficiency curves for simulated gravitational-wave signals: (a)
linearly-polarized Gaussian waves; (b) band-limited white-noise bursts with two
independent polarization components. Note that four curves in the latter plot
are nearly identical: 100–110 Hz, 0.1 s; 100–200 Hz, 0.1 s; 250–260 Hz, 0.1 s;
and 250–350 Hz, 0.01 s. Statistical errors are comparable to the size of the plot
symbols.
power at frequencies above 64 Hz (the lower end of the nominal search range), is at
the limit of what the search method can detect. For some of the other waveforms,
the efficiency levels off at a value slightly less than 1.0 due to the application of the
auxiliary-channel vetoes, which randomly coincide in time with some of the simulated
signals. This effect is most pronounced for the longest-duration simulated signals due
to the veto logic used in this analysis, which rejects a trigger if there is any overlap
between the reconstructed trigger duration and a vetoed time interval. The 70-Hz
sine-Gaussian with Q=100 has a duration longer than 1 s and is reconstructed quite
poorly; it is omitted from figure 9c and from the following results.
The analytic expressions of the fits are used to determine the signal strength hrss
for which efficiencies of 50% and 90% are reached. These fits are subject to statistical
Search for gravitational-wave bursts in LIGO data 23
Table 2. hrss values corresponding to 50% and 90% detection efficiencies for
simulated sine-Gaussian signals with various central frequencies and Q values.
The 70 Hz sine-Gaussian with Q=100 is not detected reliably.
hrss (10
−21 Hz−1/2)
50% efficiency 90% efficiency
Central
frequency (Hz) Q=3 Q=8.9 Q=100 Q=3 Q=8.9 Q=100
70 3.4 5.8 — 19.2 52.0 —
100 1.8 1.7 2.6 10.4 9.4 17.7
153 1.5 1.4 1.7 8.2 8.3 8.7
235 1.6 1.7 1.9 11.0 9.8 12.6
361 2.4 2.7 3.2 11.5 16.7 20.9
554 3.3 3.2 3.2 16.1 17.9 20.4
849 5.9 4.9 4.5 28.4 28.9 24.9
1053 8.3 7.2 6.6 39.3 37.5 37.5
Table 3. hrss values corresponding to 50% and 90% detection efficiencies for
simulated Gaussian signals with various widths. The waveform with τ=6.0 ms
does not reach an efficiency of 90% within the range of signal amplitudes
simulated.
hrss (10
−21 Hz−1/2)
τ (ms) 50% efficiency 90% efficiency
0.05 6.6 33.9
0.1 4.4 25.3
0.25 3.0 14.3
0.5 2.2 13.5
1.0 2.2 10.6
2.5 3.4 20.5
4.0 8.3 43.3
6.0 39.0 —
errors from the limited number of simulations performed to produce the efficiency data
points. Also, the overall amplitude scale is subject to the uncertainty in the calibration
of the interferometer response, conservatively estimated to be 10% [30]. We increase
the nominal fitted hrss values by the amount of these systematic uncertainties to arrive
at conservative hrss values at efficiencies of 50% and 90%, summarized in tables 2,
3, and 4. The sine-Gaussian hrss values are also displayed graphically in figure 11,
showing how the frequency dependence generally follows that of the instrumental
noise.
Event rate limits as a function of waveform type and signal amplitude can be
represented by an “exclusion diagram”. Each curve in an exclusion diagram indicates
what the rate limit would be for a population of signals with a fixed hrss, as a
function of hrss. The curves in figure 12 illustrate, using selected sine-Gaussian and
Gaussian waveforms that were also considered in the S1 and S2 analyses, that the
amplitude sensitivities achieved by this S4 analysis are at least an order of magnitude
better than the sensitivities achieved by the S2 analysis. For instance, the 50%
efficiency hrss value for 235 Hz sine-Gaussians with Q=8.9 is 1.5 × 10
−20 Hz−1/2
for S2 and 1.7× 10−21 Hz−1/2 for S4. (Exclusion curves were not generated for the S3
Search for gravitational-wave bursts in LIGO data 24
Table 4. hrss values corresponding to 50% and 90% detection efficiencies for
simulated “white noise burst” signals with various base frequencies, bandwidths,
and durations.
hrss (10
−21 Hz−1/2)
Base frequency Bandwidth Duration
(Hz) (Hz) (s) 50% eff. 90% eff.
100 10 0.1 1.8 4.7
100 100 0.1 1.9 4.1
100 100 0.01 1.3 2.9
250 10 0.1 1.8 4.5
250 100 0.1 2.4 5.4
250 100 0.01 1.8 4.3
1000 10 0.1 6.5 15.8
1000 100 0.1 7.9 16.7
1000 100 0.01 5.5 12.7
1000 1000 0.1 19.2 42.6
1000 1000 0.01 9.7 22.3
1000 1000 0.001 9.5 23.7
100 1000
Sensitivities for Sine−Gaussian Waveforms
Frequency (Hz)
90% for Q=3
90% for Q=8.9
90% for Q=100
50% for Q=3
50% for Q=8.9
50% for Q=100
H2 noise
L1 noise
H1 noise
Figure 11. Sensitivity of the analysis pipeline for sine-Gaussian waveforms as a
function of frequency and Q. Symbols indicate the hrss values corresponding to
50% and 90% efficiency, taken from table 2. The instrumental sensitivity curves
from figure 2 are shown for comparison.
analysis, but the S3 sensitivity was 9 × 10−21 Hz−1/2 for this particular waveform.)
The improvement is greatest for lower-frequency sine-Gaussians and for the widest
Gaussians, due to the reduced low-frequency detector noise and the explicit extension
of the search band down to 64 Hz.
Search for gravitational-wave bursts in LIGO data 25
]Hz [strain/rssh
-2210 -2110 -2010 -1910 -1810 -1710 -1610
70 Hz
153 Hz
235 Hz
554 Hz
849 Hz
1053 Hz
S4 LIGO-TAMA
(a) Sine-Gaussians with Q=8.9
]Hz [strain/rssh
-2210 -2110 -2010 -1910 -1810 -1710 -1610
0.05 ms
0.1 ms
0.25 ms
1.0 ms
2.5 ms
4.0 ms
(b) Gaussians
Figure 12. Exclusion diagrams (rate limit at 90% confidence level, as a function
of signal amplitude) for (a) sine-Gaussian and (b) Gaussian simulated waveforms
for this S4 analysis compared to the S1 and S2 analyses (the S3 analysis did not
state a rate limit). These curves incorporate conservative systematic uncertainties
from the fits to the efficiency curves and from the interferometer response
calibration. The 849 Hz curve labeled “LIGO-TAMA” is from the joint burst
search using LIGO S2 with TAMA DT8 data [8], which included data subsets
with different combinations of operating detectors with a total observation time
of 19.7 days and thereby achieved a lower rate limit. The hrss sensitivity of the
LIGO-TAMA search was nearly constant for sine-Gaussians over the frequency
range 700–1600 Hz.
Search for gravitational-wave bursts in LIGO data 26
8. Astrophysical reach estimates
In order to set an astrophysical scale to the sensitivity achieved by this search, we
can ask what amount of mass converted into gravitational-wave burst energy at a
given distance would be strong enough to be detected by the search pipeline with 50%
efficiency. We start with the expression for the instantaneous energy flux emitted by a
gravitational wave source in the two independent polarizations h+(t) and h×(t) [31],
d2EGW
(ḣ+)
2 + (ḣ×)
, (8)
and follow the derivations in [32]. Plausible astrophysical sources will, in general, emit
gravitational waves anisotropically, but here we will assume isotropic emission in order
to get simple order-of-magnitude estimates. The above formula, when integrated over
the signal duration and over the area of a sphere at radius r (assumed not to be at
a cosmological distance), yields the total energy emitted in gravitational waves for a
given signal waveform. For the case of a sine-Gaussian with frequency f0 and Q ≫ 1,
we find
EGW =
(2πf0)
2h2rss . (9)
Taking the waveform for which we have the best hrss sensitivity, a 153 Hz sine-
Gaussian with Q=8.9, and assuming a typical Galactic source distance of 10 kpc, the
above formula relates the 50%-efficiency hrss = 1.4× 10
−21 Hz−1/2 to 10−7 solar mass
equivalent emission into a gravitational-wave burst from this hypothetical source and
under the given assumptions. For a source in the Virgo galaxy cluster, approximately
16 Mpc away, the same hrss would be produced by an energy emission of roughly
0.25M⊙c
2 in a burst with this highly favourable waveform.
We can draw more specific conclusions about detectability for models of
astrophysical sources which predict the absolute energy and waveform emitted. Here
we consider the core-collapse supernova simulations of Ott et al. [15] and a binary black
hole merger waveform calculated by the Goddard numerical relativity group [11] (as a
representative example of the similar merger waveforms obtained by several groups).
While the Monte Carlo sensitivity studies in section 7 did not include these particular
waveforms, we can relate the modeled waveforms to qualitatively similar waveforms
that were included in the Monte Carlo study and thus infer the approximate sensitivity
of the search pipeline for these astrophysical models.
Ott et al. simulated core collapse for three progenitor models and calculated
the resulting gravitational wave emission, which was dominated by oscillations of
the protoneutron star core driven by accretion [15]. Their s11WW model, based
on a non-spinning 11-M⊙ progenitor, produced a total gravitational-wave energy
emission of 1.6× 10−8M⊙c
2 with a characteristic frequency of ∼654 Hz and duration
of several hundred milliseconds. If this were a sine-Gaussian, it would have a Q
of several hundred; table 2 shows that our sensitivity does not depend strongly on
Q, so we might expect 50% efficiency for a signal at this frequency with hrss of
∼3.7 × 10−21 Hz−1/2. However, the signal is not monochromatic, and its increased
time-frequency volume may degrade the sensitivity by up to a factor of ∼2. Using
this EGW and hrss ≈ 7 × 10
−21 Hz−1/2 in equation 9, we find that our search
has an approximate “reach” (distance for which the signal would be detected with
50% efficiency by the analysis pipeline) of ∼0.2 kpc for this model. The m15b6
model, based on a spinning 15-M⊙ progenitor, yields a very similar waveform and
Search for gravitational-wave bursts in LIGO data 27
essentially the same reach. The s25WW model, based on a 25-M⊙ progenitor, was
found to emit vastly more energy in gravitational waves, 8.2 × 10−5M⊙c
2, but with
a higher characteristic frequency of ∼937 Hz. With respect to the Monte Carlo
results in section 7, we may consider this similar to a high-Q sine-Gaussian, yielding
hrss ≈ 5.5×10
−21 Hz−1/2, or to a white noise burst with a bandwidth of ∼100 Hz and
a duration of > 0.1 s, yielding hrss ≈ 8 × 10
−21 Hz−1/2. Using the latter, we deduce
an approximate reach of 8 kpc for this model.
A pair of merging black holes emits gravitational waves with very high efficiency;
for instance, numerical evolutions of equal-mass systems without spin have found the
radiated energy from the merger and subsequent ringdown to be 3.5% or more of the
total mass of the system [11]. From figure 8 of that paper, the frequency of the signal
at the moment of peak amplitude is seen to be
fpeak ≈
15 kHz
(Mf/M⊙)
, (10)
where Mf is the final mass of the system. Very roughly, we can consider the
merger+ringdown waveform to be similar to a sine-Gaussian with central frequency
fpeak and Q ≈ 2 for purposes of estimating the reach of this search pipeline for binary
black hole mergers. (Future analyses will include Monte Carlo efficiency studies using
complete inspiral-merger-ringdown waveforms.) Thus, a binary system of two 10-M⊙
black holes (i.e. Mf ≈ 20M⊙) has fpeak ≈ 750 Hz, and from table 2 we can estimate
the hrss sensitivity to be ∼5.5×10
−21 Hz−1/2. Using EGW = 0.035Mfc
2, we conclude
that the reach for such a system is roughly 1.4 Mpc. Similarly, a binary system with
Mf = 100M⊙ has fpeak ≈ 150 Hz, a sensitivity of ∼1.5×10
−21 Hz−1/2, and a resulting
reach of roughly 60 Mpc.
9. Discussion
The search reported in this paper represents the most sensitive search to date for
gravitational-wave bursts in terms of strain amplitude, reaching hrss values below
10−20 Hz−1/2, and covers a broad frequency range, 64–1600 Hz, with a live observation
time of 15.5 days.
Comparisons with previous LIGO [16, 17] and LIGO-TAMA [8] searches have
already been shown graphically in figure 12. The LIGO-TAMA search targeted
millisecond-duration signals with frequency content in the 700–2000 Hz frequency
regime (i.e., partially overlapping the present search) and had a detection efficiency
of at least 50% (90%) for signals with hrss greater than ∼ 2 × 10
−19 Hz−1/2
(10−18 Hz−1/2). Among other searches with broad-band interferometric detectors [33,
34, 35], the most recent one by the TAMA collaboration reported an upper limit of
0.49 events per day at the 90% confidence level based on an analysis of 8.1 days of the
TAMA300 instrument’s ninth data taking run (DT9) in 2003–04. The best sensitivity
of this TAMA search was achieved when looking for narrow-band signals at TAMA’s
best operating frequency, around 1300 Hz, and it was at hrss ≈ 10
−18 Hz−1/2 for 50%
detection efficiency [35]. Although we did not measure the sensitivity of the S4 LIGO
search with narrow-band signals at 1300 Hz, LIGO’s noise at that frequency range
varies slowly enough so that we do not expect it to be significantly worse than the
sensitivity for 1053 Hz sine-Gaussian signals described in section 7, which stands at
about 7× 10−21 Hz−1/2.
Search for gravitational-wave bursts in LIGO data 28
Comparisons with results from resonant mass detectors were detailed in our
previous publications [16, 17]. The upper limit of ∼ 4×10−3 events per day at the 95%
confidence level on the rate of gravitational wave bursts set by the IGEC consortium
of five resonant mass detectors still represents the most stringent rate limit for hrss
signal strengths of order 10−18 Hz−1/2 and above [36]. This upper limit quickly falls off
and becomes inapplicable to signals weaker than 10−19 Hz−1/2 (see figure 14 in [17].)
Furthermore, with the improvement in our search sensitivity, the signal strength of the
events corresponding to the slight excess seen by the EXPLORER and NAUTILUS
resonant mass detectors in their 2001 data [37] falls well above the 90% sensitivity of
our current S4 search: as described in [17], the optimal orientation signal strength of
these events assuming a Gaussian morphology with τ=0.1 ms corresponds to a hrss
of 1.9 × 10−19 Hz−1/2. For such Gaussians our S4 search all-sky 90% sensitivity is
2.5 × 10−20 Hz−1/2 (see Table 3) and when accounting for optimal orientation, this
improves by roughly a factor of 3, to 9.3×10−21 Hz−1/2. The rate of the EXPLORER
and NAUTILUS events was of order 200 events/year (or 0.55 events per day) [37, 38].
A steady flux of gravitational-wave bursts at this rate is excluded by our present
measurement at the 99.9% confidence level. Finally, in more recent running of the
EXPLORER and NAUTILUS detectors, an analysis of 149 days of data collected in
2003 set an upper limit of 0.02 events per day at the 95% confidence level and with a
hrss sensitivity of ∼ 2× 10
−19 Hz−1/2 [39].
The S5 science run, which began in November 2005 and is expected to continue
until late 2007, has a goal of collecting a full year of coincident LIGO science-mode
data. Searches for gravitational-wave bursts using S5 data are already underway and
will be capable of detecting any sufficiently strong signals which arrive during that
time, or else placing an upper limit on the rate of such signals on the order of a few
per year. Furthermore, the detector noise during the S5 run has reached the design
goals for the current LIGO interferometers, and so the amplitude sensitivity of S5
burst searches is expected to be roughly a factor of two better than the sensitivity of
this S4 search.
Another direction being pursued with the S5 data is to make appropriate use
of different detector network configurations. In addition to the approach used
in the S4 analysis reported here, which requires a signal to appear with excess
power in a time-frequency map in all three LIGO interferometers, data from two-
detector combinations is also being analyzed to maximize the total observation
time. Furthermore, using LIGO data together with simultaneous data from other
interferometers can significantly improve confidence in a signal candidate and allow
more properties of the signal to be deduced. The GEO 600 interferometer has joined
the S5 run for full-time observing in May 2006, and we look forward to the time
when VIRGO begins operating with sensitivity comparable to the similarly-sized
LIGO interferometers. Members of the LSC are currently implementing coherent
network analysis methods using maximum likelihood approaches for optimal detection
of arbitrary burst signal (see, for example, [40]) and for robust signal consistency
tests [41, 42]. Such methods will make the best use of the data collected from the
global network of detectors to search for gravitational-wave bursts.
Search for gravitational-wave bursts in LIGO data 29
Acknowledgments
The authors gratefully acknowledge the support of the United States National Science
Foundation for the construction and operation of the LIGO Laboratory and the
Science and Technology Facilities Council of the United Kingdom, the Max-Planck-
Society, and the State of Niedersachsen/Germany for support of the construction
and operation of the GEO600 detector. The authors also gratefully acknowledge the
support of the research by these agencies and by the Australian Research Council, the
Council of Scientific and Industrial Research of India, the Istituto Nazionale di Fisica
Nucleare of Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria
d’Economia, Hisenda i Innovació of the Govern de les Illes Balears, the Scottish
Funding Council, the Scottish Universities Physics Alliance, The National Aeronautics
and Space Administration, the Carnegie Trust, the Leverhulme Trust, the David
and Lucile Packard Foundation, the Research Corporation, and the Alfred P. Sloan
Foundation. This document has been assigned LIGO Laboratory document number
LIGO-P060016-C-Z.
References
[1] Sigg D (for the LSC) 2006 Class. Quantum Grav. 23 S51–6
[2] Lück H et al 2006 Class. Quantum Grav. 23 S71–8
[3] Acernese F et al 2006 Class. Quantum Grav. 23 S63–9
[4] Ando M and the TAMA Collaboration 2005 Class. Quantum Grav. 22 S881–9
[5] Fritschel P 2003 Gravitational-Wave Detection: Proc. SPIE vol 4856 ed Cruise M and Saulson
P (SPIE) p 282–91
[6] Acernese F et al 2006 Class. Quantum Grav. 23 S635–42
[7] Kuroda K et al 2003 Proc. 28th Int. Cosmic Ray Conf. ed Kajita T et al (Universal Academy
Press) p 3103
[8] Abbott B et al (LSC) and Akutsu T et al (TAMA Collaboration) 2006 Phys. Rev. D 73 102002
[9] Blanchet L, Damour T, Esposito-Farèse G and Iyer BR 2004 Phys. Rev. Lett. 93 091101
[10] Jaranowski P, Królak A and Schutz BF 1998 Phys. Rev. D 58 063001
[11] Baker J G, Centrella J, Choi D, Koppitz M and van Meter J 2006 Phys. Rev. D 73 104002
[12] Dimmelmeier H, Font J A and Müller E 2001 Astrophys. J. Lett. 560 L163–6
[13] Ott C D, Burrows A, Livne E and Walder R 2004 Astrophys. J. 600 834–64
[14] Burrows A, Livne E, Dessart L, Ott C D and Murphy J 2006 Astrophys. J. 640 878–90
[15] Ott C D, Burrows A, Dessart L and Livne E 2006 Phys. Rev. Lett. 96 201102
[16] Abbott B et al (LSC) 2004 Phys. Rev. D 69 102001
[17] Abbott B et al (LSC) 2005 Phys. Rev. D 72 062001
[18] Abbott B et al (LSC) 2006 Class. Quantum Grav. 23 S29–39
[19] Lazzarini A and Weiss R 1995 LIGO Science Requirements Document (SRD) LIGO technical
document LIGO-E950018-02-E
[20] Klimenko S and Mitselmakher G 2004 Class. Quantum Grav. 21 S1819–30
[21] The version of WaveBurst used for this analysis may be found at
http://ldas-sw.ligo.caltech.edu/cgi-bin/cvsweb.cgi/Analysis/WaveBurst/S4/?cvsroot=GDS
with the CVS tag “S4”
[22] Daubechies I 1992 Ten Lectures on Wavelets (Philadelphia: SIAM)
[23] Klimenko S, Yakushin I, Rakhmanov M and Mitselmakher G 2004 Class. Quantum Grav. 21
S1685–94
[24] Cadonati L and Márka S 2005 Class. Quantum Grav. 22 S1159–67
[25] Cadonati L 2004 Class. Quantum Grav. 21 S1695–703
[26] The version of CorrPower used for this analysis may be found at
http://www.lsc-group.phys.uwm.edu/cgi-bin/cvs/viewcvs.cgi/matapps/src/searches/burst/CorrPower/?cvsroot=lscsoft
with the CVS tag “CorrPower-080605”
[27] Chatterji S, Blackburn L, Martin G and Katsavounidis E 2004 Class. Quantum Grav. 21 S1809–
[28] The version of KleineWelle used for this analysis may be found at
http://ldas-sw.ligo.caltech.edu/-cgi-bin/-cvsweb.cgi/-Analysis/-WaveBurst/-S4/-?cvsroot=GDS
http://www.lsc-group.phys.uwm.edu/-cgi-bin/-cvs/-viewcvs.cgi/-matapps/-src/-searches/-burst/-CorrPower/-?cvsroot=lscsoft
Search for gravitational-wave bursts in LIGO data 30
http://ldas-sw.ligo.caltech.edu/cgi-bin/cvsweb.cgi/gds/Monitors/kleineWelle/?cvsroot=GDS
dated January 20, 2005
[29] Beauville F et al 2006 “A comparison of methods for gravitational wave burst searches from
LIGO and Virgo”, submitted to Phys. Rev. D, preprint gr-qc/0701026
[30] Dietz A, Garofoli J, González G, Landry M, O’Reilly B and Sung M 2006 LIGO technical
document LIGO-T050262-01-D
[31] Shapiro S L and Teukolsky S A 1983 Black Holes, White Dwarfs, and Neutron Stars (New York:
John Wiley & Sons)
[32] Riles K 2004 LIGO technical document LIGO-T040055-00-Z
[33] Nicholson D et al 1996 Phys. Lett. A 218 175
[34] Forward R L 1978 Phys. Rev. D 17 379
[35] Ando M et al (TAMA Collaboration) 2005 Phys. Rev. D 71 082002
[36] Astone P et al (International Gravitational Event Collaboration) 2003 Phys. Rev. D 68 022001
[37] Astone P et al 2002 Class. Quantum Grav. 19 5449
[38] Coccia E, Dubath F and Maggiore M 2004 Phys. Rev. D 70 084010
[39] Astone P et al 2006 Class. Quantum Grav. 23 S169–78
[40] Klimenko S, Mohanty S, Rakhmanov M and Mitselmakher G 2005 Phys. Rev. D 72 122002
[41] Wen L and Schutz B 2005 Class. Quantum Grav. 22 S1321–35
[42] Chatterji S, Lazzarini A, Stein L, Sutton P J, Searle A and Tinto M 2006 Phys. Rev. D 74
082005
http://ldas-sw.ligo.caltech.edu/-cgi-bin/-cvsweb.cgi/-gds/-Monitors/-kleineWelle/-?cvsroot=GDS
http://arxiv.org/abs/gr-qc/0701026
	Introduction
	Instruments and data collection
	Trigger generation
	Signal consistency tests
	H1/H2 amplitude consistency test
	Cross-correlation consistency tests
	Additional selection criteria for event candidates
	Data quality conditions
	Auxiliary-channel vetoes
	Gamma cut
	Search results
	Amplitude sensitivity of the search
	Astrophysical reach estimates
	Discussion
